Trigonometry · StudentHub Lesson

Trigonometry Basics: SOH-CAH-TOA

Use sine, cosine and tangent ratios to find missing sides and angles in right triangles.

25 minIntermediate

What you will learn

Identify opposite, adjacent and hypotenuse sides
Recall the SOH-CAH-TOA mnemonic
Calculate a missing side using trig ratios
Calculate a missing angle using inverse trig functions
Apply trigonometry to real-world problems

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Topic notes

Main Idea

Trigonometry relates the angles of a right triangle to the ratios of its sides using sine, cosine and tangent.

Key Concepts

SOH-CAH-TOA: Sin=Opposite/Hypotenuse, Cos=Adjacent/Hypotenuse, Tan=Opposite/Adjacent
Identify sides relative to the angle being used
Inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) find angles from ratios

Definitions

Opposite: side across from the reference angle
Adjacent: side next to the reference angle (not the hypotenuse)
Hypotenuse: longest side, opposite the right angle

Formulas

sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent

Examples

Angle 30°, hypotenuse 10: opposite = 10 × sin(30°) = 5
Opposite 4, adjacent 3: θ = tan⁻¹(4/3) ≈ 53.1°

Common Mistakes

Mixing up opposite and adjacent sides
Using the wrong ratio for the given information
Forgetting to use inverse functions when solving for angles

Key concepts

SOH-CAH-TOAOpposite, adjacent, hypotenuseSine, cosine, tangent ratiosInverse trig functionsRight triangle problem solving

Important terms

Sine: Ratio of the opposite side to the hypotenuse.
Cosine: Ratio of the adjacent side to the hypotenuse.
Tangent: Ratio of the opposite side to the adjacent side.

Worked examples

Problem

Find the opposite side when angle = 30° and hypotenuse = 10

1. Use sin(θ)=opposite/hypotenuse
2. sin(30°)=opposite/10
3. opposite=10×0.5

Answer: 5

Problem

Find angle θ when opposite = 4 and adjacent = 3

1. Use tan(θ)=opposite/adjacent
2. tan(θ)=4/3
3. θ=tan⁻¹(4/3)

Answer: ≈53.1°

Quick revision

SOH-CAH-TOA mnemonic
Identify the angle before labelling sides
Sine = opp/hyp
Cosine = adj/hyp
Tangent = opp/adj
Use inverse functions for angles

Check your understanding

Question 1 · Multiple choice

Which ratio is tangent?

Question 2 · Multiple choice

If sin(θ) = opposite/hypotenuse, what does cos(θ) equal?

Question 3 · Short answer

Find the hypotenuse if the angle is 45° and the adjacent side is 7 (cos45°≈0.707).

Question 4 · Short answer

Find angle θ if tan(θ)=1.

Question 5 · True or false

Basic SOH-CAH-TOA ratios only apply to right triangles.

Done with Trigonometry Basics: SOH-CAH-TOA?

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