Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
5 Trigonometric Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 5 Trigonometric Identities 5.1 Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Sum and Difference Identities for Cosine 5.4 Sum and Difference Identities for Sine and Tangent 5.5 Double-Angle Identities 5.6 Half-Angle Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2 5.1 Fundamental Identities Fundamental Identities ▪ Using the Fundamental Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Fundamental Identities Reciprocal Identities Quotient Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Fundamental Identities Pythagorean Identities Negative-Angle Identities Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Note In trigonometric identities, θ can be an angle in degrees, a real number, or a variable. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Example 1 If FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT and θ is in quadrant II, find each function value. Pythagorean identity (a) sec θ In quadrant II, sec θ is negative, so Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) (b) sin θ Quotient identity Reciprocal identity from part (a) Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) (b) cot(– θ) Reciprocal identity Negative-angle identity Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 Caution To avoid a common error, when taking the square root, be sure to choose the sign based on the quadrant of θ and the function being evaluated. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Example 2 EXPRESSING ONE FUNCITON IN TERMS OF ANOTHER Write cos x in terms of tan x. Since sec x is related to both cos x and tan x by identities, start with Take reciprocals. Reciprocal identity Take the square root of each side. 1 tan2 x cos x 2 1 tan x The sign depends on the quadrant of x. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 Example 3 REWRITING AN EXPRESSION IN TERMS OF SINE AND COSINE 1 cot 2 Write in terms of sin θ and cos θ, and 2 1 csc then simplify the expression so that no quotients appear. cos2 1 cot 2 1 sin2 2 1 csc 1 1 sin2 cos2 2 1 sin 2 sin 1 2 1 sin 2 sin Quotient identities Multiply numerator and denominator by the LCD. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 REWRITING AN EXPRESSION IN TERMS OF SINE AND COSINE (cont’d) Example 3 sin2 cos2 2 sin 1 Distributive property 1 2 cos Pythagorean identities sec 2 Reciprocal identity Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 Caution When working with trigonometric expressions and identities, be sure to write the argument of the function. For example, we would not write An argument such as θ is necessary in this identity. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14