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Trigonometry: Finding angles and sides

Trigonometry: Finding angles and sides

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Explainer Video

Tutor: Bilal

Summary

Trigonometry: Finding angles and sides

In a nutshell

​sin⁡()\sin()sin()​, cos⁡()\cos()cos(), tan⁡()\tan()tan() and their corresponding ratios can be used to find missing lengths and angles in a right-angled triangle.



Calculating a side

PROCEDURE

1.
Label the sides and angle of the triangle.
2.
Work out what trigonometric ratio you have to use (SOH, CAH or TOA).
3.
Write down and rearrange the formula to make the missing side the subject.
4.
Substitute in the values for the other side and the angle to get the length.

​

Example 1

Find the value of the side labelled xxx in the diagram given to 1 decimal place.

Maths; Trigonometry; KS4 Year 10; Trigonometry: Finding angles and sides


Label the sides and angle:

Maths; Trigonometry; KS4 Year 10; Trigonometry: Finding angles and sides


OOO​ and HHH are the given sides. So, use SOH:

sin⁡(50)=OH\sin(50) = \frac{O}{H}sin(50)=HO​


So:

 O=H×sin⁡(50)O = H \times \sin(50)O=H×sin(50)​​


Substitute in the values:

O=10×sin⁡(50)=7.660444....O = 10 \times \sin(50) = 7.660444....O=10×sin(50)=7.660444....​​

            

Round to 1 decimal place, like the question asks:

O=7.7m (1 d.p.)‾\underline{O = 7.7m \ (1 \ d.p.)}O=7.7m (1 d.p.)​



Calculating an angle

Calculating the missing angle of a right-angled triangle is very similar to calculating the missing length. However, it is important to first learn about inverse trigonometric functions.


The inverse trigonometric functions

Inverse trigonometric functions invert the trigonometric function, giving the value of the angle xxx​.

Function

What it's called

Examples

​sin⁡−1()\sin^{-1}()sin−1()​​

Inverse sine

If sin⁡(x)=0.5\sin(x) = 0.5sin(x)=0.5, then x=sin⁡−1(0.5)x = \sin^{-1}(0.5)x=sin−1(0.5)​​

​cos⁡−1()\cos^{-1}()cos−1()​​

Inverse cosine

If cos⁡(x)=0.8\cos(x) = 0.8cos(x)=0.8​, then x=cos⁡−1(0.8)x = \cos^{-1}(0.8)x=cos−1(0.8)​​​

​tan⁡−1()\tan^{-1}()tan−1()​​

Inverse tangent

If tan⁡(x)=0.42\tan(x) = 0.42tan(x)=0.42​, then x=tan⁡−1(0.42)x = \tan^{-1}(0.42)x=tan−1(0.42)​​​​


Note: The inverse function sin⁡−1() \sin^{-1}()sin−1()​​ is NOT the same as 1sin⁡()\frac{1}{\sin()}sin()1​​! It is just notation to tell us that it is the inverse of the sin⁡()\sin()sin() function. The same goes for the other two trigonometric functions.


Calculating the missing angle

The process of finding the missing angle of a right-angled triangle is very similar to finding the missing side.

PROCEDURE

1.
Label the sides and angle of the triangle.
2.
Work out what trigonometric ratio you have to use (SOH, CAH or TOA).
3.
Write down the corresponding formula.
4.
Substitute in the values for the sides.
5.
Perform the appropriate inverse trigonometric function to get the angle.

Example 2

Find the value of xxx in the missing diagram to 1 decimal place.

Maths; Trigonometry; KS4 Year 10; Trigonometry: Finding angles and sides


First, label the diagram:

Maths; Trigonometry; KS4 Year 10; Trigonometry: Finding angles and sides


OOO​ and AAA are the important sides here. Hence, use TOA:

tan⁡(x)=oppositeadjacent=OA\tan(x) = \frac{opposite}{adjacent}=\frac{O}{A}tan(x)=adjacentopposite​=AO​​


Substitute in the values:

tan⁡(x)=512\tan(x)=\frac{5}{12}tan(x)=125​​


Use the corresponding inverse trigonometric function:

If tan(x)=512tan(x) = \frac{5}{12}tan(x)=125​ then x=tan−1(512)x = tan^{-1}(\frac{5}{12})x=tan−1(125​)


x=22.61986....x= 22.61986....x=22.61986....​​


​Round to 1 decimal place, like the question asks: ​

​x=22.6° (1 d.p.)‾\underline{x = 22.6 \degree \ (1 \ d.p.)}x=22.6° (1 d.p.)​



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Learn with Basics

Learn the basics with theory units and practise what you learned with exercise sets!
Length:
Trigonometric ratios: SOH CAH TOA

Unit 1

Trigonometric ratios: SOH CAH TOA

Finding missing lengths in a triangle

Unit 2

Finding missing lengths in a triangle

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Optional

Trigonometry: Finding angles and sides

Unit 3

Trigonometry: Finding angles and sides

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FAQs - Frequently Asked Questions

What do the inverse trigonometric functions do?

They help you find the angle of a triangle when using the trigonometric identities.

When do I need to use trigonometry to find the missing length?

If you're given an angle and only 1 side, then you have to use the trigonometric ratios.

How do I know which trigonometric ratio to use?

Label your triangle, then see what side you're given and what side you have to find and then check for the corresponding trigonometric ratio.

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