Periodic Functions Iii

Week 1

MAB112

GR

Periodic Functions III

Introduction In this section of trigonometry, you will use trigonometrical ratios and identities: The ratios include: Ratio

Abbr.

Fraction

Sine

sin

sin θ =

Cosine

cos

cos θ =

Tangent

tan

tan θ =

Cosecant

csc

csc θ =

Secant

sec

sec θ =

Cotangent

cot

cot θ =

opp adj opp hyp hyp adj

Other forms hyp

n/a

hyp

n/a

adj

n/a

opp

csc θ = 1 sin θ

adj

sec θ = 1 cos θ

opp

cot θ = 1 tan θ

θ

θ

The main identity is that sin2 𝑥 + cos2 𝑥 = 1. The above identity is proven using the unit circle. The co-ordinates of a point P on the circumference of the circle is given by (cos θ, sin θ) where θ is the angle made by the radius to point P and the positive direction of the x-axis. The unit circle is defined as having a radius of one, so hyp = 1. For P, x=adj and y=opp as seen. We can say that: sin 𝜃 = cos 𝜃 =

𝑜𝑝𝑝 𝑎𝑑𝑗

𝑦

ℎ𝑦𝑝 = 1 = 𝑦 𝑥

ℎ𝑦𝑝 = 1 = 𝑥

θ

Pythagoras theorem: 𝑎𝑑𝑗 2 + 𝑜𝑝𝑝2 = ℎ𝑦𝑝2 or 𝑥 2 + 𝑦 2 = 12 sin2 𝜃 + cos2 𝜃 = 12 = 1

Questions Q1. Given that sin2 𝑥 + cos 2 𝑥 = 1, show that: sec 2 𝑥 = 1 + tan2 𝑥 Q2. Simplify the following trigonometric expressions: tan 𝜃+1

(i) cos 𝜃+sin 𝜃−cos 3 𝜃−sin 3 𝜃

csc 𝑥+cot 2 𝑥 csc 𝑥

(ii) tan 𝑥−sin 𝑥 cos 𝑥+cot 𝑥−sin 3 𝑥 sec 𝑥