Sine ratio
sin θ = opposite ÷ hypotenuse
In a right-angled triangle, the sine of an angle compares the side opposite it with the hypotenuse.
Symbols
- θ
- the angle (not the right angle)
- opposite
- side across from θ
- hypotenuse
- longest side
When it applies
Right-angled triangles, for acute angles θ. Check your calculator is in degrees or radians to match the angle.
Worked example
In a right triangle the side opposite θ is 3 cm and the hypotenuse is 6 cm. Find θ.
- 1.sin θ = 3 ÷ 6 = 0.5
- 2.θ = sin⁻¹(0.5) = 30°
Answer: θ = 30°
Cosine ratio
cos θ = adjacent ÷ hypotenuse
Compares the side next to the angle (not the hypotenuse) with the hypotenuse.
Symbols
- θ
- the angle
- adjacent
- side next to θ, not the hypotenuse
- hypotenuse
- longest side
When it applies
Right-angled triangles, acute θ. Angle mode (degrees/radians) must match.
Worked example
A right triangle has hypotenuse 10 m and angle θ = 60°. Find the adjacent side.
- 1.adjacent = 10 × cos 60°
- 2.cos 60° = 1/2
- 3.adjacent = 10 × 0.5 = 5
Answer: 5 m
Tangent ratio
tan θ = opposite ÷ adjacent
Compares the two shorter sides of a right triangle.
Symbols
- θ
- the angle
- opposite
- side across from θ
- adjacent
- side next to θ
When it applies
Right-angled triangles, acute θ. tan 90° is undefined.
Worked example
The adjacent side is 12 cm and θ = 45°. Find the opposite side.
- 1.opposite = 12 × tan 45°
- 2.tan 45° = 1
- 3.opposite = 12
Answer: 12 cm
Pythagorean identity
sin²θ + cos²θ = 1
True for every angle. It lets you find cos θ from sin θ (or the reverse), choosing the sign from the quadrant.
Symbols
- θ
- any angle
- sin²θ
- (sin θ)²
When it applies
All angles. Taking a square root gives ±; pick the sign that matches the quadrant of θ.
Worked example
sin θ = 3/5 and θ is acute. Find cos θ and tan θ.
- 1.cos²θ = 1 − (3/5)² = 1 − 9/25 = 16/25
- 2.θ acute, so cos θ is positive: cos θ = 4/5
- 3.tan θ = sin θ ÷ cos θ = (3/5) ÷ (4/5) = 3/4
Answer: cos θ = 4/5, tan θ = 3/4
Reference only — no interactive Mathover tool for this formula yet.
Degrees to radians
radians = degrees × π ÷ 180
A full turn is 360° or 2π radians, so 180° = π radians. Multiply by 180/π to go back.
Symbols
- π
- pi ≈ 3.14159
When it applies
Any angle. Calculus formulas for sin and cos assume radians.
Worked example
Convert 135° to radians.
- 1.135 × π ÷ 180
- 2.135 ÷ 180 = 3/4
- 3.3π/4 ≈ 2.3562
Answer: 3π/4 ≈ 2.3562 rad
Sine rule
a ÷ sin A = b ÷ sin B = c ÷ sin C
In any triangle, each side divided by the sine of its opposite angle gives the same value.
Symbols
- a, b, c
- side lengths
- A, B, C
- angles opposite those sides
When it applies
Any triangle. Use it when you know a side and its opposite angle. Finding an angle can give two possible answers (the ambiguous case).
Worked example
In triangle ABC, A = 30°, B = 45° and a = 5 cm. Find b.
- 1.b = a × sin B ÷ sin A
- 2.b = 5 × (√2/2) ÷ (1/2)
- 3.b = 5√2 ≈ 7.07
Answer: b = 5√2 cm ≈ 7.07 cm
Reference only — no interactive Mathover tool for this formula yet.
Cosine rule
c² = a² + b² − 2ab cos C
Generalises Pythagoras to any triangle. When C = 90°, cos C = 0 and it becomes a² + b² = c².
Symbols
- a, b
- two sides
- C
- the angle between them
- c
- the side opposite C
When it applies
Any triangle. Use when you know two sides and the included angle, or all three sides.
Worked example
a = 5 cm, b = 8 cm and C = 60°. Find c.
- 1.c² = 25 + 64 − 2(5)(8)cos 60°
- 2.cos 60° = 1/2, so 2(5)(8)(1/2) = 40
- 3.c² = 89 − 40 = 49
- 4.c = 7
Answer: c = 7 cm
Reference only — no interactive Mathover tool for this formula yet.