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Trigonometry Worksheets

Related Pages Math Worksheets Free Printable Worksheets

There are six sets of Trigonometry worksheets:

• Trig Ratios: Sin, Cos, Tan
• Sin & Cos of Complementary Angles
• Find Missing Sides
• Find Missing Angles
• Area of Triangle using Sine
• Law of Sines and Cosines

Sine, Cosine, & Tangent Worksheets

In these free math worksheets, students learn how to find the trig ratios: sine, cosine, and tangent.

There are five sets of Sine, Cosine, & Tangent worksheets: Trigonometry Worksheet (Learn Adjacent, Opposite, & Hypotenuse) Trigonometric Ratios Worksheets (Sine Ratio, Cosine Ratio, Tangent Ratio)

How to find Sine, Cosine, & Tangent worksheets? In the context of trigonometry, the sides of a right triangle are often described in relation to an angle within the triangle. The common terms used are:

Hypotenuse: The side opposite the right angle. It is the longest side of the right triangle.

Opposite: The side opposite a specified angle. In other words, if you’re looking at one of the non-right angles, the side opposite that angle is called the “opposite” side.

Adjacent: The side adjacent to a specified angle. It is the side that is next to the angle but is not the hypotenuse.

Sine (sin), cosine (cos), and tangent (tan) are three fundamental trigonometric functions that describe the relationships between the sides and angles of a right triangle. These functions are widely used in mathematics and various scientific fields.

Sine (sin): In a right triangle, the sine of an angle (θ) is the ratio of the length of the side opposite the angle to the length of the hypotenuse. sin(θ)= Opposite/Hypotenuse

Cosine (cos): The cosine of an angle (θ) in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. cos(θ)= Adjacent/Hypotenuse

Tangent (tan): The tangent of an angle (θ) in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. tan(θ)= Opposite/Adjacent

These functions are defined not only in the context of right triangles but also more broadly using the unit circle and as periodic functions. They have various applications in physics, engineering, computer science, and other fields where understanding the relationships between angles and sides is crucial.

Click on the following worksheet to get a printable pdf document. Scroll down the page for more Sine, Cosine, & Tangent Worksheets .

More Sine, Cosine, & Tangent Worksheets

Printable (Answers on the second page.) Trigonometry Worksheet #1 (Adjacent, Opposite, & Hypotenuse) Trig Ratios Worksheet #2 (Sin, Cos, Tan Ratios) Trig Ratios Worksheet #3 (Sin, Cos, Tan Ratios) Trig Ratios Worksheet #4 (Sin, Cos, Tan Ratios) Trig Ratios Worksheet #5 (Sin, Cos, Tan Ratios)

Online Trigonometry (sine, cosine, tangent) Trigonometry (sine, cosine, tangent) Trigonometry (using a calculator) Inverse Trigonometry (using a calculator) Trigonometry (find an unknown side) Trigonometry (find an unknown angle) Using Sine Using Cosine Using Tangent Using Sine, Cosine or Tangent Trigonometry Applications Problems Law of Sines or Sine Rule Law of Sines Law of Cosines or Cosine Rule Law of Cosines

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Trigonometry Worksheets

Trigonometry worksheets promote an understanding of trigonometry concepts. Students can use these worksheets to improve their skills in solving trigonometric problems. The questions included in the worksheet are based on solving trigonometric identities, deriving trigonometry formulas, understanding trigonometry elements such sines, cosines, etc.

Benefits of Trigonometry Worksheets

Trigonometry finds use in several real life applications such as architecture, music, data analysis, etc. Hence, students must be well versed in the topic. This can only happen with clear concepts and immense practice that is provided by trigonometry worksheets.

As the scope of trigonometry is very vast, thus the worksheets have a gradual increase in the level of difficulty. This makes sure that a student doesn't get confused and can assimilate information effectively. Another added feature of these worksheets is that they are flexible, enabling students to work at their own pace.

Download Trigonometry Worksheet PDFs

They combine fun with studies so that students can have a holistic learning experience.These math worksheets should be practiced regularly and are free to download in PDF formats.

Visual maths worksheets, each maths worksheet is differentiated and visual.

Trigonometry Worksheets

Maths Worksheets / Trigonometry Worksheets

Trigonometry is a mathematical method used to define relations between elements of a triangle. Our maths trigonometry worksheets with answers will help your child or student to grasp and understand basic and more advanced ways of solving trigonometric equations. Our comprehensive resources include Pythagoras and trigonometry worksheets with answers, trigonometry area of triangle worksheets and transformations of trig graphs worksheets – all designed to make trigonometry fun and interesting.

