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Chapter 13: Trigonometric Identities and Equations 13.1: Trigonometric Identities Trigonometric identity: an equation involving trigonometric functions that is true for all values for which every expression in the equation is defined Examples: 1. Find the exact value of each expression if 0o < π < 90o Using the Trigonometric Identities. a. if cot π = 2, find tan π 2 c. if cos π = 3 find sin π 4 b. if sin π = 5 , find cos π 2 d. if cos π = 3 find csc π 2. Find the exact value of each expression if 270o < π < 360o Using the Trigonometric Identities. 1 a. If cos π = 2, find sin π 3 b. If cot π = β 2, find cos π. 2. Simplify each expression. a. tan π cos2 π c. cos π csc π tan π b. csc 2 π β cot 2 π cot2 π d. 1 + 1βππ π 2 π 3. When unpolarized light passes through polarized sunglass lenses, the intensity of the light is cut in half. If the light passes through another polarized lens with its axis at an angle of π to the first, the intensity of the πΌ light is again diminished. The intensity of the emerging light can be found by using the formula πΌ = πΌπ β cscπ2 π, where πΌπ is the intensity of the light incoming to the second polarized lens, I is the intensity of the emerging light, and π is the angle between the axes of polarization. a. Simplify the formula in terms of cos π. b. Use the simplified formula to determine the intensity of light that passes through a second polarizing lens with axis at 30o to the original. 13.2: Verifying Trigonometric Identities Verifying Identities by Transforming One Side: 1. Simplify one side of an equation until the two sides of the equation are the same. It is often easier to work with the more complicated side of the equation. 2. Transform that expression into the form of the simpler side. Suggestions for Verifying Identities: ο· Substitute one or more basic trigonometric identities to simplify the expression. ο· Factor or multiply as necessary. You may have to multiply both the numerator and denominator by the same trigonometric expression. ο· Write each side of the identity in terms of sine and cosine only. Then simplify each side as much as possible. ο· The properties of equality do not apply to identities as with equations. Do not perform operations to the quantities on each side of an unverified identity. Examples: 1. Verify that each equation is an identity. a. c. sec2 π tan π = cot π + tan π sec π = sin π tan π+cot π e. tan2 π csc 2 π = 1 + tan2 π b. (1 + sin π)(1 β sin π) = cos2 π d. f. sin2 (π₯) tan2 π₯ 1βcos π 1+cos π = cos 2 π₯ = (csc π β cot π)2 13.3: Sum and Differences of Angles Identities Sum Identities: ο· ο· sin(A + B) = sin A cos B + cos A sin B cos(A + B) = cos A cos B β sin A sin B ο· tan (A + B) = tan π΄+tan π΅ 1βtan π΄ tan π΅ Difference Identities: ο· ο· sin(A β B) = sin A cos B β cos A sin B cos(A β B) = cos A cos B + sin A sin B ο· tan (A β B) = tan π΄ β tan π΅ 1+tan π΄ tan π΅ Examples: 1. Find the exact value of each expression. Using Sum and Difference of Angles Identities. 5π a. cos 165o b. cos 105o c. cos 12 d. sin (β15o) e. sin 135o f. sin (β210o) 5π g. tan 75 ° h. csc 12 2. Verify that each equation is an identity. a. sin (90 + π) = cos π 3π b. cos ( 2 β π) = β sin π π c. tan (π + ) = β cot π 2 13.4: Double-Angle and Half-Angle Identities Double-Angle Identities: the following hold true for all values of π Tan2π= cos 2π = cos2 π β sin2 π cos 2π = 2cos2 π β 1 cos 2π = 1 β 2sin2 π sin 2π = 2sin π cos π 2 tan π 1βπ‘ππ2 π Half-Angle Identities: the following identities hold true for all values of π π 1βcos π 2 2 sin = ±β π 1+cos π 2 2 cos = ±β π 1βcos π 2 1+cos π tan = ±β Examples: π π 1. Find the exact values of sin 2π, cos 2π, sin 2, and cos 2 . 5 π a. cos π = β 13 ; 2 < π < π c. sin π = 1 4 ; 0° < π < 90° b. sin π = d. cos π = 4 5 3 5 ; 90° < π < 180° ; 270° < π < 360° , cos π β β 1 8 e. tan π = β 15 ; 90° < π < 180° 2. Find the exact value of each expression. π a. sin 8 b. tan 15° 3. Verify that each equation is an identity. a. tan π = 1βcos 2π sin 2π b. (sin π + cos π)2 = 1 + 2 sin π cos π 13.5: Solving Trigonometric Equations trigonometric equations: trig identities are equations that are true for certain values of the variable Examples: 1. Solve each equation if 0o < π < 360o. a. 2sin π + 1 = 0 b. cos π = β3 2 c. sin 2π = β β3 2 2. Solve each equation for all values of π if π is measured in radians. a. 4sin2 π β 1 = 0 π π b. sin 2 + cos 2 = β2 c. cos 2π + 4cos π = β3 3. Solve each equation for all values of π if π is measured in degrees. a. cos 2π β sin2π + 2 = 0 b. cos π β 2cos πsin π = 0 4. Solve each equation for all values of π if π is measured in radians. a. sin2 2π + cos2 π = 0 b. cos 8π = 1 c. sin π tan π β tan π = 0 c. 2cos2 π = cos π