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SOHCAHTOA

Here we will learn how to use SOHCAHTOA and trigonometry to find unknown sides and angles in right angled triangles. You’ll learn how to label the sides of right-angled triangles, what sin, cos and tan are, what their inverses are (sin-1, cos-1, tan-1) and how we can use them.

Look out for the trigonometry practice problems, worksheets and exam questions at the end.

What is SOHCAHTOA?

SOHCAHTOA is a mnemonic that gives us an easy way to remember the three main trigonometric ratios. They are sine (sin), cosine (cos) and tangent (tan). 

We can use these to work out missing sides and angles in right-angled triangles

SOHCAHTOA labelled sides

SOH stands for:

\[\sin ( \theta ) = \frac{Opposite}{Hypotenuse}\]

CAH stands for:

\[\cos ( \theta ) = \frac{Adjacent}{Hypotenuse}\]

TOA stands for:

\[\tan ( \theta ) = \frac{Opposite}{Adjacent}\]

We can abbreviate these to the SOHCAHTOA triangles:

SOHCAHTOA triangles

What is SOHCAHTOA?

What is SOHCAHTOA?

How to label the sides of a right angled triangle

SOHCAHTOA labelled triangle

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

The opposite side is the side that is opposite the angle.

The adjacent side is the side that is adjacent (next to) the angle.

E.g.
We have a right-angled triangle with a side of length 8 cm and a side of length x cm

Label the sides of the right-angled triangle.

SOHCAHTOA how to example

  1. Label the longest side of the triangle H (hypotenuse)

SOHCAHTOA how to example

2Label the side opposite the angle O (opposite)

3Label the side next to the angle A (adjacent)

SOHCAHTOA how to example

Top tip: once you have labeled the hypotenuse (H) and the opposite (O), the adjacent (A) must be the only side left!

How to label the sides of a right angled triangle

How to label the sides of a right angled triangle

How to use SOHCAHTOA to find the unknown sides of a right angled triangle

We can use trigonometry to work out the unknown sides of a right-angled triangle by using SOHCAHTOA.

  1. Label the sides of the right-angled triangle that we have information about.
  2. Choose the trig ratio we need.
  3. Substitute the values from the triangle into the function. 
  4. Calculate the unknown side, rearranging if necessary.

SOHCAHTOA worksheet

SOHCAHTOA worksheet

SOHCAHTOA worksheet

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

DOWNLOAD FREE
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SOHCAHTOA worksheet

SOHCAHTOA worksheet

SOHCAHTOA worksheet

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

DOWNLOAD FREE

SOHCAHTOA – unknown side examples

Example 1: find a missing side 

Calculate the side labeled x

SOHCAHTOA example 1

  1. Label the sides of the right-angled triangle that we have information about. Circle the labels so not to confuse them with the side lengths.

SOHCAHTOA example 1 labelled triangle

2Choose the trig ratio we need.

SOHCAHTOA triangles

The trigonometric function that involves the opposite (O) and the hypotenuse (H) is sin.

\[\sin(\theta)= \frac{Opposite}{Hypotenuse}\]

3Substitute the values from the triangle into the function.

SOHCAHTOA example 1 labelled triangle

\[\sin(40)= \frac{x}{8}\]

4Calculate the unknown side, rearranging if necessary. We can either solve the equation or use the SOHCAHTOA triangles.

SOHCAHTOA example 1 workings

x=5.1 (1 d.p)

Always check that the answer is sensible:

SOHCAHTOA example 1 labelled triangle

As 5.1 is smaller than the 8 (the hypotenuse) it is a sensible answer.

Example 2: find a missing side

Calculate the side labeled x

SOHCAHTOA example 2

SOHCAHTOA example 2 labelled sides

SOHCAHTOA triangles

The trigonometric function that involves the adjacent (A) and the hypotenuse (H) iscos.

\[\cos(\theta) = \frac{Adjacent}{Hypotenuse}\]

SOHCAHTOA example 2 labelled sides

\[\cos(36) = \frac{x}{15}\]

SOHCAHTOA example 2 workings

x=12.1 (1.d.p)

Always check that the answer is sensible:

SOHCAHTOA example 2 labelled sides

As 12.1 is smaller than the 15 (the hypotenuse) it is a sensible answer.

Example 3: find a missing side

Calculate the side labeled x

SOHCAHTOA example 3

SOHCAHTOA example 3 labelled sides

SOHCAHTOA triangles

The trigonometric function that involves the opposite (O) and the adjacent (A) is tan.

\[\tan(\theta) = \frac{Opposite}{Adjacent}\]

SOHCAHTOA example 3 labelled sides
\[\tan(36) = \frac{10}{x}\]

SOHCAHTOA example 3 workings

x=13.8 (1.d.p)

Always check that the answer is sensible:

SOHCAHTOA example 3 labelled sides

A good sketch tells us this answer looks reasonable.

Practice SOHCAHTOA questions (finding an unknown side)

1. Calculate the length of the side labeled x . Give your answer to 1 d.p.

 

SOHCAHTOA practice Q1

 

 

7.8 cm

GCSE Quiz True

18.4 cm

GCSE Quiz False

60.4 cm

GCSE Quiz False

16.9 cm

GCSE Quiz False

Label the triangle:

 

 

SOHCAHTOA practice Q1

 

 

We know H and we want to work out O so use sin.