Discovering Trigonometry (Investigation)

Trigonometry (A) Missing Lengths

Trigonometry (B) Missing Angles and Lengths

Trigonometry (C) Word Problems

Trigonometry (C) Word Problems (With Clues)

3D Trigonometry and Pythagoras (B)

Finding Exact Trig Values

3D Trigonometry and Pythagoras (A)

Area of Triangles

Area of Triangles (With Clues)

Choosing the Correct Trig Rule

Sine Rule Cosine Rule and Area Mixed Exercise

The Cosine Rule

The Sine Rule

Trigonometric Graphs (A)

Trigonometric Graphs (B)

Using Exact Trig Values

Describing Transformations of Trig Graphs

Solving Trigonometric Equations

Transformations of Trig Graphs (A)

Transformations of Trig Graphs (B)

Printable Trigonometry Worksheets with Answers

Why is trigonometry important.

When learning about trigonometry, many students struggle to understand how it is relevant in real life.

Studying the properties of triangles and calculating missing angles can seem like an arbitrary task, but it actually has many applications in the real world. From astronomy, to aviation, medicine and more, trigonometry allows many industries to function.

Essential Knowledge for Future Scientists

A solid knowledge of trigonometry is essential for any student who aspires to work in a scientific discipline.

Students will need to master trigonometry at GCSE level in order to study A-Level maths and sciences, subjects which they will need to secure a university place for engineering, medicine, and the sciences. Although many computer applications perform trigonometric calculations automatically, all the staff working in these industries will have an excellent knowledge of trigonometry and use it regularly.

Nurture Students’ Transferable Skills

Even if students do not wish to pursue a career in these technical and scientific fields, practicing trigonometry questions helps them develop transferable skills valued by all employers.

Using Trigonometry improves students’ problem solving abilities and promotes critical thinking, skills which will prove useful throughout their studies and into adult life. Providing students with trigonometry worksheets helps to nurture these skills, allowing them to be successful not just at their maths exams but right the way through their career.

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Trigonometry Worksheets for High School

Explore the surplus collection of trigonometry worksheets that cover key skills in quadrants and angles, measuring angles in degrees and radians, conversion between degrees, minutes and radians, understanding the six trigonometric ratios, unit circles, frequently used trigonometric identities, evaluating, proving and verifying trigonometric expressions and the list go on...

List of Trigonometry Worksheets

Explore the trigonometry worksheets in detail.

Grasp and retain trigonometric concepts with ease employing these visually appealing charts for quadrants and angles, right triangle trigonometric ratio chart, trigonometric ratio tables, allied angles and unit circle charts to mention a few.

Identify the quadrant encompassing the terminal side of the angle with this set of quadrants and angles worksheets. Draw the indicated angle on the coordinate plane, measure the angles in the quadrant and represent as degrees and radians and a lot more.

Conversion of Degrees and Radians

Introduce the two ways to measure an angle, namely degrees and radians with this set of worksheets. Adequate worksheets are provided to assist in practicing prompt conversions of degrees to radians and vice-versa.

To specifically and accurately measure the size of an angle in degrees, it is further broken down into degrees, minutes and seconds. This worksheet stack consists of ample exercises to practice conversion between degrees, minutes and seconds.

Determine the reference angles in degrees and radians, find the coterminal angles for the indicated angles, and positive and negative coterminal angles with this assemblage of reference and coterminal angles worksheets.

Trigonometric Ratios | Right Triangle Trigonometry

Kick start your learning with these trig ratio worksheets. Identify the legs, side and angles, introduce the six trigonometric ratios both primary trig ratios and reciprocal trig ratios and much more with these trigonometric ratio worksheets.

Included here are fundamental identities like quotient, reciprocal, cofunction and Pythagorean identities, sum and difference identities, sum-to-product, product-to-sum, double angle and half angle identities and ample trig expression to be simplified, proved and verified using the trigonometric formulas.

Unit Circle Worksheets

Packed in these unit circle worksheets are exercises to find the coordinates of a point on the unit circle, determine the corresponding angle measure, use the unit circle to find the six trigonometric ratios and a lot more.

Trigonometric Ratios of Allied Angles

Allied angle worksheets here enclose exercises like finding the exact value of the trigonometric ratio offering angle measures in degrees or radians, evaluating trig ratios of allied angles and proving the trigonometric statements to mention just a few.

Evaluating Trigonometric Expressions

These worksheets outline the concept of evaluating trigonometric expressions involving primary, reciprocal and fundamental trigonometric ratios, evaluating expressions using a calculator, evaluate using allied angles and more!

Evaluating Trigonometric Functions Worksheets

With this set of evaluating trigonometric functions worksheets at your disposal, you have no dearth of practice exercises. Begin with substituting the specified x-values in trigonometric functions and solve for f(x).

Inverse Trigonometric Function Worksheets

Utilize this adequate supply of inverse trigonometric ratio worksheets to find the exact value of inverse trig ratios using charts and calculators, find the measure of angles, solve the equations, learn to evaluate inverse and the composition of trigonometric functions and a lot more.

