Trigonometry Cloud County Community College Spring, 2012 Instructor: Timothy L. Warkentin

Chapter 3: Trigonometric Identities and Equations 3.1 Verification of Trigonometric Identities 3.2 Sum, Difference, and Cofunction Identities 3.3 Double- and Half-Angle Identities 3.4 Identities Involving the Sum of Trigonometric Functions 3.5 Inverse Trigonometric Functions 3.6 Trigonometric Equations

Chapter 3 Overview The concentration in Chapter 3 is on trigonometric identities which ultimately are used to solve trigonometric equations. It is worth noting again that the emphasis in trigonometry is on the properties of the functions and not on the computation of their values.

3.1: Verification of Trigonometric Identities 1 An Identity is tautologically true for all values in its domain. The Fundamental Trigonometric Identities:

3.1: Verification of Trigonometric Identities 2 Strategies for verifying trigonometric identities ( no operations that assume the identity can be used): Examples 1-5 –Transform one side into the other or transform both sides independently into identical expressions. –Distribute or expand factors. –Factor across terms. –Obtain a common denominator. –Use the fundamental identities. –Rewrite both sides in terms of only sines and cosines. –Multiply both numerator and denominator by conjugates.

3.2: Sum, Difference, and Cofunction Identities 1 Sum and Difference Identities: Examples 1-6 Cofunction Identities: Cofunctions of complimentary angles are equal.

3.3: Double- and Half-Angle Identities 1 Double-Angle Identities. Examples 1-3 Power-Reducing Identities (very useful in Calculus). Example 4

3.3: Double- and Half-Angle Identities 2 Half-Angle Identities. Examples 5-7

3.4: Identities Involving the Sum of Trigonometric Functions 1 This section is not covered.

3.5: Inverse Trigonometric Functions 1 Trigonometric functions can be thought of as accepting angles and returning ratios. Inverse trigonometric functions would therefore accept ratios and return angles. Inverse notation: Using Domain Surgery (include the origin and preserve the range) to create 1-to-1 trigonometric functions. Sine, Cosine and Tangent – Example 1 Inverse identities for cotangent, secant and cosecant:

3.5: Inverse Trigonometric Functions 2 Composition of trigonometric functions and their inverses. Examples 2-5 Graphing inverse trigonometric functions (TI calculator). Example 7 Inverse trigonometric function applications. Example 8

3.6: Trigonometric Equations 1 Strategies for solving trig equations: Examples 1-6 –Graph each side of the equation to locate intersection points (approximate these to check against any exact solutions found). –Use the Zero Product Property (ZPP). –Square each side of the equation (this can introduce extraneous solutions). –Use the Quadratic Formula with a trigonometric function as the independent value sought. –Use the fundamental identities. –Check the solution domain for multiple solutions (there may be an infinite number of solutions). –Dividing both sides of the equation by factors that contain the variable may cause the loss of solutions. –Check any solutions against physical constraints on the domain. Example 7 Trigonometric regression with TI calculators. Example 8