3.12

Topic

Equivalent Representations of Trigonometric Functions

Practice by unit: Trigonometric and Polar Functions

Sub-standards

• 3.12.A.1 — Rewrite trigonometric expressions in equivalent forms with the Pythagorean identity. The Pythagorean Theorem can be applied to right triangles with points on the unit circle at coordinates (cos , sin ), resulting in the Pythagorean identity: 2 2 sin + cos = 1.
• 3.12.A.2 — The Pythagorean identity can be algebraically manipulated into other forms involving trigonometric functions, such as 2 2 tan = sec − 1, and can be used to establish other trigonometric relationships, such as 2arcsinx = arccos 1 ( − x ), with appropriate domain restrictions.
• 3.12.B.1 — Rewrite trigonometric expressions in equivalent forms with sine and cosine sum identities. The sum identity for sine is sin( + =) sin cos + cos sin .
• 3.12.B.2 — The sum identity for cosine is cos( + =) cos cos − sin sin .
• 3.12.B.3 — The sum identities for sine and cosine can also be used as difference and double-angle identities.
• 3.12.B.4 — Properties of trigonometric functions, known trigonometric identities, and other algebraic properties can be used to verify additional trigonometric identities. continued on next page AP Precalculus Course and Exam Description Course Framework V.1 | 98 Return to Table of Contents© 2023 College Board UNIT 3
• 3.12.C.1 — Solve equations using equivalent analytic representations of trigonometric functions. A specific equivalent form involving trigonometric expressions can make information more accessible.
• 3.12.C.2 — Equivalent trigonometric forms may be useful in solving trigonometric equations and inequalities. UNIT 3 Required Course Content r 2 2 y r = x + y and = arctan x for x > 0 or = arctan + for x < 0. x ( ) y( )