Law of Sines Worksheets

Navigate through this law of sines worksheets that encompass an array of topics like finding the missing side and the unknown angles, solving triangles, an ambiguous case in a triangle, finding the area of SAS triangle and more.

Law of Cosines Worksheets

Incorporate the law of cosines worksheets to elevate your understanding of the concept and practice to find the missing sides of a triangle, finding the unknown angles (SAS & SSS), solving triangles and much more.

Solving Triangles Worksheets

Access this huge collection of solving triangles worksheets to comprehend the topics like solving triangles, finding the area of the triangle, solving the triangle using the given area and much more worksheets are included.

Principal Solutions of Trig Equations Worksheets

Reinforce the concept of principal solutions of trigonometric equations with this adequate supply of worksheets like solving linear trigonometric equations, solving trigonometric equations in quadratic form and much more.

General Solutions of Trig Equations Worksheets

Employ this assortment of general solutions of trigonometric equations worksheets that feature ample of exercises to hone your skills in solving different types of trigonometric equations to obtain the general solutions.

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Sine and Cosine Worksheets with Answer Keys

Feel free to download and enjoy these free worksheets on functions and relations. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Sine, Cosine, tangent, to find side length
• Sine, Cosine, Tangent Chart
• Inverse Trig Functions
• Real World Applications of SOHCATOA
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Trigonometry mixed homework including problem solving

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

Last updated

26 April 2018

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Trigonometry questions designed to test students ability to apply their knowledge of basic trigonometry using the sine, cosine and tangent ratios. Includes problem solving questions. Solutions provided!

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Trigonometry

Here we will learn about trigonometry including how to use SOHCAHTOA, inverse trigonometric functions, exact trigonometric values and the hypotenuse. We’ll also learn about the sine rule, the cosine rule, how to find the area of a triangle using ½abSinC , 3 D trigonometry and how to use the sine, cosine and tangent graphs.

There are also trigonometry worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is trigonometry?

Trigonometry is the relationship between angles and side lengths within triangles; it is derived from the greek words “trigōnon” meaning triangle and “metron” meaning measure.

Trigonometry was originally used by the Babylonians, over 1500 years before the Greek form that we use today. It is used widely in science and engineering and product design. 

The higher GCSE curriculum expands the use of trigonometric functions for non right-angle triangles, developing from the fundamental knowledge of the three trigonometric ratios (expressed as the mnemonic SOHCAHTOA) and exact trigonometric values in right angle triangles.

See also: 15 Trigonometry questions

What is trigonometry

Trigonometry worksheet

Get your free trigonometry worksheet of 20+ questions and answers. Includes reasoning and applied questions.

What is SOHCAHTOA?

SOHCAHTOA is the abbreviation used to describe the three trigonometric ratios for the sine , cosine and tangent functions.

To determine which trigonometric function you need to use to answer a question, it depends on the location of the angle and the sides of the triangle that will be used. The trigonometric functions apply to right-angle triangles.

• If you know the hypotenuse and the opposite side of the angle, you would use the sine function.
• If you know the hypotenuse and the adjacent side (next to) the angle, you would use the cosine function.
• If you know the opposite and adjacent sides to the angle, you use the tangent function.

We can use SOHCAHTOA to calculate lengths and angles in 2D and 3D shapes by recognising right-angle triangles. E.g. We can find the length A C of a parallelogram or the length A H in the cuboid below.

Step-by-step guide: SOHCAHTOA

What is the hypotenuse?

The hypotenuse is the longest side of a right angle triangle. It is the side opposite the right angle.

The hypotenuse does not occur for other types of triangles unless we know more information (such as an isosceles triangle can be made from 2 identical right angle triangles, back-to-back).

Labelling the other sides of the triangle

Once we know which angle we are using, we can label the sides opposite (O) , adjacent ( A ) and hypotenuse (H) . We know the hypotenuse is opposite the right angle. The opposite side is opposite the angle we are using. The adjacent side is next to the angle we are using. The triangle below is labelled based on using the angle θ .

Step-by-step guide: Hypotenuse

Example 1: find a side given the angle and the hypotenuse

ABC is a right angle triangle. The size of angle ACB = 60º and the length BC = 16cm .

Calculate the value of x .

Labelling the sides OAH in relation to the angle 60º , we can use the hypotenuse, and we need to find the adjacent side. We therefore need to use the cosine function. 

Inverse trigonometric functions

What are inverse trigonometric functions.

Inverse trigonometric functions allow us to calculate the size of angle θ for a right-angle triangle.

The inverse trigonometric functions look like this:

Step-by-step guide: Trigonometric functions

Example 2: find the angle using inverse trigonometric functions

Calculate the size of angle θ correct to 2 decimal places.

