Trigonometric Identities. Copyright 2017, 2013, 2009 Pearson Education, Inc.

5.3 Sum and Difference Identities Difference Identity for Cosine Sum Identity for Cosine Cofunction Identities Applications of the Sum and Difference Identities Verifying an Identity Copyright 2017, 2013, 2009 Pearson Education, Inc. 2

Difference Identity for Cosine Point Q is on the unit circle, so the coordinates of Q are (cos B, sin B). The coordinates of S are (cos A, sin A). The coordinates of R are (cos(a B), sin (A B)). Copyright 2017, 2013, 2009 Pearson Education, Inc. 3

Sum Identity for Cosine To find a similar expression for cos(a + B) rewrite A + B as A ( B) and use the identity for cos(a B). Cosine difference identity Negative-angle identities Copyright 2017, 2013, 2009 Pearson Education, Inc. 6

Example 1(a) FINDING EXACT COSINE FUNCTION VALUES Find the exact value of cos 15. ( ) cos15 = cos 45 30 o o o = cos45 cos30 + sin45 cos30 o o o o 2 3 2 1 = + 2 2 2 2 + = 6 2 2 Copyright 2017, 2013, 2009 Pearson Education, Inc. 8

Example 2 USING COFUNCTION IDENTITIES TO FIND θ Find one value of θ or x that satisfies each of the following. (a) cot θ = tan 25 (b) sin θ = cos ( 30 ) Copyright 2017, 2013, 2009 Pearson Education, Inc. 12

Example 2 USING COFUNCTION IDENTITIES TO FIND θ (continued) Find one value of θ or x that satisfies the following. 3π (c) csc = sec x 4 3π csc = sec x 4 3π π csc = csc x 4 2 3π π = 4 2 x π x = 4 Copyright 2017, 2013, 2009 Pearson Education, Inc. 13

Note Because trigonometric (circular) functions are periodic, the solutions in Example 2 are not unique. We give only one of infinitely many possibilities. Copyright 2017, 2013, 2009 Pearson Education, Inc. 14

Applying the Sum and Difference Identities If either angle A or B in the identities for cos(a + B) and cos(a B) is a quadrantal angle, then the identity allows us to write the expression in terms of a single function of A or B. Copyright 2017, 2013, 2009 Pearson Education, Inc. 15

Example 3 REDUCING cos (A B) TO A FUNCTION OF A SINGLE VARIABLE Write cos(180 θ) as a trigonometric function of θ alone. o o o cos(180 θ) = cos180 cosθ + sin180 sinθ = ( 1)cos θ + (0)sinθ Copyright 2017, 2013, 2009 Pearson Education, Inc. 16

Example 4 FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t Suppose that are in quadrant II. Find cos(s + t). Method 1 Sketch an angle s in quadrant II such that Since let y = 3 and r = 5. and both s and t The Pythagorean theorem gives Since s is in quadrant II, x = 4 and Copyright 2017, 2013, 2009 Pearson Education, Inc. 17

Example 4 FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.) Sketch an angle t in quadrant II such that Since 12 x cos t = =, let x = 12 and 13 r r = 13. The Pythagorean theorem gives Since t is in quadrant II, y = 5 and Copyright 2017, 2013, 2009 Pearson Education, Inc. 18

Example 4 FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.) Method 2 We use Pythagorean identities here. To find cos s, recall that sin 2 s + cos 2 s = 1, where s is in quadrant II. 2 3 2 + cos s = 1 5 9 2 + cos s = 1 25 2 16 cos s = 25 coss = 4 5 sin s = 3/5 Square. Subtract 9/25 cos s < 0 because s is in quadrant II. Copyright 2017, 2013, 2009 Pearson Education, Inc. 20

Example 4 FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.) To find sin t, we use sin 2 t + cos 2 t = 1, where t is in quadrant II. 2 2 12 sin t + = 1 13 2 144 sin t + = 1 Square. 169 2 25 sin t = 169 5 sint = 13 From this point, the problem is solved using cos t = 12/13 Subtract 144/169 sin t > 0 because t is in quadrant II. (see Method 1). Copyright 2017, 2013, 2009 Pearson Education, Inc. 21

Example 5 APPLYING THE COSINE DIFFERENCE IDENTITY TO VOLTAGE Common household electric current is called alternating current because the current alternates direction within the wires. The voltage V in a typical 115-volt outlet can be expressed by the function where is the angular speed (in radians per second) of the rotating generator at the electrical plant, and t is time measured in seconds. (Source: Bell, D., Fundamentals of Electric Circuits, Fourth Edition, Prentice-Hall, 1988.) Copyright 2017, 2013, 2009 Pearson Education, Inc. 22

Example 5 APPLYING THE COSINE DIFFERENCE IDENTITY TO VOLTAGE (continued) (a) It is essential for electric generators to rotate at precisely 60 cycles per sec so household appliances and computers will function properly. Determine for these electric generators. Each cycle is 2 radians at 60 cycles per sec, so the angular speed is = 60(2 ) = 120 radians per sec. Copyright 2017, 2013, 2009 Pearson Education, Inc. 23

Example 5 APPLYING THE COSINE DIFFERENCE IDENTITY TO VOLTAGE (continued) (c) Determine a value of so that the graph of is the same as the graph of Using the negative-angle identity for cosine and a cofunction identity gives Therefore, if Copyright 2013, 2009, 2005 Pearson Education, Inc. 25

Example 6 VERIFYING AN IDENTITY Verify that the following equation is an identity. 3π sec x = csc x 2 Work with the more complicated left side. 3π 1 sec x = 2 3π cos x 2 1 = 3π 3π cos cos x+ sin sin x 2 2 Copyright 2013, 2009, 2005 Pearson Education, Inc. 26

Example 6 VERIFYING AN IDENTITY (continued) 1 1 = 3π 3π cos cos x+ sin sin x 0gcos x+ ( 1)sin 2 2 1 = sin x x = csc x The left side is identical to the right side, so the given equation is an identity. Copyright 2013, 2009, 2005 Pearson Education, Inc. 27