SOHCAHTOA (Trigonometry)

Welcome to Trigonometry! At Grade 5, you will learn how to connect the angles inside a right-angled triangle to the lengths of its sides.

What you'll learn:

How to correctly label the sides of a right-angled triangle.
What the acronym "SOH CAH TOA" means and how to choose the right formula.
How to calculate a missing side length.
How to calculate a missing angle using inverse trigonometric functions.

1. Labelling the Triangle

Before doing any calculations, you must label your triangle correctly. The names of the sides change depending on which angle you are looking at. Let's call our angle $θ\theta$ (the Greek letter "theta").

$The hypotenuse is opposite the right angle, while the opposite and adjacent sides depend on the chosen angle \theta.$

Right-angled triangle labelled with Hypotenuse, Opposite, and Adjacent

Definition

The Three Sides

Hypotenuse (H): Always the longest side, and always directly opposite the right angle.
Opposite (O): The side directly opposite the angle you are working with ( $θ\theta$ ).
Adjacent (A): The side next to the angle $θ\theta$ . It connects the right angle and the angle $θ\theta$ .

2. The Formulas

There are three main trigonometric ratios: sine, cosine, and tangent (shortened to sin, cos, and tan on your calculator). We use the memory trick SOH CAH TOA to remember the formulas.

Key Idea

SOH CAH TOA

SOH: $sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
CAH: $cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
TOA: $tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Tip

Formula Triangles

If you struggle to rearrange formulas, you can put SOH, CAH, and TOA into formula triangles, just like you would for Speed, Distance, Time. Put the middle letter (O, A, or O) at the top of the triangle. Cover up what you want to find!

Formula triangles show how to rearrange SOH, CAH and TOA by covering the quantity you want to find.

3. Finding a Missing Side

To find a missing side length, follow a simple 4-step process: label the sides, pick your formula, substitute the numbers, and solve.

Example

Finding a side (numerator)

A right-angled triangle has an angle of 35°. The hypotenuse is 20 cm. Find the length of the side opposite the 35° angle, labelled $x$ .

For the 35° angle, the unknown side x is opposite and the 20 cm side is the hypotenuse.

Label the sides. We want the Opposite (which is $x$ ), and we know the Hypotenuse (which is 20).
Choose the formula. We have O and H, which means we need SOH.
Write out the formula and substitute the values:
$sin⁡(35∘)=x20\sin(35^\circ) = \frac{x}{20}$
Solve for $x$ by multiplying both sides by 20. Type this into your calculator to get the final answer:
$d.p.)\begin{aligned} x &= 20 \times \sin(35^\circ) \\ x &= 11.47\text{ cm (to 2 d.p.)} \end{aligned}$

When the unknown is on the bottom

Sometimes, the side you are trying to find ends up on the bottom (the denominator) of the fraction. This just adds one extra step to your solving process.

Example

Finding a side (denominator)

A right-angled triangle has an angle of 42°. The side opposite the angle is 11 cm. Find the length of the adjacent side, $x$ .

For the 42° angle, 11 cm is the opposite side and x is the adjacent side.

Label the sides. We know the Opposite (11) and we want the Adjacent ( $x$ ).
Choose the formula. O and A means we use TOA.
Substitute the values:
$tan⁡(42∘)=11x\tan(42^\circ) = \frac{11}{x}$
To solve, first multiply by $x$ to get it off the bottom, then divide by $tan⁡(42∘)\tan(42^\circ)$ . Essentially, $x$ and $tan⁡(42∘)\tan(42^\circ)$ swap places:
$d.p.)\begin{aligned} x \times \tan(42^\circ) &= 11 \\ x &= \frac{11}{\tan(42^\circ)} \\ x &= 12.22\text{ cm (to 2 d.p.)} \end{aligned}$

4. Finding a Missing Angle

If you know two side lengths, you can find a missing angle. You still use SOH CAH TOA, but at the end, you need to use the inverse trigonometric functions ( $sin^{-1}$ , $cos^{-1}$ , or $tan^{-1}$ ) on your calculator. You usually get these by pressing the SHIFT or 2ND button before the sin, cos, or tan button.