The two sides that can be used to calculate the value of θ are the opposite and the hypotenuse and so we apply the sine function to θ to get

In order to calculate θ , we rearrange the equation by using the inverse sine function.

We therefore have:

Exact trigonometric values

What are exact trigonometric values.

Exact trigonometric values are found when the relationship between the sides and the angles in a triangle have a specific relationship. Summarising these values, we obtain the exact trigonometric values for sine, cosine and tangent for 0 ≤ θ ≤ 90º .

Example 3: using exact trigonometric values

ABC is an equilateral triangle. M is the midpoint of AC . Calculate the exact size of the angle θ .

AC = 6cm so MC = 3cm

We therefore have the triangle:

Using the table above,

Pythagoras or trigonometry?

We need to be able to interpret problems and recognise whether we need to use Pythagoras’ Theorem in 2 D, 3 D, or one of the three trigonometric ratios.

This flow chart describes the information you need to know about a shape in order to solve the problem.

It is important to recognise that with most of these problems, you may need to use the Pythagorean Theorem, or trigonometry, or both within the same question so you must be confident with these topics individually to access this topic fully. 

Below is a summary of methods that can be used for right angled triangles:

Example 4: find the hypotenuse using trigonometry

Calculate the length of the hypotenuse of a right triangle, x , to 1 decimal place.

The two important sides in this question are the opposite side ( O ) to the angle and the hypotenuse ( H ) so we need to use the sine function to calculate the value of x .

Solving non right angle triangles

Sine rule (the law of sines), what is the sine rule.

The sine rule (or the law of sines) is a relationship between the size of an angle in a triangle and the opposite side. There are three relationships in a triangle as there are 3 angles with their opposing sides but you will only need to use two.

Pythagoras’ Theorem cannot be used to find the third side of a non-right angled triangle. Instead we can use the sine rule or the cosine rule, depending on the information we know about the triangle. 

To find a missing angle:   \frac{\sin (A)}{a}=\frac{\sin (B)}{b}

To find a missing side:   \frac{a}{\sin (A)}=\frac{b}{\sin (B)}

Step-by-step guide: Sine rule

Example 5: Finding a missing side of a triangle using the sine rule

Calculate the length AB . Write your answer to 2 decimal places.

Label each angle A, B and C and each side a, b and c:

Here we know side a and we want to find the length of c , therefore we can state:

Here, the length AB = 7.00cm (2dp) .

Cosine rule (the law of cosines)

What is the cosine rule.

The cosine rule (or the law of cosines) is a formula which can be used to calculate the missing sides of a triangle or to find a missing angle. To do this we need to know the two arrangements of the formula and what each variable represents.

To find a missing side:   a^{2}=b^{2}+c^{2}-2bc\cos(A)

   To find a missing angle:       A=\cos^{-1}(\frac{b^2+c^2-a^2}{2bc})

Step-by-step guide: Cosine rule

Example 6: find the missing side using the cosine rule

Find the length of x for triangle ABC , correct to 2 decimal places.

The vertices are already labelled with A located on the angle we are using so we only need to label the opposite sides of a, b, and c .

Here, we need to find the missing side a , therefore we need to state the cosine rule with a 2 as the subject:

1/2abSin(C) (area of a triangle)

What is 1/2absin(c) .

\frac{1}{2}abSin(C) is a formula to calculate the area of any triangle.

Step-by-step guide: Area of a triangle trig

Example 7: area using A=1/2ab S in(C)

Calculate the area of the triangle ABC . Write your answer to 2 decimal places.

Here, we label each side a, b, and c .

3D trigonometry

What is 3d trigonometry.

3 D trigonometry is the application of the trigonometric skills developed for 2 dimensional triangles.

To find missing sides or angles in 3 dimensional shapes, we need to be very clear with the rules and formulae to find these different angles and side lengths. The flowchart below can help determine which function you need to use:

Once you can justify which rule or formulae you need to use, you may need to carry out this process again for another triangle in the question.

Top tip: Look out for common angles or common sides.

Step-by-step guide: 3D trigonometry

Example 8: find the missing angle in a triangular prism

Calculate the size of the angle in the triangular prism ABCDEF.

We can see that triangle ABF and triangle ACF share the side AF . We can use triangle ACF to calculate the length of AF , which will then help us calculate the size of angle θ .

This triangle contains no information about the angles so we need to use Pythagoras Theorem.

This is a right angled triangle involving angles so we need to use SOHCAHTOA.

Since we know O and A , we need to use tan.

Trigonometric graphs

The trigonometric functions sine, cosine and tangent can be represented by graphs. 

For example, as the angle changes, so does the value of sine. This can be plotted on a graph.

Let’s look at this in more detail below.

Sine, cosine and tangent graphs

What are sine, cosine, and tangent graphs.

