Worksheet 4.8 Properties of Trigonometric Functions

Exercises:
1. Sketch the following graphs stating the period for each.
(a) y = cos 4x
(b) y = 2 cos 4 x − π
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(c) y = tan 2 x + π
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(d) y = 1 + sin
x−π
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(e) y = 2 − cos x + π
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(f) y = | cos x |
(g) y = cos |x|
2. Solve the following equations for 0 ≤ x ≤ 2π.
(a) 3 cos 2 x − cos x = 0
(b) 2 sin 2 x + sin x − 1 = 0
(c) 4 cos 3 x − 4 cos 2 x − 3 cos x + 3 = 0
(d) tan 2 x + 2 tan x + 1 = 0
Section 3 Trigonometric Identities
This section states and proves some common trig identities.
Pythagorean Identities
1○ cos 2 θ + sin 2 θ = 1
2○ 1 + tan 2 θ = sec 2 θ
3○ 1 + cot 2 θ = csc 2 θ Proof of 1○: Consider a circle of radius 1 centred at the origin.
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1
y
θ
x
. (x,
y)
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• Let θ be the angle measured anticlockwise
for the positive x-axis.
• Using trig ratios we see that x = cos θ and
y = sin θ.
• By Pythagoras’ Theorem x 2 + y 2 = 1 i.e.
cos 2 θ + sin 2 θ = 1.
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