School curriculum· 35 min· 21 questions
Learn first: Trigonometric identities
The formulas in this topic link the functions of two angles. With them you can find values for angles missing from the table (15°, 75°, 105°), simplify expressions, solve trigonometric equations and find the greatest and least values of a function. All the formulas follow from one — the formula for cos(α − β) — so understanding the logic matters more than memorising: if you forget a formula, you can rebuild it in a line or two. Tangent and cotangent are written tg and ctg, as in Uzbek and Russian textbooks.
Take the points A(cos α; sin α) and B(cos β; sin β) on the unit circle. The angle between the vectors OA and OB is α − β, and both have length 1. Write their scalar (dot) product in two ways:
cos(α − β) = cos α cos β + sin α sin β — the source of every formula
The rest follow by substitution: replacing β with −β gives cos(α + β); the reduction formula sin x = cos(π/2 − x) gives sin(α ± β); dividing sin by cos gives tg(α ± β); setting β = α gives the double-angle formulas.
sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β ∓ sin α sin β
tg(α ± β) = (tg α ± tg β)/(1 ∓ tg α tg β)
Note: the sign in the cos formula is reversed — minus for a sum, plus for a difference. The sin formula has "mixed" products (sin · cos), the cos formula "matching" ones (cos · cos, sin · sin).
These formulas work in both directions: sin 40° cos 20° + cos 40° sin 20° = sin 60° = √3/2.
Evaluate sin 75°.
Similarly cos 15° = cos(45° − 30°) = (√6 + √2)/4 and tg 15° = 2 − √3.
tg α = 2, tg β = 3. Find tg(α + β).
Set β = α:
sin 2α = 2 sin α cos α
cos 2α = cos²α − sin²α = 2cos²α − 1 = 1 − 2sin²α
tg 2α = 2tg α/(1 − tg²α)
sin 3α = 3sin α − 4sin³α, cos 3α = 4cos³α − 3cos α
The three forms of cos 2α are obtained from each other with sin²α + cos²α = 1 — choose whichever suits the problem.
Lowering the power (much needed in integrals and equations) — express sin²α and cos²α from the cos 2α formula:
sin²α = (1 − cos 2α)/2, cos²α = (1 + cos 2α)/2
Half-angle formulas come from replacing α with α/2: sin²(α/2) = (1 − cos α)/2. For example, sin 15° = √((1 − cos 30°)/2) = (√6 − √2)/4 (15° is in quadrant I, so the sign is +).
Learn to recognise ready-made patterns: 2 sin 15° cos 15° = sin 30° = 1/2; cos²15° − sin²15° = cos 30° = √3/2; 1 − 2sin²22.5° = cos 45° = √2/2.
sin α = 3/5 with α in quadrant I. Find sin 2α and cos 2α.
Evaluate cos 20° · cos 40° · cos 80°.
sin α + sin β = 2 sin((α + β)/2) cos((α − β)/2)
sin α − sin β = 2 cos((α + β)/2) sin((α − β)/2)
cos α + cos β = 2 cos((α + β)/2) cos((α − β)/2)
cos α − cos β = −2 sin((α + β)/2) sin((α − β)/2)
They come from adding the formulas of §2: sin(x + y) + sin(x − y) = 2 sin x cos y, then putting x + y = α, x − y = β. Their main use is in equations: turn a sum into a product and set each factor equal to zero.
Simplify (cos 3α + cos α)/(sin 3α − sin α).
sin α cos β = ½(sin(α + β) + sin(α − β))
cos α cos β = ½(cos(α − β) + cos(α + β))
sin α sin β = ½(cos(α − β) − cos(α + β))
If an equation contains a product, it is often convenient to turn it into a sum and compare the parts; in integration a product is hard to integrate, but a sum is easy.
For example, sin 15° sin 75° = ½(cos 60° − cos 90°) = ½ · ½ = 1/4.
An expression of the form a sin x + b cos x can be written as a single sine:
a sin x + b cos x = √(a² + b²) · sin(x + φ), where cos φ = a/√(a² + b²), sin φ = b/√(a² + b²)
Proof: expand sin(x + φ) with the formula from §2. Consequence: the greatest value of the expression is √(a² + b²) and the least is −√(a² + b²). For example, √3 sin x + cos x = 2 sin(x + π/6), since √(3 + 1) = 2, cos φ = √3/2, sin φ = 1/2.
Find the greatest and least values of y = 3 sin x + 4 cos x.