Trigonometric graphs are a visual representation of the sine, cosine and tangent functions. The horizontal axis represents the angle, usually written as θ , and the vertical axis is the trig function.

See below for all three trigonometric graphs for all angles of θ between -360º and 360º (-360 < θ < 360).

The graph of y = sin(θ)

Step-by-step guide: Sin graph

The graph of y = cos(θ)

Step-by-step guide: Cos graph

The graph of y = tan(θ)

Step-by-step guide: Tan graph

Example 9: state the value of tan(θ) with θ known

Use the graph of y = tan(θ) to estimate the value of y when θ = 120º .

Here we draw the vertical line at 120º until it reaches the tangent curve and then a horizontal line towards the y -axis.

As the scale for each mark on the y- axis is 0.25 , the value for tan(120) is approximately equal to -1.7 (1dp) .

Common misconceptions

• Labelling a triangle incorrectly

E.g. This triangle has been incorrectly labelled with the side next to the angle.

This will have an impact on the formula for the sine rule, the cosine rule, and the area of the triangle.

• Using the incorrect trigonometric function

If the triangle is incorrectly labelled it can lead to the use of the incorrect standard or inverse trigonometric function.

• Rounding the decimal too early

This can lose accuracy marks. Always use as many decimal places as possible throughout the calculation, then round the solution.

• Pythagoras’ Theorem or trigonometry?

Use the flowchart to help you to recognise when to use Pythagoras’ Theorem and when to use trigonometry. Remember, you may need to use both.

• Using the sine rule instead of cosine rule

In order to use the sine rule we need to have pairs of opposite angles and sides.

• Not using the included angle

For the cosine rule and the area of a triangle using A=1/2absin(C) , the angle is included between the two sides. Using any other angle will result in an incorrect solution.

• Using A = b × h ÷ 2

If the vertical height of a triangle is not available then we cannot calculate the area by halving the base times the height.

• Using the inverse trig function instead, inducing a mathematical error

If the inverse trig function is used instead of the standard trig function, the calculator may return a maths error as the solution does not exist.

• Sine and cosine graphs switched

The sine and cosine graphs are very similar and can easily be confused with one another. A tip to remember is that you “sine up” from 0 for the sine graph so the line is increasing whereas you “cosine down” from 1 so the line is decreasing for the cosine graph.

• Asymptotes are drawn incorrectly for the graph of the tangent function

The tangent function has an asymptote at 90º because this value is undefined. As the curve repeats every 180º , the next asymptote is at 270º and so on.

• The graphs are sketched using a ruler

Each trigonometric graph is a curve and therefore the only time you are required to use a ruler is to draw a set of axes. Practice sketching each curve freehand and label important values on each axis.

Practice trigonometry questions

1. Calculate the length of the side BC:

This is a right angled triangle involving angles so use SOHCAHTOA.

First we need to label the sides O, A and H.

We know A and we want to find H so we need to use cos.

2. Using your knowledge of exact trigonometric values, work out the size of the angle marked . \theta

Using the exact trigonometric values,

3. Work out the size of angle \theta .

This is not a right angled triangle and we know an angle and its opposite side so we need to use the sine rule.

4. Calculate the area of the following triangle:

We do not know the base or the height so we need to use:

5. Calculate the length of AE.

The triangles AGH and AEH share the line AH . Using the triangle AGH we can calculate the length of the line AH.

Using the triangle AEH we can calculate the length of AE .

6. Write down the coordinates of a minimum point on the graph of y=cos(\theta) for 0^{\circ} \leqslant \theta \leqslant 360^{\circ}

The minimum point occurs at (180, -1)

Trigonometry GCSE questions

1. Below is a sketch of a football pitch ABCD .

(a)   Player F is standing exactly 60m perpendicular to Player E on the goal line and 75m from the corner where Player A is standing.

Player A kicks the football directly to Player F . Calculate the bearing of F from A . Write your answer correct to 2 decimal places.

(b)   Player F then passes the ball to Player G (the goal keeper). Player G is standing at the midpoint of BC at a bearing of 060^{\circ} from F .

How far is Player G from Player F ? Write your answer to 2 decimal places.

Bearing of F from A = 180 – angle EAF

Angle AEF = \sin^{-1}(\frac{60}{75})

Angle AEF = 53.13^{\circ}

Bearing of F from  A = 180 – 53.13 =126.87^{\circ}

FG = \frac{45}{sin(60)}

FG = 51.96m

2.  Triangle ABE and ACD are similar with AB:BC = 1:3. Using the information on the diagram, calculate the area of the shaded region BCDE .

State the units in your answer.

