C2: Trigonometrical Identities Learning Objective: to be able to use simple trigonometrical relationships to solve problems

Starter: Solve for 0 ≤ x ≤ 360° tan x = √3 sin x = sin 15° cos (50° + 2x) = -1

Trigonometric identities Two important identities that must be learnt are: sin2θ + cos2θ ≡ 1 The symbol ≡ means “is identically equal to” although an equals sign can also be used. An identity, unlike an equation, is true for every value of the given variable so, for example: sin24° + cos24° ≡ 1, sin267° + cos267° ≡ 1, sin2π + cos2π ≡ 1, etc.

Trigonometric identities We can prove these identities by considering a right-angled triangle: x y r θ Also: But by Pythagoras’ theorem x2 + y2 = r2 so:

Task 1: Exercise 10A

Trigonometric identities One use of these identities is to simplify trigonometric equations. For example: Solve sin θ = 3 cos θ for 0° ≤ θ ≤ 360°. Dividing through by cos θ: Using a calculator, the principal solution is θ = 71.6° (to 1 d.p.) 71.6° So the solutions in the given range are: 251.6° θ = 71.6°, 251.6° (to 1 d.p.)

Trigonometric identities Solve 2cos2θ – sin θ = 1 for 0 ≤ θ ≤ 360°. We can use the identity cos2θ + sin2 θ = 1 to rewrite this equation in terms of sin θ. 2(1 – sin2 θ) – sin θ = 1 2 – 2sin2θ – sin θ = 1 2sin2θ + sin θ – 1 = 0 (2sin θ – 1)(sin θ + 1) = 0 Note that when sin θ = 1 or –1 there will only be one solution in the range 0° ≤ θ ≤ 360°. So: sin θ = 0.5 or sin θ = –1 If sin θ = 0.5, θ = 30°, 150° If sin θ = –1, θ = 270°

Task 2 : Solve for 0 ≤ x ≤ 360° sin x – 1 = cos2 x 6cos x – 5sin x = 0 cos x + sin x = 3 sin x sin2 x + 3cos2 x = 2 answer in radians 3sin2 x – 2cos x + 1 = 0 answer in radians 4sin x cos x – 2sin x – 2cos x + 1 = 0 (3sin x – 1)(2tan x + 3) = 0 3sin x + 3 = cos2 x

Task 3: Exercise 10D