Trigonometry IIIFundamental Trigonometric Identities.

By Mr Porter

Summary of Definitions

Reciprocal RelationshipsHypotenuse

Adjacent

Opposite

θ

α

Complementary Relationships Negative Angle

Pythagorean Identities of Trigonometry.

For any angle θ

θ

r

x

y

Now and

Using Pythagoras’ Theorem

Likewise,

Examples: Simplify the following

a) Write down the identities

Options:(1) replace the ‘1’ with a trig expression(2) Rearrange an identity and replace

In this case, rearrange the 1st identity sin2θ = 1 – cos2θ, and cos2θ = 1 – sin2θ

b) Write down the identities

Sometimes, we need to take small steps!

Use the 3rd identity to replace denominator

Now, replace cot and cosec with their sin and cos equivalents.

Fraction rearrange

Extension student would continue to the next step.

Examples: Simplify the following

c) Write down the identities

Use the 2nd identity, rearranged.

Use the reciprocal trig angles.

d)Write down the identities

No matches, FACTORISE!

Now use an identity (try number 1).

Use the complementary trig angles.

Exercise

a) Simplify

b) Simplify

c) Simplify

d) Simplify

Trigonometric Identity Proofs.a) Prove that

LHS Break into terms of sin and cos

Common denominator.

Expand numerator

Rearrange numerator

Write down the identities

Trigonometric Identity Proofs.

b) Prove

LHS Break into terms of sin and cos

Common denominator.

Write down the identities

Trigonometric Identity Proofs.

d) Prove

LHS Break into terms of sin and cos and rearrange

Factorise

Express brackets as a common denominator.

Use identity

Expand brackets

Use definitions

This was NOT an easy question!

Exercise

a) Prove

b) Prove

c) Prove

d) Prove