Area (ABE) = ½ × a × b × sin(C) = ½ × 4 × 6 × sin(30)

Sin(30) = ½

Area (ABE) = 6cm^2

As AB:BC=1:3, BC= 4 × 3 = 12cm and DE = 6 × 3 = 18cm

AC = 12+4 = 16cm and AD = 6+18 = 24cm

Area (ACD) = ½ × a × b × sin(C) = ½ × 16 × 24 × sin(30)

Area (ACD) = 96cm^2

Area BCDE = 96 – 6 = 90cm^2

3.  (a)  The cuboid ABCDEFGH is shown below. ADEF is a square face with the side length of 2m , and the length AB = 8cm.

Calculate the length of the line AE. Write your answer as a surd in its simplest form.

(b)  Given that the point X lies on the line EH so that XH = 3EX and angle XAB = 54.7^{\circ} , calculate the length of the line BX .

AE^2 = AF^2 + EF^2

AE^2 = 2^2 + 2^2 = 8

AE = \sqrt{8} = 2\sqrt{2}

EX = 8 / 4 = 2m

AX = \sqrt{2^{2}+2\sqrt{2}^{2}}

AX = 2\sqrt{3}

Cosine rule stated to find BX :

BX^{2}=AX^{2}+AB^{2}-2\times{AX}\times{AB}\times\cos(A)

BX^{2}=(2\sqrt{3})^{2}+8^{2}-2\times{2\sqrt{3}}\times{8}\times\cos(54.7)

BX^{2}=43.97m

Learning checklist

You have now learned how to:

• use trigonometric ratios in similar triangles to solve problems involving right-angled triangles
• recognise, sketch and interpret graphs of trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size
• apply trigonometric ratios to find angles and lengths in right-angled triangles and, where possible, general triangles in 2 and 3 dimensional figures
• know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90° ; know the exact value of tan θ for θ = 0°, 30°, 45°, 60°
• know and apply the sine rule, and cosine rule, to find unknown lengths and angles
• know and apply area = \frac{1}{2}abSinC to calculate the area, sides or angles of any triangle

The next lessons are

• Pythagoras theorem
• Circle theorems

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Solving Real-Life Problems Using Trigonometry

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TRIGONOMETRY WORD PROBLEMS WORKSHEET WITH ANSWERS

1. The angle of elevation of the top of the building at a distance of 50 m from its foot on a horizontal plane is found to be 60 °. Find the height of the building.

2.  A ladder placed against a wall such that it reaches the top of the wall of height 6 m and the ladder is inclined at an angle of 60°. Find how far the ladder is from the foot of the wall.

3.  A string of a kite is 100 meters long and t he inclination of the string  with the ground is  60°. Find the height of the kite, assuming that there is no slack in the string.

4.  From the top of the tower 30 m height a man is observing the base of a tree at an angle of depression measuring 30 ° . Find the distance between the tree and the tower.

5.  A man wants to determine the height of a light house. He measured the angle at A and found that tan A = 3/4. What is the height of the light house if A is 40 m from the base?

6. A ladder is leaning against a vertical wall makes an angle of 20° with the ground. The foot of the ladder is 3 m from the wall. Find the length of ladder.

7. A kite is flying at a height of 65 m attached to a string. If the inclination of the string with the ground is 31°,  find the length of string.

8. The length of a string between a kite and a point on the ground is 90 m. If the string makes an angle θ with the ground level such that tan θ = 15/8, how high will the kite be ?

9.  An airplane is observed to be approaching a point that is at a distance of 12 km from the point of observation and makes an angle of elevation of 50 °. Find the height of the airplane above the ground.

10.  A balloon is connected to a meteorological station by a cable of length 200 m inclined at 60 °  angle with the ground. Find the  height of the balloon from the ground (Imagine that there is no slack in the cable).

1. Answer :

Draw a sketch.

Here, AB represents height of the building, BC represents distance of the building from the point of observation.

In the right triangle ABC, the side which is opposite to the angle 60° is known as opposite side (AB), the side which is opposite to 90° is called hypotenuse side (AC) and the remaining side is called adjacent side (BC).

Now we need to find the length of the side AB.

tanθ = Opposite side/Adjacent side

tan60° = AB/BC

√3 x 50 = AB

Approximate value of √3 is 1.732

AB = 50 (1.732)

     AB = 86.6 m

So, the height of the building is 86.6 m.

2. Answer :

Here AB represents height of the wall, BC represents the distance between the wall and the foot of the ladder and AC represents the length of the ladder.

In the right triangle ABC, the side which is opposite to angle 60° is known as opposite side (AB), the side which is opposite to 90° is called hypotenuse side (AC) and remaining side is called adjacent side (BC).

Now, we need to find the distance between foot of the ladder and the wall. That is, we have to find the length of BC.

tanθ = opposite side/adjacent side

BC = (6/√3) x (√3/√3)

BC = (6√3)/3

Approximate value of √3 is 1.732.

BC = 2 (1.732)

BC = 3.464 m 

So, the distance between foot of the ladder and the wall is 3.464 m.

