Trigonometry III Fundamental Trigonometric Identities. By Mr Porter

Summary of Definitions Reciprocal Relationships Hypotenuse Adjacent Opposite θ α Complementary RelationshipsNegative 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 1 st identity sin 2 θ = 1 – cos 2 θ, and cos 2 θ = 1 – sin 2 θ b) Write down the identities Sometimes, we need to take small steps! Use the 3 rd 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 2 nd 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