Trigonometry Ratios

Learn sine, cosine, and tangent for right-angled triangles

CAPS Grade 10 Mathematics

Trigonometry is the study of relationships between angles and sides in triangles. The three main ratios are sine, cosine, and tangent - remember them using SOH CAH TOA!

Right Triangle: Opposite, Adjacent, Hypotenuse
Opposite (O) Adjacent (A) Hypotenuse (H) θ
The hypotenuse is always opposite the right angle. Opposite is across from angle θ. Adjacent is next to angle θ.

What You'll Learn

Identify opposite, adjacent, and hypotenuse sides
Use SOH CAH TOA to remember ratios
Find unknown side lengths using trig ratios
Find unknown angles using inverse trig functions
Solve real-world problems (elevation/depression)

Quiz 1: SOH CAH TOA

What does SOH stand for

A) sin = opposite/adjacent
B) sin = opposite/hypotenuse
C) sin = adjacent/hypotenuse
D) sin = hypotenuse/opposite

1. Triangle Sides

H

Hypotenuse

  • Longest side of a right triangle
  • Always opposite the right angle (90°)
O

Opposite

  • Side directly across from the angle θ
  • "Opposite" means facing the angle
A

Adjacent

  • Side next to the angle θ
  • Not the hypotenuse

Quiz 2: Identify Sides

In a right triangle, which side is ALWAYS the longest

A) Opposite
B) Adjacent
C) Hypotenuse
D) Depends on the angle
SOH CAH TOA - Visual Memory Aid
Opposite (O) Adjacent (A) Hypotenuse (H) θ SOH: sin θ = O/H CAH: cos θ = A/H TOA: tan θ = O/A
Use SOH CAH TOA to remember which sides go with each ratio.

2. The Three Ratios

SOH CAH TOA - Memory Aid

SOH: sin θ = Opposite / Hypotenuse
CAH: cos θ = Adjacent / Hypotenuse
TOA: tan θ = Opposite / Adjacent
S

Sine (sin)

sin θ = opposite / hypotenuse
  • Use when you know opposite and hypotenuse
C

Cosine (cos)

cos θ = adjacent / hypotenuse
  • Use when you know adjacent and hypotenuse
T

Tangent (tan)

tan θ = opposite / adjacent
  • Use when you know opposite and adjacent

Quiz 3: Which Ratio

To find the opposite side when you know the hypotenuse and angle, use:

A) sin
B) cos
C) tan
D) Pythagoras

3. Finding Unknown Sides

Example 1: Using Sine

Problem

In a right triangle, angle θ = 30° and hypotenuse = 10 cm. Find the opposite side.

Step-by-Step Solution
  1. Use SOH: sin θ = opposite / hypotenuse
  2. sin 30° = opposite / 10
  3. 0.5 = opposite / 10
  4. opposite = 10 × 0.5 = 5 cm
Answer: opposite side = 5 cm

Example 2: Using Cosine

Problem

In a right triangle, angle θ = 60° and hypotenuse = 20 m. Find the adjacent side.

Step-by-Step Solution
  1. Use CAH: cos θ = adjacent / hypotenuse
  2. cos 60° = adjacent / 20
  3. 0.5 = adjacent / 20
  4. adjacent = 20 × 0.5 = 10 m
Answer: adjacent side = 10 m
Angle of Elevation
60° Angle of Elevation 20 m Height
Angle of elevation: looking UP from the horizontal line to an object above.

4. Finding Unknown Angles

Example 3: Using Inverse Tangent

Problem

In a right triangle, opposite = 4 cm, adjacent = 3 cm. Find angle θ.

Step-by-Step Solution
  1. Use TOA: tan θ = opposite / adjacent = 4/3
  2. tan θ = 1.3333...
  3. θ = tan⁻¹(4/3)
  4. θ ≈ 53.13°
Answer: θ ≈ 53.13°
Angle of Depression
30° Angle of Depression 50 m Distance
Angle of depression: looking DOWN from the horizontal line to an object below.

Quiz 4: Find Side

If tan θ = 0.75 and adjacent = 8 cm, what is the opposite side

A) 4 cm
B) 6 cm
C) 8 cm
D) 10 cm

5. Angles of Elevation & Depression

↑

Angle of Elevation

  • Looking UP from horizontal
  • Example: looking at top of a building
↓

Angle of Depression

  • Looking DOWN from horizontal
  • Example: looking down from a cliff

Example 4: Building Height

Problem

A person stands 20 m from a building. The angle of elevation to the top is 60°. Find the building height.

Step-by-Step Solution
  1. adjacent = 20 m, angle = 60°, need opposite (height)
  2. Use TOA: tan 60° = height / 20
  3. tan 60° = √3 ≈ 1.732
  4. height = 20 × 1.732 = 34.64 m
Answer: Building height ≈ 34.64 m

Quiz 5: Find Angle

If sin θ = 0.5, what is θ (θ acute)

A) 30°
B) 45°
C) 60°
D) 90°

Quiz 6: Real-World

A ladder 10 m long leans against a wall at 60° to the ground. How high up the wall does it reach

A) 5 m
B) 8.66 m
C) 10 m
D) 20 m

6. Common Mistakes

Mistake:

Labeling opposite and adjacent incorrectly - opposite is ACROSS from the angle

Mistake:

Using the wrong ratio - always check which sides you know

Mistake:

Calculator in radian mode instead of degree mode

Mistake:

Forgetting to draw a diagram - always sketch first

7. Practice Questions

Q1

Find sin θ if opposite = 3, hypotenuse = 5

Answer: sin θ = 3/5 = 0.6
Q2

Find angle θ if tan θ = 1

Answer: θ = 45°
Q3

A flagpole casts a 15 m shadow when sun elevation is 50°. Find height.

Answer: height = 15 × tan 50° ≈ 17.88 m

8. Summary

SOH CAH TOA

sin θ = O/H
cos θ = A/H
tan θ = O/A

Inverse Functions

θ = sin⁻¹(O/H)
θ = cos⁻¹(A/H)
θ = tan⁻¹(O/A)

Key Tips

  • Always draw a diagram
  • Label O, A, H correctly
  • Use SOH CAH TOA to choose ratio
  • Check calculator is in DEGREE mode
Back to Trigonometry Next: Special Angles