Trigonometric graphs, triangles and equations

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

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The idea

Each trigonometric function repeats after its period and takes values only in its range. A number multiplying x changes the period; a number multiplying or added to the function changes the range.

Any triangle can be solved with two rules. Use the sine rule when you know a side and its opposite angle; use the cosine rule for two sides with the angle between them, or for three sides. In triangle ABC, side a is opposite angle α, b opposite β and c opposite γ.

An inverse function gives one angle only, taken from its principal range. A trigonometric equation has many solutions, so find the basic angle, then every angle in the stated interval or the general solution.

Rules to remember

Period: sin, cos, sec, cosec 2π; tan, cot π. For sin kx or cos kx the period is 2π/|k|; for tan kx it is π/|k|
Range: sin x and cos x lie in [−1, 1], so a + b sin x lies in [a − |b|, a + |b|]; tan x takes all real values
Sine rule: a/sin α = b/sin β = c/sin γ = 2R (R = circum-radius)
Cosine rule: a² = b² + c² − 2bc cos α, or cos α = (b² + c² − a²)/(2bc)
Area Δ = ½bc sin α = √(s(s − a)(s − b)(s − c)) with s = (a + b + c)/2; in-radius r = Δ/s; circum-radius R = abc/(4Δ)
Principal ranges: sin⁻¹x in [−π/2, π/2]; cos⁻¹x in [0, π]; tan⁻¹x in (−π/2, π/2)
General solutions (n ∈ Z): sin x = sin α gives x = nπ + (−1)ⁿα; cos x = cos α gives x = 2nπ ± α; tan x = tan α gives x = nπ + α

Common mistake

Dividing an equation by sin x or cos x and losing solutions. In sin 2x = sin x, write 2 sin x cos x − sin x = 0 and factorise, so that sin x = 0 is kept as well as cos x = ½.

Worked example

In triangle ABC, b = 5, c = 8 and α = 60°. The length of side a is

Two sides and the angle between them are known, so use the cosine rule: a² = b² + c² − 2bc cos α.
a² = 25 + 64 − 2 × 5 × 8 × cos 60° = 89 − 80 × ½ = 89 − 40 = 49.
a = √49 = 7.

Answer: a = 7

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