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# Trig made Easy - Copy - THE UNIVERSITY OF AKRON Department...

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THE UNIVERSITY OF AKRON Department of Theoretical and Applied Mathematics LESSON 6: TRIGONOMETRIC IDENTITIES by Thomas E. Price Directory Table of Contents Begin Lesson Copyright c 1999-2001 [email protected] Last Revision Date: August 17, 2001

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Table of Contents 1. Introduction 2. The Elementary Identities 3. The sum and difference formulas 4. The double and half angle formulas 5. Product Identities and Factor formulas 6. Exercises Solutions to Exercises
1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as a 2 - b 2 = ( a - b )( a + b ) or a 3 + b 3 = ( a + b )( a 2 - ab + b 2 ) . Identities such as these are used to simplifly algebriac expressions and to solve alge- briac equations. For example, using the third identity above, the expression a 3 + b 3 a + b simpliflies to a 2 - ab + b 2 . The first identiy verifies that the equation ( a 2 - b 2 ) = 0 is true precisely when a = ± b. The formulas or trigonometric identities introduced in this lesson constitute an integral part of the study and applications of trigonometry. Such identities can be used to simplifly complicated trigonometric expressions. This lesson contains several examples and exercises to demonstrate this type of procedure. Trigonometric identities can also used solve trigonometric equations. Equations of this type are introduced in this lesson and examined in more detail in Lesson 7 . For student’s convenience, the identities presented in this lesson are sumarized in Appendix A

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2. The Elementary Identities Let ( x, y ) be the point on the unit circle centered at (0 , 0) that determines the angle t rad . Recall that the definitions of the trigonometric functions for this angle are sin t = y tan t = y x sec t = 1 y cos t = x cot t = x y csc t = 1 x . These definitions readily establish the first of the elementary or fundamental identities given in the table below. For obvious reasons these are often referred to as the reciprocal and quotient identities. These and other identities presented in this section were introduced in Lesson 2 Sections 2 and 3 . sin t = 1 csc t cos t = 1 sec t tan t = 1 cot t = sin t cos t csc t = 1 sin t sec t = 1 cos t cot t = 1 tan t = cos t sin t . Table 6.1: Reciprocal and Quotient Identities. 4
Section 2: The Elementary Identities 5 Example 1 Use the reciprocal and quotient formulas to verify sec t cot t = csc t. Solution: Since sec t = 1 cos t and cot t = cos t sin t we have sec t cot t = 1 cos t cos t sin t = 1 sin t = csc t. Example 2 Use the reciprocal and quotient formulas to verify sin t cot t = cos t. Solution: We have sin t cot t = sin t cos t sin t = cos t.

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Section 2: The Elementary Identities 6 x y ) , ( y x t - t ) , ( y x - Several fundamental identities follow from the sym- metry of the unit circle centered at (0 , 0). As indicated in the figure, if ( x, y ) is the point on this circle that determines the angle t rad , then ( x, - y ) is the point that determines the angle ( - t ) rad . This suggests that sin( - t ) = - y = - sin t and cos( - t ) = x = cos t . Such functions are called odd and even respectively 1 . Sim-
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Trig made Easy - Copy - THE UNIVERSITY OF AKRON Department...

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