3. Answer :

Here AB represents height of kite from the ground, BC represents the distance of kite from the point of observation.

In the right triangle ABC the side which is opposite to angle 60° is known as opposite side (AB), the side which is opposite to 90° is called hypotenuse side (AC) and remaining side is called adjacent side (BC).

Now we need to find the height of the side AB.

sinθ = opposite side/hypotenuse side

sinθ = AB/AC

sin60° = AB/100

√3/2 = AB/100

(√3/2) x 100 = AB

AB = 50√3 m

So, the height of kite from the ground 50√3 m.

4. Answer :

Here AB represents height of the tower, BC represents the distance between foot of the tower and the foot of the tree.

Now we need to find the distance between foot of the tower and the foot of the tree (BC).

tan30° = AB/BC

1/√3 = 30/BC

BC = 30(1.732)

 BC = 51.96 m

So, the distance between the tree and the tower is 51.96 m.

5. Answer :

Here BC represents height of the light house, AB represents the distance between the light house from the point of observation.

In the right triangle ABC the side which is opposite to the angle A is known as opposite side (BC), the side which is opposite to 90° is called hypotenuse side (AC) and remaining side is called adjacent side (AB).

Now we need to find the height of the light house (BC).

tanA = opposite side/adjacent side

tanA = BC/AB

Given : tanA = 3/4.

3/4 = BC/40

Multiply each side by 40.

So, the height of the light house is 30 m.

6. Answer :

Here AB represents height of the wall, BC represents the distance of the wall from the foot of the ladder.

In the right triangle ABC, the side which is opposite to the angle 20° is known as opposite side (AB),the side which is opposite to 90° is called hypotenuse side (AC) and remaining side is called adjacent side (BC).

Now we need to find the length of the ladder (AC).

cosθ = adjacent side/hypotenuse side

Cosθ = BC/AC

Cos 20° = 3/AC

0.9397 = 3/AC

AC = 3/0.9396

So, the length of the ladder is about 3.193 m.

7. Answer :

Here AB represents height of the kite. In the right triangle ABC the side which is opposite to angle 31° is known as opposite side (AB), the side which is opposite to 90° is called hypotenuse side (AC) and the remaining side is called adjacent side (BC).

Now we need to find the length of the string AC.

sin31° = AB/AC

0.5150 = 65/AC

AC = 65/0.5150

AC = 126.2 m

Hence, the length of the string is 126.2 m.

8. Answer :

Here AB represents height of the balloon from the ground. In the right triangle ABC the side which is opposite to angle θ is known as opposite side (AB), the side which is opposite to 90° is called hypotenuse side (AC) and remaining side is called adjacent side (BC).

tanθ = 15/8 ----> cotθ = 8/15

cscθ = √(1+ cot 2 θ)

cscθ = √(1 + 64/225)

cscθ = √(225 + 64)/225

cscθ = √289/225

cscθ = 17/15 ----> sinθ = 15/17

But, sinθ = opposite side/hypotenuse side = AB/AC.

AB/AC = 15/17

AB/90 = 15/17

So, the height of the tower is 79.41 m.

9. Answer :

Here AB represents height of the airplane from the ground. In the right triangle ABC the side which is opposite to angle 50° is known as opposite side (AB), the side which is opposite to 90° is called hypotenuse side (AC) and remaining side is called adjacent side (BC).

From the figure given above, AB stands for the height of the airplane above the ground.

sin50° = AB/AC

0.7660 = h/12

0.7660 x 12 = h

So, the height of the airplane above the ground is 9.192 km.

10. Answer :

Here AB represents height of the balloon from the ground. In the right triangle ABC the side which is opposite to angle 60° is known as opposite side (AB), the side which is opposite to 90° is called hypotenuse (AC) and the remaining side is called as adjacent side (BC).

From the figure given above, AB stands for the height of the balloon above the ground.

sin60° = AB/200

√3/2 = AB/200

AB = (√3/2) x 200

AB = 100(1.732)

AB = 173.2 m

So, the height of the balloon from the ground is 173.2 m.

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Trigonometry : Solving Word Problems with Trigonometry

Study concepts, example questions & explanations for trigonometry, all trigonometry resources, example questions, example question #1 : solving word problems with trigonometry.

You can draw the following right triangle using the information given by the question:

Since you want to find the height of the platform, you will need to use tangent.

You can draw the following right triangle from the information given by the question.

In order to find the height of the flagpole, you will need to use tangent.

You can draw the following right triangle from the information given in the question:

In order to find out how far up the ladder goes, you will need to use sine.

In right triangle ABC, where angle A measures 90 degrees, side AB measures 15 and side AC measures 36, what is the length of side BC?

This triangle cannot exist.

