Which ratio uses opposite and hypotenuse?
Use SOHCAHTOA.
SOH means sin = opposite ÷ hypotenuse.
Trigonometry
Trigonometry uses sine, cosine and tangent to find missing sides and angles in right-angled triangles. It links closely to Pythagoras’ theorem and coordinates.
Overview
Trigonometry is used in right-angled triangles to connect angles with side lengths.
The three main trig ratios are sine, cosine and tangent.
SOH CAH TOA
Most exam mistakes happen because students choose the wrong sides, not because the calculation itself is hard.
What you should understand after this topic
- Identify opposite, adjacent and hypotenuse
- Know when to use sine, cosine or tangent
- Find missing sides in right-angled triangles
- Find missing angles in right-angled triangles
- Use inverse trigonometric functions correctly on a calculator
Key Definitions
Hypotenuse
The longest side, opposite the right angle.
Opposite
The side directly opposite the chosen angle.
Adjacent
The side next to the chosen angle that is not the hypotenuse.
Sine
\(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
Cosine
\(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)
Tangent
\(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)
Inverse Trigonometric Functions
Used to find an angle, for example \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\).
Chosen Angle
The reference angle used to identify the opposite and adjacent sides.
Key Rules
Right angle first
Trigonometry here only works in right-angled triangles.
Hypotenuse first
Always identify the longest side before anything else.
Angle changes the labels
Opposite and adjacent depend on which angle you choose.
Choose ratio from known sides
Pick the trig ratio that uses the two sides involved.
SOHCAHTOA Guide
Sine (SOH)
\( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
Cosine (CAH)
\( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
Tangent (TOA)
\( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)
Inverse Trigonometric Functions
Use \( \sin^{-1} \), \( \cos^{-1} \), or \( \tan^{-1} \) to find angles.
How to Solve
Step 1: Identify the right-angled triangle
Trigonometry only works in right-angled triangles.
Exam tip: Always check for the \(90^\circ\) angle first.
The hypotenuse is the longest side.
Step 2: Label opposite and adjacent
These depend on the chosen angle.
Opposite = across from the angle.
Adjacent = next to the angle (not the hypotenuse).
Important: Labels change if you choose a different angle.
Step 3: Use SOH CAH TOA
\( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
\( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
\( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)
Sine (SOH)
Opposite / Hypotenuse
Cosine (CAH)
Adjacent / Hypotenuse
Tangent (TOA)
Opposite / Adjacent
Step 4: Choose the correct ratio
Identify known sides.
Identify the missing side.
Choose the ratio linking them.
Step 5: Finding an angle
Use inverse trig functions.
\( \theta = \sin^{-1}(0.6) \)
Use \(\sin^{-1}\), \(\cos^{-1}\), \(\tan^{-1}\).
Step 6: Calculator skills
Degree mode
Always use degrees.
Inverse buttons
\(\sin^{-1}, \cos^{-1}, \tan^{-1}\)
Do not round early
Round at the end only.
Check answers
Side cannot exceed hypotenuse.
Step 7: Exam method
See Pythagoras for alternative method.
- Check right angle.
- Label sides.
- Choose SOH CAH TOA.
- Substitute values.
- Rearrange.
- Use calculator.
- Write units.
Example Questions
Edexcel
Exam-style questions focusing on identifying the correct trigonometric ratio.
Edexcel
The right-angled triangle shows the opposite side and hypotenuse.
Which trigonometric ratio uses opposite and hypotenuse?
Edexcel
The right-angled triangle shows the adjacent side and hypotenuse.
Which trigonometric ratio uses adjacent and hypotenuse?
AQA
Exam-style questions focusing on calculating missing sides using sine and cosine.
AQA
In the right-angled triangle, θ = 30° and the hypotenuse is 14 cm.
Find the opposite side x.
AQA
In the right-angled triangle, θ = 60° and the hypotenuse is 20 cm.
Find the adjacent side x.
OCR
Exam-style questions focusing on choosing the correct ratio and finding angles.
OCR
A student wants to use tangent, but the only known side is the hypotenuse.
Explain why tangent cannot be used.
OCR
In the right-angled triangle, the opposite side is 9 cm and the hypotenuse is 15 cm.
Find the angle θ.
OCR
In a right-angled triangle, the opposite side is 12 cm and the adjacent side is 16 cm.
Write down the trigonometric equation needed to find θ.
Exam Checklist
Step 1
Check that the triangle is right-angled.
Step 2
Mark the chosen angle and identify opposite, adjacent and hypotenuse.
Step 3
Choose sine, cosine or tangent from the sides involved.
Step 4
Rearrange carefully or use inverse trig for angles.
Most common exam mistakes
Wrong labels
Mixing up opposite and adjacent.
Wrong ratio
Choosing sine when cosine or tangent is needed.
Wrong calculator function
Using \(\sin\) instead of \(\sin^{-1}\) when finding an angle.
Wrong mode
Calculator set to radians instead of degrees.