Example

Finding a missing angle

A right-angled triangle has an adjacent side of 9 cm and an opposite side of 5 cm. Calculate the size of the angle $θ\theta$ .

$The two given sides are opposite and adjacent to the unknown angle \theta, so tangent is the relevant ratio.$

Label the sides. We have the Opposite (5) and the Adjacent (9).
Choose the formula. O and A means we need TOA.
Substitute the values:
$tan⁡(θ)=59\tan(\theta) = \frac{5}{9}$
Use the inverse tan function to find the angle. Type this exactly into your calculator:
$d.p.)\begin{aligned} \theta &= \tan^{-1}\left(\frac{5}{9}\right) \\ \theta &= 29.1^\circ \text{ (to 1 d.p.)} \end{aligned}$

Common Mistake

Dividing by the word 'tan'

Remember that "tan" is a function, not a number. You cannot isolate $θ\theta$ by dividing by the word "tan". You must use the inverse function $tan^{-1}$ to extract the angle!

Exam technique

In the exam

Check your calculator mode! Make sure your calculator is set to Degrees (there should be a small 'D' or 'Deg' at the top of the screen, not an 'R' or 'G'). If it is in radians, every answer will be wrong.
Don't round early: Keep the full, long decimal numbers on your calculator screen during your working out. Only round your very final answer.
Read the question carefully: Exams usually ask you to round to 1 decimal place or 3 significant figures. Check what the question asks for so you don't lose the final accuracy mark.

Self review

Check yourself

Can you label the Hypotenuse, Opposite, and Adjacent sides from memory?
Do you know which two sides are involved in the "CAH" formula?
If you have an equation like $cos⁡(θ)=0.5\cos(\theta) = 0.5$ , which calculator button do you press to find $θ\theta$ ?

SOHCAHTOA (Trigonometry)

Welcome to Trigonometry! At Grade 5, you will learn how to connect the angles inside a right-angled triangle to the lengths of its sides.

What you'll learn:

How to correctly label the sides of a right-angled triangle.
What the acronym "SOH CAH TOA" means and how to choose the right formula.
How to calculate a missing side length.
How to calculate a missing angle using inverse trigonometric functions.

1. Labelling the Triangle

$The hypotenuse is opposite the right angle, while the opposite and adjacent sides depend on the chosen angle \theta.$

Right-angled triangle labelled with Hypotenuse, Opposite, and Adjacent

Definition

The Three Sides

Hypotenuse (H): Always the longest side, and always directly opposite the right angle.
Opposite (O): The side directly opposite the angle you are working with ( $θ\theta$ ).
Adjacent (A): The side next to the angle $θ\theta$ . It connects the right angle and the angle $θ\theta$ .

2. The Formulas

There are three main trigonometric ratios: sine, cosine, and tangent (shortened to sin, cos, and tan on your calculator). We use the memory trick SOH CAH TOA to remember the formulas.

Key Idea

SOH CAH TOA

SOH: $sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
CAH: $cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
TOA: $tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Tip

Formula Triangles

Formula triangles show how to rearrange SOH, CAH and TOA by covering the quantity you want to find.

3. Finding a Missing Side

To find a missing side length, follow a simple 4-step process: label the sides, pick your formula, substitute the numbers, and solve.

Example

Finding a side (numerator)

A right-angled triangle has an angle of 35°. The hypotenuse is 20 cm. Find the length of the side opposite the 35° angle, labelled $x$ .

For the 35° angle, the unknown side x is opposite and the 20 cm side is the hypotenuse.

Label the sides. We want the Opposite (which is $x$ ), and we know the Hypotenuse (which is 20).
Choose the formula. We have O and H, which means we need SOH.
Write out the formula and substitute the values:
$sin⁡(35∘)=x20\sin(35^\circ) = \frac{x}{20}$
Solve for $x$ by multiplying both sides by 20. Type this into your calculator to get the final answer:
$d.p.)\begin{aligned} x &= 20 \times \sin(35^\circ) \\ x &= 11.47\text{ cm (to 2 d.p.)} \end{aligned}$

When the unknown is on the bottom

Sometimes, the side you are trying to find ends up on the bottom (the denominator) of the fraction. This just adds one extra step to your solving process.