 

 

\begin{aligned} \sin(\theta)&=\frac{O}{H}\\ \sin(23)&=\frac{x}{20}\\ 20 \times \sin(23) &= x\\ 7.81462257 &= x\\ x&=7.8\mathrm{cm} \end{aligned}

2. Calculate the length of the side labeled x . Give your answer to 2 d.p.

 

SOHCAHTOA practice Q2

 

22.42 cm

GCSE Quiz False

37.50 cm

GCSE Quiz False

20.18 cm

GCSE Quiz True

11.15 cm

GCSE Quiz False

Label the triangle:

 

 

SOHCAHTOA practice Q2 explained

 

 

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

 

 

\begin{aligned} \cos(\theta)&=\frac{A}{H}\\ \cos(42)&=\frac{15}{x}\\ x \times \cos(42) &=15\\ x &= \frac{15}{\cos(42)}\\ x &=20.18449094\\ x &=20.18 \mathrm{cm} \end{aligned}

3. Calculate the length of the side labeled x . Give your answer to 2 s.f.

 

 

SOHCAHTOA practice Q3

 

 

6.0 cm

GCSE Quiz False

4.8 cm

GCSE Quiz False

11 cm

GCSE Quiz False

2.8 cm

GCSE Quiz True

Label the triangle:

 

 

SOHCAHTOA practice Q3 explained

 

 

We know O and we want to work out A so use tan.

 

 

\begin{aligned} \tan(\theta)&=\frac{O}{A}\\ \tan(62)&=\frac{5.3}{x}\\ x \times \tan(62) &= 5.3\\ x &=\frac{5.3}{\tan(62)}\\ x &= 2.818059988\\ x &= 2.8 \mathrm{cm} \end{aligned}

Inverse trigonometric functions

In order to work out missing angles in right-angled triangles we need use the inverse trigonometric functions:

\[\sin ^{-1} \qquad \cos ^{-1} \qquad \tan ^{-1}\]

We can find these on the calculator by pressing SHIFT and then sin, cos or tan.

Step by step guide: Inverse trigonometric functions (coming soon)

How to use SOHCAHTOA to find the unknown angles of right angled triangles

We can use trigonometry to work out the unknown angles of a right-angled triangle by using SOHCAHTOA.

  1. Label the sides of the right-angled triangle that we have information about.
  2. Choose the trig ratio we need.
  3. Substitute the values from the triangle into the function.
  4. Using inverse trig functions, work out the missing angle θ.

SOHCAHTOA – unknown angle examples

Example 4: find a missing angle

Calculate the angle labeled θ.

SOHCAHTOA example 4

SOHCAHTOA example 4 labelled triangle

SOHCAHTOA triangles

The trigonometric function that involves the opposite (O) and the hypotenuse (H) is sin.

\[\sin(\theta) = \frac{Opposite}{Hypotenuse}\]

SOHCAHTOA example 4 labelled triangle
\[\sin(\theta) = \frac{5}{9}\]

SOHCAHTOA example 4 workings

Remember we can get sin-1 on the calculator by pressing SHIFT and then sin.

Always check that the answer is sensible: We can estimate from the diagram that θ is an acute angle. 

SOHCAHTOA example 4 labelled triangle

As 33.7o is less than 90o it is a sensible answer.

Example 5: find a missing angle

Calculate the angle labeled θ.

SOHCAHTOA example 5

SOHCAHTOA example 5 labelled triangle

SOHCAHTOA triangles

The trigonometric function that involves the adjacent (A) and the hypotenuse (H) is cos.

\[\cos(\theta) = \frac{Adjacent}{Hypotenuse}\]

SOHCAHTOA example 5
\[\cos(\theta) = \frac{9}{20}\]

SOHCAHTOA example 5 workings

Remember we can get cos-1 on the calculator by pressing SHIFT and then cos.

Always check that the answer is sensible: We can estimate from the diagram that θ is an acute angle.

SOHCAHTOA example 5 labelled triangle

As 63.7o is less than 90o it is a sensible answer.

Example 6: find a missing angle

Calculate the angle labeled θ.

SOHCAHTOA example 6

SOHCAHTOA example 6 labelled sides

SOHCAHTOA example 6 triangles

The trigonometric function that involves the opposite (O) and the adjacent (A) is tan.

\[\tan(\theta) = \frac{Opposite}{Adjacent}\]

SOHCAHTOA example 6 labelled triangle
\[\tan(\theta) = \frac{15}{7}\]

SOHCAHTOA example 6 workings

Remember we can get tan-1 on the calculator by pressing SHIFT and then tan.

Always check that the answer is sensible: We can estimate from the diagram that θ is an acute angle.

SOHCAHTOA example 6

As 65o is less than 90o it is a sensible answer.

Common misconceptions

  • Make sure you calculator is in ‘degrees’
    All angles at GCSE are measured in degrees. Make sure your calculator screen is showing a small ‘D’ at the top before using it for trigonometry. If it is showing an ‘R’ it will measure the angle in radians which are not needed until A Level. 

