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.
Angles are measured in degrees or in radians, where π radians = 180°. The formulas for arc length and sector area work only with the angle in radians.
An identity is true for every angle. The test expects you to know the basic identities and the compound, double and half angle formulas by heart, and to spot which one turns the question into known values. Exact values such as sin 15° or tan 75° come from writing the angle as a sum or difference of 30°, 45° and 60°.
For angles like π ± θ or π/2 ± θ, find the quadrant to fix the sign. The ratio stays the same for π ± θ and changes to its co-ratio for π/2 ± θ.
Getting the sign wrong in cos(α ± β). The sign in the middle is opposite to the one in the bracket: cos(α + β) = cos α cos β − sin α sin β.
Answer: 63/65
Try each one, then open the answer.
Answer: C. 1 − cos 2θ = 2 sin2θ and sin 2θ = 2 sin θ cos θ, so the ratio is sin θ/cos θ = tan θ. (1 + cos 2θ)/sin 2θ would give cot θ.
Answer: C. tan(θ/2) = (1 − cos θ)/sin θ with θ = 45°: (1 − 1/√2)/(1/√2) = √2 − 1. √2 + 1 is tan 67.5°, and 1 − 1/√2 forgets to divide by sin 45°.
Answer: B. cos 2θ = (1 − tan2θ)/(1 + tan2θ) = (1 − 9)/(1 + 9) = −4/5. The value 3/5 is sin 2θ = 2 tan θ/(1 + tan2θ).
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