Example Question #5 : Solving Word Problems With Trigonometry

A support wire is anchored 10 meters up from the base of a flagpole, and the wire makes a 25 o angle with the ground. How long is the wire, w? Round your answer to two decimal places.

23.81 meters

28.31 meters

21.83 meters

To make sense of the problem, start by drawing a diagram. Label the angle of elevation as 25 o , the height between the ground and where the wire hits the flagpole as 10 meters, and our unknown, the length of the wire, as w. 

Now, we just need to solve for w using the information given in the diagram. We need to ask ourselves which parts of a triangle 10 and w are relative to our known angle of 25 o . 10 is opposite this angle, and w is the hypotenuse. Now, ask yourself which trig function(s) relate opposite and hypotenuse. There are two correct options: sine and cosecant. Using sine is probably the most common, but both options are detailed below.

We know that sine of a given angle is equal to the opposite divided by the hypotenuse, and cosecant of an angle is equal to the hypotenuse divided by the opposite (just the reciprocal of the sine function). Therefore:

To solve this problem instead using the cosecant function, we would get:

The reason that we got 23.7 here and 23.81 above is due to differences in rounding in the middle of the problem. 

Example Question #6 : Solving Word Problems With Trigonometry

When the sun is 22 o above the horizon, how long is the shadow cast by a building that is 60 meters high?

To solve this problem, first set up a diagram that shows all of the info given in the problem. 

Next, we need to interpret which side length corresponds to the shadow of the building, which is what the problem is asking us to find. Is it the hypotenuse, or the base of the triangle? Think about when you look at a shadow. When you see a shadow, you are seeing it on something else, like the ground, the sidewalk, or another object. We see the shadow on the ground, which corresponds to the base of our triangle, so that is what we'll be solving for. We'll call this base b.

Therefore the shadow cast by the building is 150 meters long.

If you got one of the incorrect answers, you may have used sine or cosine instead of tangent, or you may have used the tangent function but inverted the fraction (adjacent over opposite instead of opposite over adjacent.)

Example Question #7 : Solving Word Problems With Trigonometry

From the top of a lighthouse that sits 105 meters above the sea, the angle of depression of a boat is 19 o . How far from the boat is the top of the lighthouse?

423.18 meters

318.18 meters

36.15 meters

110.53 meters

To solve this problem, we need to create a diagram, but in order to create that diagram, we need to understand the vocabulary that is being used in this question. The following diagram clarifies the difference between an angle of depression (an angle that looks downward; relevant to our problem) and the angle of elevation (an angle that looks upward; relevant to other problems, but not this specific one.) Imagine that the top of the blue altitude line is the top of the lighthouse, the green line labelled GroundHorizon is sea level, and point B is where the boat is.

Merging together the given info and this diagram, we know that the angle of depression is 19 o  and and the altitude (blue line) is 105 meters. While the blue line is drawn on the left hand side in the diagram, we can assume is it is the same as the right hand side. Next, we need to think of the trig function that relates the given angle, the given side, and the side we want to solve for. The altitude or blue line is opposite the known angle, and we want to find the distance between the boat (point B) and the top of the lighthouse. That means that we want to determine the length of the hypotenuse, or red line labelled SlantRange. The sine function relates opposite and hypotenuse, so we'll use that here. We get:

Example Question #8 : Solving Word Problems With Trigonometry

Angelina just got a new car, and she wants to ride it to the top of a mountain and visit a lookout point. If she drives 4000 meters along a road that is inclined 22 o to the horizontal, how high above her starting point is she when she arrives at the lookout?

9.37 meters

1480 meters

3708.74 meters

10677.87 meters

1616.1 meters

As with other trig problems, begin with a sketch of a diagram of the given and sought after information.

Angelina and her car start at the bottom left of the diagram. The road she is driving on is the hypotenuse of our triangle, and the angle of the road relative to flat ground is 22 o . Because we want to find the change in height (also called elevation), we want to determine the difference between her ending and starting heights, which is labelled x in the diagram. Next, consider which trig function relates together an angle and the sides opposite and hypotenuse relative to it; the correct one is sine. Then, set up:

Therefore the change in height between Angelina's starting and ending points is 1480 meters. 

Example Question #9 : Solving Word Problems With Trigonometry

Two buildings with flat roofs are 50 feet apart. The shorter building is 40 feet tall. From the roof of the shorter building, the angle of elevation to the edge of the taller building is 48 o . How high is the taller building?

To solve this problem, let's start by drawing a diagram of the two buildings, the distance in between them, and the angle between the tops of the two buildings. Then, label in the given lengths and angle. 

Example Question #10 : Solving Word Problems With Trigonometry

Two buildings with flat roofs are 80 feet apart. The shorter building is 55 feet tall. From the roof of the shorter building, the angle of elevation to the edge of the taller building is 32 o . How high is the taller building?

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