Trigonometric identities and exact values

Exact values

• sin 30° = 1/2, cos 30° = √3/2 and tan 30° = 1/√3 (or √3/3).
• sin 45° = cos 45° = √2/2 (or 1/√2), and tan 45° = 1.
• sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3.
• sin 0° = 0, cos 0° = 1, sin 90° = 1 and cos 90° = 0.
• Learn them from the two special triangles: half an equilateral triangle with sides 1, √3 and 2, and a right-angled isosceles triangle with sides 1, 1 and √2.

The two identities

• tan θ ≡ sin θ ÷ cos θ.
• sin²θ + cos²θ ≡ 1, which also gives sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ.
• If sin θ = 3/5 and θ is acute, then cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5 and tan θ = 3/4.

Using the identities

• Simplify: (1 − sin²θ) ÷ cos θ = cos²θ ÷ cos θ = cos θ.
• Prove: (sin θ + cos θ)² ≡ sin²θ + 2 sin θ cos θ + cos²θ ≡ 1 + 2 sin θ cos θ.
• Solve: sin x = 2cos x means tan x = 2, so x = 63.4° and 243.4° (to 1 decimal place).

Key terms

Identity

An equation true for every value of the variable, shown by ≡.

Exact value

A value written with fractions or surds, not a rounded decimal.

Acute angle

An angle less than 90°.

sin²θ

(sin θ)², the square of sin θ.