  • Right triangle 
    Right-angled triangles are sometimes called right triangles.

  • When to use SOHCAHTOA
    We can use SOHCAHTOA to find a missing side of a right angled triangle when we have another side and a given angle.

    We can use SOHCAHTOA to find a missing angle of a right angled triangle when we have two given sides.

    If we have two sides and we want to find the third we can use the Pythagorean Theorem a2+b2=c2 .

  •  Other trig identities 
    The trigonometric identities we need to know at GCSE are sin, cos and tan however there are lots more.
    Here are some you will see at A Level:

    \frac{1}{\sin(\theta)} = \text{cosec}(\theta)

    \frac{1}{\cos(\theta)} = \text{sec}(\theta)

    \frac{1}{\tan(\theta)} = \text{cot}(\theta)

Practice SOHCAHTOA questions (finding an unknown angle)

1. Calculate the size of the angle labeled \theta . Give your answer to 3 s.f.

 

 

Practice SOHCAHTOA unknown angle Q1

 

43.3^{\circ}
GCSE Quiz False

46.7^{\circ}
GCSE Quiz True

36.0^{\circ}
GCSE Quiz False

0.0126^{\circ}
GCSE Quiz False

Label the triangle:

 

 

Practice SOHCAHTOA unknown angle Q1 explained

 

 

We know O and H so use sin.

 

 

\begin{aligned} \sin(\theta)&=\frac{O}{H}\\ \sin(\theta)&=\frac{8}{11}\\ \theta&=\sin^{-1}(\frac{8}{11})\\ \theta &=46.65824177\\ \theta&=47.7^{\circ} \end{aligned}

2. Calculate the size of the angle labeled \theta . Give your answer to 2 s.f.

 

 

Practice SOHCAHTOA unknown angle Q2

 

64^{\circ}
GCSE Quiz True

1.1^{\circ}
GCSE Quiz False

26^{\circ}
GCSE Quiz False

23^{\circ}
GCSE Quiz False

Label the triangle:

 

 

 

 

We know A and H so use cos.

 

 

\begin{aligned} \cos(\theta)&=\frac{A}{H}\\ \cos(\theta)&=\frac{5.4}{12.3}\\ \theta&=\cos^{-1}(\frac{5.4}{12.3})\\ \theta&=63.95835004\\ \theta&=64^{\circ} \end{aligned}

3. Calculate the size of the angle labeled \theta . Give your answer to 1 d.p.

 

 

Practice SOHCAHTOA unknown angle Q2 explained

60.0^{\circ}
GCSE Quiz False

0.5^{\circ}
GCSE Quiz False

30.0^{\circ}
GCSE Quiz False

26.6^{\circ}
GCSE Quiz True

Label the triangle:

 

 

Practice SOHCAHTOA unknown angle Q3 explained

 

 

 

We know O and A so use tan.

 

 

\begin{aligned} \tan(\theta)&=\frac{O}{A}\\ \tan(\theta)&=\frac{12.6}{25.2}\\ \theta &= \tan^{-1}(\frac{12.6}{25.2})\\ \theta&=26.56505118\\ \theta&= 26.6^{\circ} \end{aligned}

SOHCAHTOA GCSE exam questions

 

GCSE SOHCAHTOA Q1

 

1. Find the size of angle a to 2.d.p.

(3 marks)

Show answer
20-14=6cm

(1)

 

\cos(a) = \frac{6}{13}

(1)

 

\begin{aligned} a&=\cos^{-1}(\frac{6}{13})\\ a&=62.51^{\circ} \end{aligned}

(1)

2. Rosie is flying a kite. She holds the kite string 1m from the ground. The kite string is 10m long and the kite flies at an angle of 40^{\circ} . Calculate the height of the kite above the ground.

 

SOHCAHTOA kite question GCSE

 

(3 marks)

Show answer
\begin{aligned} \sin(40)&=\frac{opp}{10}\\ 10\sin(40)&=opp \end{aligned}

(1)

 

opp=6.43m

(1)

 

total height:

6.43+1=7.43m

(1)

3. Length AB = 15cm

Angle ABE = 51^{\circ}

Angle ADE = 32^{\circ}

 

GCSES SOHCAHTOA Q3

 

Work out the length of DE.

(4 marks)

Show answer

Length AE:

 

\begin{aligned} \tan(51)&=\frac{AE}{15}\\ 15\tan(51)&=AE \end{aligned}

(1)

 

AE = 18.52

(1)

 

Length DE:

 

\begin{aligned} \tan(32)&=\frac{18.52}{x}\\ x&=\frac{18.52}{tan(32)} \end{aligned}

(1)

 

x=29.64cm

(1)

Learning checklist

You have now learned how to:

  • Apply trigonometric ratios to find lengths in right-angled triangles in 2 dimensions
  • Apply trigonometric ratios to find angles in right-angled triangles in 2 dimensions

The next lessons are

  • Sine rule
  • Cosine rule
  • Area of a triangle (½abSinC)
  • Exact trig values
  • Inverse trig functions
  • Sine, cosine & tangent graphs

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