Trigonometric Ratios: SOH CAH TOA
Learn the three primary trigonometric ratios and how to use them to find missing sides and angles in right-angled triangles.
What Are Trigonometric Ratios?
Trigonometric ratios are relationships between the sides of a right-angled triangle relative to one of its acute angles. They allow us to calculate unknown sides or angles when we know some measurements.
The three primary ratios are Sine (sin), Cosine (cos), and Tangent (tan). We use the mnemonic SOH CAH TOA to remember them.
The Right-Angled Triangle
The sides are named relative to the angle θ (theta).
SOH CAH TOA — The Three Ratios
sin(θ) = O H
Sine = Opposite / Hypotenuse
cos(θ) = A H
Cosine = Adjacent / Hypotenuse
tan(θ) = O A
Tangent = Opposite / Adjacent
Memory Tip: "Some Old Horses Can Always Hear Their Owners Approach" or simply remember the letters: SOH-CAH-TOA.
Worked Examples
Find the length of side x (opposite).
Step 1: Identify sides. We know the hypotenuse (12 cm) and want the opposite side (x). This means we use SOH (sin).
Step 2: Write the formula. sin(35°) = x / 12
Step 3: Rearrange. x = 12 × sin(35°)
Step 4: Calculate. x = 12 × 0.5736 = 6.88 cm (2 d.p.)
Find the length of side y (hypotenuse).
Step 1: Identify sides. We know the adjacent (8 cm) and want the hypotenuse (y). This means we use CAH (cos).
Step 2: Write the formula. cos(50°) = 8 / y
Step 3: Rearrange. y = 8 / cos(50°)
Step 4: Calculate. y = 8 / 0.6428 = 12.45 cm (2 d.p.)
Find angle θ.
Step 1: Identify sides. We know the opposite (7 cm) and adjacent (10 cm). This means we use TOA (tan).
Step 2: Write the formula. tan(θ) = 7/10 = 0.7
Step 3: Use inverse tan. θ = tan-1(0.7)
Step 4: Calculate. θ = 34.99° ≈ 35.0° (1 d.p.)
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see feedback.
Question 1
In a right-angled triangle, the angle is 40°, and the hypotenuse is 15 cm. Which ratio would you use to find the opposite side?
Question 2
Calculate: sin(30°) × 20. Round to 2 decimal places.
Question 3
A ladder leans against a wall. The angle with the ground is 65° and the ladder is 6 m long (hypotenuse). How far up the wall does it reach? (Round to 2 d.p.)
Question 4
The opposite side is 9 cm and the adjacent side is 12 cm. What is the angle θ? (Round to 1 d.p.)
Question 5
Which formula correctly rearranges cos(θ) = A/H to find the adjacent side?
Key Formulas Summary
| Ratio | Formula | To find the side | To find the angle |
|---|---|---|---|
| sin(θ) | O / H | O = H × sin(θ) | θ = sin-1(O/H) |
| cos(θ) | A / H | A = H × cos(θ) | θ = cos-1(A/H) |
| tan(θ) | O / A | O = A × tan(θ) | θ = tan-1(O/A) |