Common Mistakes
These are common mistakes students make when using trigonometry in GCSE Maths.
Mixing up opposite and adjacent
Incorrect
A student labels sides incorrectly in the triangle.
Correct
Always identify sides relative to the given angle: opposite is across from the angle, adjacent is next to it, and hypotenuse is the longest side.
Forgetting labels depend on the angle
Incorrect
A student keeps the same labels when the reference angle changes.
Correct
Opposite and adjacent depend on the chosen angle. If the angle changes, the labels must be reassigned.
Using the wrong trig ratio
Incorrect
A student chooses sine, cosine or tangent incorrectly.
Correct
Use SOHCAHTOA: \(\sin = \frac{\text{opp}}{\text{hyp}}\), \(\cos = \frac{\text{adj}}{\text{hyp}}\), \(\tan = \frac{\text{opp}}{\text{adj}}\). Match the ratio to the sides you are using.
Not using inverse trig for angles
Incorrect
A student uses sin, cos or tan when finding an angle.
Correct
When finding an angle, use inverse functions: \(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\).
Calculator in the wrong mode
Incorrect
A student gets an incorrect answer due to radians mode.
Correct
Ensure your calculator is set to degrees (DEG mode) for GCSE questions unless stated otherwise.
Try It Yourself
Practise solving problems using trigonometric ratios.
Foundation
Foundation Practice
Use sine, cosine and tangent to find missing sides in right-angled triangles.
Find x to 1 decimal place.
Use tan because you have adjacent and need opposite.
tan 35° = x ÷ 10, so x = 10 × tan 35° = 7.0 cm.
Find x to 1 decimal place.
You have opposite and need hypotenuse, so use sin.
sin 40° = 8 ÷ x, so x = 8 ÷ sin 40° = 12.4 cm.
Find x to 1 decimal place.
You have hypotenuse and need adjacent, so use cos.
cos 50° = x ÷ 12, so x = 12 × cos 50° = 7.7 cm.
Which ratio uses opposite and adjacent?
TOA.
TOA means tan = opposite ÷ adjacent.
Find x to 1 decimal place.
Use tan: opposite ÷ adjacent.
tan 28° = x ÷ 14, so x = 14 × tan 28° = 7.4 cm.
Which side is always opposite the right angle?
It is the longest side.
The hypotenuse is always opposite the right angle.
In SOHCAHTOA, what does CAH mean?
C stands for cosine.
CAH means cos = adjacent ÷ hypotenuse.
Find x to 1 decimal place.
Adjacent = hypotenuse × cos(angle).
x = 15 × cos 62° = 7.0 cm.
Which trigonometric ratio should you use when you know the adjacent side and need the opposite side?
Use TOA.
Use tan because tan = opposite ÷ adjacent.
Higher
Higher Practice
Find missing angles and solve multi-step trigonometry problems.
Find angle x to 1 decimal place.
Use tan x = opposite ÷ adjacent.
tan x = 7 ÷ 9, so x = tan⁻¹(7/9) = 37.9°.
Find angle x to 1 decimal place.
Use cos x = adjacent ÷ hypotenuse.
cos x = 12 ÷ 20 = 0.6, so x = cos⁻¹(0.6) = 53.1°.
A ladder is 5 m long and makes an angle of 70° with the ground. How high up the wall does it reach? Give your answer to 1 decimal place.
The ladder is the hypotenuse and the height is opposite the angle.
height = 5 × sin 70° = 4.7 m.
Find x to 1 decimal place.
Use tan because opposite and adjacent are involved.
tan 32° = x ÷ 18, so x = 18 × tan 32° = 11.2 m.
Find angle x when opposite = 8 cm and hypotenuse = 13 cm. Give your answer to 1 decimal place.
Use sin x = opposite ÷ hypotenuse.
sin x = 8/13, so x = sin⁻¹(8/13) = 38.0°.
Which calculation finds angle x if adjacent = 6 and hypotenuse = 10?
Adjacent and hypotenuse use cosine.
cos x = adjacent ÷ hypotenuse, so x = cos⁻¹(6/10).
A ramp rises 1.5 m over a horizontal distance of 6 m. Find the angle the ramp makes with the ground to 1 decimal place.
Use tan angle = rise ÷ run.
tan x = 1.5 ÷ 6, so x = tan⁻¹(0.25) = 14.0°.
A student uses sine with adjacent and hypotenuse. What is wrong?
CAH uses adjacent and hypotenuse.
Adjacent and hypotenuse use cosine, not sine.
Find x to 1 decimal place if tan x = 3/4.
Use inverse tan.
x = tan⁻¹(3/4) = 36.9°.
A triangle has angle 25° and hypotenuse 14 cm. Which calculation finds the opposite side?
SOH: opposite = hypotenuse × sin angle.
opposite = 14 × sin 25°.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What are sine, cosine and tangent?
Ratios used to find sides or angles in right-angled triangles.
What does SOHCAHTOA mean?
It helps remember trig ratios.
When do I use trigonometry?
When you have angles and sides in triangles.