Trigonometric ratios and identities

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

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.

Rules to remember

π radians = 180°: multiply degrees by π/180 to get radians, and radians by 180/π to get degrees
Arc length l = rθ; area of sector A = ½r²θ (θ in radians)
sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
sin(α ± β) = sin α cos β ± cos α sin β; cos(α ± β) = cos α cos β ∓ sin α sin β
tan(α ± β) = (tan α ± tan β)/(1 ∓ tan α tan β)
sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2 tan θ/(1 − tan²θ)
Half angle: sin²(θ/2) = (1 − cos θ)/2; cos²(θ/2) = (1 + cos θ)/2; tan(θ/2) = sin θ/(1 + cos θ)

Common mistake

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.

Worked example

α and β are acute angles with sin α = 3/5 and cos β = 5/13. The value of sin(α + β) is

From the 3, 4, 5 triangle cos α = 4/5; from the 5, 12, 13 triangle sin β = 12/13.
sin(α + β) = sin α cos β + cos α sin β = (3/5)(5/13) + (4/5)(12/13).
= 15/65 + 48/65 = 63/65.

Answer: 63/65

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