Trigonometric Identities and Equations Unit Objectives: Verify trigonometric identities are true. Solve trigonometric equations Application problems with trigonometric functions and ratios. Today’s Objective: I can verify trigonometric identities.

An equation that is true for all values of x or 𝜃. Identity: Example: An equation that is true for all values of x or 𝜃. 𝑥 5 𝑥 3 = 𝑥 2 where 𝑥≠0 Domain of validity Basic Identities: Tangent Identities: Reciprocal Identities: sin 𝜃 cos 𝜃 tan 𝜃 = cot 𝜃 = cos 𝜃 sin 𝜃 1 1 1 sec 𝜃 = cot 𝜃 = csc 𝜃 = cos 𝜃 tan 𝜃 sin 𝜃 1 1 1 cos 𝜃 = tan 𝜃 = sin 𝜃 = sec 𝜃 cot 𝜃 csc 𝜃

Domain of validity: Domain of validity:

State the domain of validity and verify the identity Pick a side Change to sin or cos Simplify to other side 𝑥≠± 𝜋 2 ,± 3𝜋 2 ,… ( sin 𝑥) ( sec 𝑥) = tan 𝑥 =( sin 𝑥) 1 cos 𝑥 = sin 𝑥 cos 𝑥 ( sin 𝑥) ( sec 𝑥) = tan 𝑥 ( sin 𝑥) cot 𝑥 = cos 𝑥 𝑥≠0,±𝜋,±2𝜋,… =( sin 𝑥) cos 𝑥 sin 𝑥 ( sin 𝑥) cot 𝑥 = cos 𝑥 1 sin 𝑥 csc 𝑥 sec 𝑥 = cot 𝑥 csc 𝑥 sec 𝑥 = 1 sin 𝑥 ⋅ cos 𝑥 1 = cos 𝑥 sin 𝑥 = cot 𝑥 = 1 cos 𝑥 𝑥≠0,± 𝜋 2 ,±𝜋,…

Pythagorean Identities 1+ tan 2 𝜃= sec 2 𝜃 𝑥 2 + 𝑦 2 =1 =1+ sin 2 𝜃 cos 2 𝜃 1+ tan 2 𝜃 cos 2 𝜃 + sin 2 𝜃 =1 = cos 2 𝜃 cos 2 𝜃 + sin 2 𝜃 cos 2 𝜃 = = cos 2 𝜃 + sin 2 𝜃 cos 2 𝜃 1 y θ x = 1 cos 2 𝜃 = sec 2 𝜃 1+ cot 2 𝜃= csc 2 𝜃

Simplifying an Expression Verifying an identity tan 2 𝜃 − sin 2 𝜃 = tan 2 𝜃 sin 2 𝜃 = sin 2 𝜃 cos 2 𝜃 − sin 2 𝜃 = sin 2 𝜃 cos 2 𝜃 − sin 2 𝜃 cos 2 𝜃 cos 2 𝜃 tan 2 𝜃 − sin 2 𝜃 = sin 2 𝜃 − sin 2 𝜃 cos 2 𝜃 cos 2 𝜃 = sin 2 𝜃 (1− cos 2 𝜃 ) cos 2 𝜃 = sin 2 𝜃 sin 2 𝜃 cos 2 𝜃 = tan 2 𝜃 sin 2 𝜃 Simplifying an Expression = 1 sin 𝜃 ∙ sin 𝜃 cos 𝜃 = 1 cos 𝜃 csc 𝜃 tan 𝜃 = sec 𝜃 Pg. 908 #9-60 by 3s

Trigonometric Identities Day 2 Today’s Objective: I can verify identities.

Trigonometric Identities csc 𝜃 = 1 sin 𝜃 𝐜𝐨𝐬 𝟐 𝜽 + 𝐬𝐢𝐧 𝟐 𝜽 =𝟏 sin 2 𝜃 =1− cos 2 𝜃 cos 2 𝜃 =1− sin 2 𝜃 sec 𝜃 = 1 cos 𝜃 𝐬𝐞𝐜 𝟐 𝜽 =𝟏 + 𝐭𝐚𝐧 𝟐 𝜽 tan 2 𝜃 = sec 2 𝜃 −1 tan 𝜃 = sin 𝜃 cos 𝜃 sec 2 𝜃 − tan 2 𝜃 =1 𝐜𝐬𝐜 𝟐 𝜽 =𝟏+ 𝐜𝐨𝐭 𝟐 𝜽 cot 𝜃 = cos 𝜃 sin 𝜃 cot 2 𝜃 = csc 2 𝜃 −1 csc 2 𝜃 − cot 2 𝜃 =1