Trigonometry: SOH CAH TOA
The Skill
Trigonometry uses ratios to find missing sides and angles in right-angled triangles.
The Three Ratios
$$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
$$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$
$$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$
SOH CAH TOA
Sin = Opposite / Hypotenuse Cos = Adjacent / Hypotenuse Tan = Opposite / Adjacent
Labelling the Triangle
From your angle θ:
- Opposite (O): The side directly across from the angle
- Adjacent (A): The side next to the angle (not the hypotenuse)
- Hypotenuse (H): The longest side, opposite the right angle
Finding a Side
- Label the sides O, A, H from the given angle
- Identify which sides are involved (given and wanted)
- Choose the correct ratio (SOH, CAH, or TOA)
- Substitute and solve
Finding an Angle
Use the inverse function: $$\theta = \sin^{-1}\left(\frac{O}{H}\right) \quad \text{or} \quad \cos^{-1}\left(\frac{A}{H}\right) \quad \text{or} \quad \tan^{-1}\left(\frac{O}{A}\right)$$
The Traps
Common misconceptions and how to avoid them.
Calculator set to radians instead of degrees "The Mode Mishap"
The Mistake in Action
Find $\sin(30°)$
Wrong: $\sin(30) = -0.988...$
Why It Happens
The calculator is in radians mode (showing "RAD" or "R"), not degrees mode. The student enters 30, thinking degrees, but the calculator interprets it as 30 radians.
The Fix
Check your calculator display — you should see "DEG" or "D", not "RAD" or "R".
To change mode on Casio Classwiz:
- Press MENU
- Select Settings
- Select Angle Unit
- Choose Degree
Correct answer: $\sin(30°) = 0.5$
Quick check: You should know that $\sin(30°) = 0.5$ — if you get anything different, check your mode!
Spot the Mistake
Find $\sin(30°)$
$\sin(30) = -0.988$
Click on the line that contains the error.
Confusing opposite and adjacent sides "The O-A Swap"
The Mistake in Action
In a right-angled triangle, angle θ = 40°. The side next to θ (not the hypotenuse) is 8cm. Find the side opposite θ.
Wrong: "Adjacent = unknown, Opposite = 8" $\tan(40°) = \frac{8}{x}$
Why It Happens
Students confuse which side is "opposite" and which is "adjacent." They may label based on how the triangle looks rather than relative to the specific angle.
The Fix
Always label from YOUR angle:
- Opposite: Directly across from θ (doesn't touch θ)
- Adjacent: Next to θ (touches θ, but isn't the hypotenuse)
- Hypotenuse: Opposite the right angle (longest side)
In this problem, from θ = 40°:
- The side "next to θ" = Adjacent = 8cm
- The side "opposite θ" = Opposite = x
$$\tan(40°) = \frac{x}{8}$$ $$x = 8 \times \tan(40°) = 6.71\text{cm}$$
Spot the Mistake
Angle = 40°. Side next to θ is 8cm. Find the opposite side.
Opposite = 8cm, Adjacent = x
$\tan(40°) = \frac{8}{x}$
Click on the line that contains the error.
Not using inverse functions to find angles "The Inverse Forget"
The Mistake in Action
In a triangle, Opposite = 6cm and Hypotenuse = 10cm. Find the angle.
Wrong: $\sin(\theta) = \frac{6}{10} = 0.6$ $\theta = 0.6°$
Why It Happens
Students correctly find sin(θ) = 0.6 but then think that IS the angle. They don't realise they need to use the inverse sine function to find θ.
The Fix
0.6 is not the angle — it's the VALUE of sin(θ).
To find the angle, use inverse sine (sin⁻¹ or arcsin):
$$\theta = \sin^{-1}(0.6) = 36.9°$$
On your calculator: Press SHIFT then SIN, then enter 0.6.
Spot the Mistake
Opposite = 6cm, Hypotenuse = 10cm. Find angle θ.
$\sin(\theta) = \frac{6}{10} = 0.6$
$\theta = 0.6°$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
In a right-angled triangle, angle A = 42° and the hypotenuse is 15cm. Find the length of the side opposite angle A.
Solution
Step 1: Label the sides from angle A
- Opposite = x (what we want)
- Hypotenuse = 15cm
Step 2: Choose the ratio We have O and H → use SOH: $\sin = \frac{O}{H}$
Step 3: Set up and solve $$\sin(42°) = \frac{x}{15}$$ $$x = 15 \times \sin(42°)$$ $$x = 15 \times 0.6691...$$ $$x = 10.04\text{cm (2 d.p.)}$$
Answer: 10.04cm
Question
In a right-angled triangle, angle B = 55° and the adjacent side is 8cm. Find the opposite side.
Solution
Step 1: Label the sides
- Adjacent = 8cm
- Opposite = x
Step 2: Choose the ratio We have A and want O → use TOA: $\tan = \frac{O}{A}$
Step 3: Set up and solve $$\tan(55°) = \frac{x}{8}$$ $$x = 8 \times \tan(55°)$$ $$x = 8 \times 1.428...$$ $$x = 11.43\text{cm (2 d.p.)}$$
Answer: 11.43cm
Question
In a right-angled triangle, the opposite side is 7cm and the adjacent side is 11cm. Find angle θ.
Solution
Step 1: Identify what we have
- Opposite = 7cm
- Adjacent = 11cm
Step 2: Choose the ratio We have O and A → use TOA: $\tan = \frac{O}{A}$
Step 3: Set up the equation $$\tan(\theta) = \frac{7}{11}$$
Step 4: Use inverse tan $$\theta = \tan^{-1}\left(\frac{7}{11}\right)$$ $$\theta = \tan^{-1}(0.6363...)$$ $$\theta = 32.5°$$
Answer: 32.5°
Level 2: Scaffolded
Fill in the key steps.
Question
Find the length of the hypotenuse. The adjacent side to the 38° angle is 12cm.
Level 3: Solo
Try it yourself!
Question
A tree is 18m tall. From a point on the ground 24m away from the base of the tree, what is the angle of elevation to the top of the tree?
Show Solution
The tree height (18m) is opposite the angle. The distance (24m) is adjacent to the angle.
$$\tan(\theta) = \frac{18}{24} = 0.75$$ $$\theta = \tan^{-1}(0.75) = 36.9°$$
Answer: 36.9°
Question
In a right-angled triangle, angle A = 52° and the side opposite angle A is 16cm. Find the hypotenuse.
Show Solution
We have Opposite = 16cm and want Hypotenuse. Use SOH: $\sin = \frac{O}{H}$
$$\sin(52°) = \frac{16}{H}$$ $$H = \frac{16}{\sin(52°)}$$ $$H = \frac{16}{0.788}$$ $$H = 20.3\text{cm (1 d.p.)}$$
Answer: 20.3cm
Examiner's View
Mark allocation: Basic trig questions are 2-3 marks. Multi-step problems are 3-4 marks.
Common errors examiners see:
- Calculator in wrong mode (radians instead of degrees)
- Labelling O and A incorrectly
- Using the wrong ratio
- Forgetting to use inverse functions for angles
What gains marks:
- Labelling the triangle clearly (O, A, H)
- Stating which ratio you're using
- Showing the rearrangement clearly
- Giving answers to appropriate accuracy
AQA Notes
AQA often asks for exact values (sin 30° = 0.5, etc.).