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7 Trigonometric Identities and Equations Copyright © 2009 Pearson Addison-Wesley 7.1-1 Trigonometric Identities and 7 Equations 7.1 Fundamental Identities 7.2 Verifying Trigonometric Identities 7.3 Sum and Difference Identities 7.4 Double-Angle and Half-Angle Identities 7.5 Inverse Circular Functions 7.6 Trigonometric Equations 7.7 Equations Involving Inverse Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 7.1-2 7.1 Fundamental Identities Fundamental Identities ▪ Using the Fundamental Identities Copyright © 2009 Pearson Addison-Wesley 1.1-3 7.1-3 Fundamental Identities Reciprocal Identities Quotient Identities Copyright © 2009 Pearson Addison-Wesley 1.1-4 7.1-4 Fundamental Identities Pythagorean Identities Negative-Angle Identities Copyright © 2009 Pearson Addison-Wesley 1.1-5 7.1-5 Note In trigonometric identities, θ can be an angle in degrees, an angle in radians, a real number, or a variable. Copyright © 2009 Pearson Addison-Wesley 1.1-6 7.1-6 FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Example 1 If and θ is in quadrant II, find each function value. (a) sec θ Pythagorean identity In quadrant II, sec θ is negative, so Copyright © 2009 Pearson Addison-Wesley 1.1-7 7.1-7 Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) (b) sin θ Quotient identity Reciprocal identity from part (a) Copyright © 2009 Pearson Addison-Wesley 1.1-8 7.1-8 Example 1 FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) (c) cot(– θ) Reciprocal identity Negative-angle identity Copyright © 2009 Pearson Addison-Wesley 1.1-9 7.1-9 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 © 2009 Pearson Addison-Wesley 1.1-10 7.1-10 Example 2 EXPRESSING ONE FUNCITON IN TERMS OF ANOTHER Express 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. ± Copyright © 2009 Pearson Addison-Wesley +±±±±± ± = + 1.1-11 The sign depends on the quadrant of x. 7.1-11 Example 3 REWRITING AN EXPRESSION IN TERMS OF SINE AND COSINE Write tan θ + cot θ in terms of sin θ and cos θ, and then simplify the expression. Quotient identities Write each fraction with the LCD. Pythagorean identity Copyright © 2009 Pearson Addison-Wesley 1.1-12 7.1-12 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. Copyright © 2009 Pearson Addison-Wesley 1.1-13 7.1-13