NUST NET (Engineering) · Maths · Trigonometry. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
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.
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.
Answer: a = 7
Try each one, then open the answer.
Answer: D. The largest angle is opposite the side 7: cos θ = (9 + 25 − 49)/(2 × 3 × 5) = −15/30 = −½, so θ = 120°. Dropping the minus sign gives 60°.
Answer: D. cot x = cos x/sin x is undefined where sin x = 0, that is at x = nπ. The odd multiples of π/2 are excluded from tan x and sec x, not from cot x.
Answer: C. cos (4π/3) = −½, and the principal value of cos−1(−½) must lie in [0, π], giving 2π/3. 4π/3 is outside the principal range.
← Trigonometric ratios and identitiesLimits and continuity →All rules on one pageStuck? Ask a question