Chapter 11: Trigonometric Identities and Equations. 11.1 Trigonometric Identities 11.2 Addition and Subtraction Formulas 11.3 Double-Angle, Half-Angle, and Product-Sum Formulas 11.4 Inverse Trigonometric Functions 11.5 Trigonometric Equations.
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11.1 Trigonometric Identities
11.2 Addition and Subtraction Formulas
11.3 Double-Angle, Half-Angle, and Product-Sum Formulas
11.4 Inverse Trigonometric Functions
11.5 Trigonometric Equations
E.g. cos 2A =cos(A + A)
= cosAcosA– sin A sin A
= cos² A – sin² A
Other forms for cos 2A are obtained by substituting either cos² A = 1 – sin² A or sin² A = 1 – cos² A to get
cos 2A = 1 – 2 sin² A or cos 2A = 2 cos² A – 1.
Example Given and sin < 0, find a) sin 2,
b) cos 2, and c) tan 2.
Solution To find sin 2, we must find sin .
Choose the negative square root since sin < 0.
Example Simplify each expression.
Example Rewrite cos2 sin as the sum or difference of two functions.
Solution By the identity for cosA sin A, with 2 = A
and = B,
Example Write sin 2– sin 4 as a product of two
Solution Use the identity for sin A– sin B, with 2 = A and 4 = B.
Choose the sign ± depending on the quadrant of the angle A/2.
Example Find the exact value of
Solution The half-angle terminates in quadrant II since