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

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

3. 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.

4. 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:

5. Task 1:
Exercise 10A

6. 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.)

7. 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°

8. 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

9. Task 3:
Exercise 10D