Example

Finding a side (denominator)

A right-angled triangle has an angle of 42°. The side opposite the angle is 11 cm. Find the length of the adjacent side, $x$ .

For the 42° angle, 11 cm is the opposite side and x is the adjacent side.

Label the sides. We know the Opposite (11) and we want the Adjacent ( $x$ ).
Choose the formula. O and A means we use TOA.
Substitute the values:
$tan⁡(42∘)=11x\tan(42^\circ) = \frac{11}{x}$
To solve, first multiply by $x$ to get it off the bottom, then divide by $tan⁡(42∘)\tan(42^\circ)$ . Essentially, $x$ and $tan⁡(42∘)\tan(42^\circ)$ swap places:
$d.p.)\begin{aligned} x \times \tan(42^\circ) &= 11 \\ x &= \frac{11}{\tan(42^\circ)} \\ x &= 12.22\text{ cm (to 2 d.p.)} \end{aligned}$

4. Finding a Missing Angle

Example

Finding a missing angle

A right-angled triangle has an adjacent side of 9 cm and an opposite side of 5 cm. Calculate the size of the angle $θ\theta$ .

$The two given sides are opposite and adjacent to the unknown angle \theta, so tangent is the relevant ratio.$

Label the sides. We have the Opposite (5) and the Adjacent (9).
Choose the formula. O and A means we need TOA.
Substitute the values:
$tan⁡(θ)=59\tan(\theta) = \frac{5}{9}$
Use the inverse tan function to find the angle. Type this exactly into your calculator:
$d.p.)\begin{aligned} \theta &= \tan^{-1}\left(\frac{5}{9}\right) \\ \theta &= 29.1^\circ \text{ (to 1 d.p.)} \end{aligned}$

Common Mistake

Dividing by the word 'tan'

Remember that "tan" is a function, not a number. You cannot isolate $θ\theta$ by dividing by the word "tan". You must use the inverse function $tan^{-1}$ to extract the angle!

Exam technique

In the exam

Check your calculator mode! Make sure your calculator is set to Degrees (there should be a small 'D' or 'Deg' at the top of the screen, not an 'R' or 'G'). If it is in radians, every answer will be wrong.
Don't round early: Keep the full, long decimal numbers on your calculator screen during your working out. Only round your very final answer.
Read the question carefully: Exams usually ask you to round to 1 decimal place or 3 significant figures. Check what the question asks for so you don't lose the final accuracy mark.

Self review

Check yourself

Can you label the Hypotenuse, Opposite, and Adjacent sides from memory?
Do you know which two sides are involved in the "CAH" formula?
If you have an equation like $cos⁡(θ)=0.5\cos(\theta) = 0.5$ , which calculator button do you press to find $θ\theta$ ?

SOHCAHTOA (Trigonometry)

1. Labelling the Triangle

2. The Formulas

3. Finding a Missing Side

When the unknown is on the bottom

4. Finding a Missing Angle

Recap questions

You've reached the end

SOHCAHTOA (Trigonometry)

1. Labelling the Triangle

2. The Formulas

3. Finding a Missing Side

When the unknown is on the bottom

4. Finding a Missing Angle

Recap questions

You've reached the end

SOHCAHTOA (Trigonometry)

1. Labelling the Triangle

2. The Formulas

3. Finding a Missing Side

When the unknown is on the bottom

4. Finding a Missing Angle

Recap questions

You've reached the end

SOHCAHTOA (Trigonometry) Revision Guide

SOHCAHTOA (Trigonometry)

1. Labelling the Triangle

2. The Formulas

3. Finding a Missing Side

When the unknown is on the bottom

4. Finding a Missing Angle

Recap questions

You've reached the end

SOHCAHTOA (Trigonometry) Revision Guide