Compound and allied angles

Maths · Trigonometric identities. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Maths topics

The idea

The sine of a sum is not the sum of the sines. To expand sin, cos or tan of α + β or α − β you need the compound angle formulas. They all come from one result, the fundamental law of trigonometry, cos(α − β) = cos α cos β + sin α sin β.

This law is proved by taking two points on the unit circle, P(cos α, sin α) and Q(cos β, sin β), and finding |PQ|² with the distance formula.

Allied angles are angles such as 90° ± θ, 180° ± θ, 270° ± θ and 360° ± θ. Their ratios reduce to a ratio of θ alone, with a sign that depends on the quadrant.

Rules to remember

cos(α − β) = cos α cos β + sin α sin β and cos(α + β) = cos α cos β − sin α sin β
sin(α + β) = sin α cos β + cos α sin β and sin(α − β) = sin α cos β − cos α sin β
tan(α + β) = (tan α + tan β)/(1 − tan α tan β) and tan(α − β) = (tan α − tan β)/(1 + tan α tan β)
180° ± θ and 360° ± θ: the function stays the same (sin stays sin, cos stays cos, tan stays tan)
90° ± θ and 270° ± θ: the function changes to its co-function (sin ↔ cos, tan ↔ cot, sec ↔ cosec)
Sign: treat θ as acute, find the quadrant of the allied angle, and give the sign that the original function has there
Examples: sin(90° + θ) = cos θ, cos(90° + θ) = −sin θ, cos(180° − θ) = −cos θ, tan(180° + θ) = tan θ

Common mistake

The cosine formulas have the opposite sign to the angle: cos(α + β) has a minus in the middle and cos(α − β) has a plus. The sine formulas keep the same sign.

Worked example

Find the exact value of cos 105°.

Write 105° = 60° + 45°, so cos 105° = cos 60° cos 45° − sin 60° sin 45°.
Substitute: (1/2)(√2/2) − (√3/2)(√2/2) = √2/4 − √6/4.
Check the sign: 105° is in the second quadrant, where cosine is negative, and √2 − √6 is negative.

Answer: cos 105° = (√2 − √6)/4 ≈ −0.259.

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Keep going

← Trigonometric ratios and fundamental identities Double, half and triple angles; sums and products →Take a Maths test