5-1 Study Guide and Intervention - MRS. FRUGE

1 NAME _____ DATE _____ PERIOD _____ Chapter 5 5 Glencoe Precalculus 5-1 Study Guide and Intervention Trigonometric Identities Basic Trigonometric Identities An equation is an identity if the left side is equal to the right side for all values of the variable for which both sides are defined. Trigonometric identities are identities that involve trigonometric functions. Reciprocal Identities Pythagorean Identities sin = 1csc csc = 1sin cos = 1sec sec = 1cos tan = 1cot cot = 1tan sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2 Example: If sin = and 0 < < 90 , find tan.

2 Use two identities to relate sin and tan . sin2 + cos2 = 1 Pythagorean Identity (35)2 + cos2 = 1 sin = 35 cos2 = 1625 Simplify. cos = 1625 or 45 Take the square root of each side. Since 0 < < 90 , cos is positive. Thus, cos = 45. Now find tan . tan = sin cos Quotient identity tan = 3545 sin = 35, cos = 45 tan = 34 Simplify. Exercises Find the value of each expression using the given information. 1. If cot = 125, find tan . 2. If sin = 14, find csc . 3. If tan = 23, find cot . 4. If sec = 2, find csc ( 2).

3 5. If cot = 43 and sin < 0, find cos and csc . 6. If sec = 4 and csc > 0, find cos and tan . 4 2 cos = , and csc = cos = , and tan = NAME _____ DATE _____ PERIOD _____ Chapter 5 6 Glencoe Precalculus 5-1 Study Guide and Intervention (continued) Trigonometric Identities Simplify and Rewrite Trigonometric Expressions You can apply trigonometric identities and algebraic techniques such as substitution, factoring, and simplifying fractions to simplify and rewrite trigonometric expressions.

4 Example: Simplify each expression. a. sec x cos x sec x cos x = 1cos cos x Reciprocal Identity = 1 cos2 cos Add. = sin2 cos Pythagorean Identity = sin x (sin cos ) Factor. = sin x tan x Quotient Identity b. csc x x + csc x cot2 x + 1sin = csc x cot2 x + csc x Reciprocal Identity = csc x (csc2 x 1) + csc x Pythagorean Identity = csc3 x csc x + csc x Distributive Property = csc3 x Simplify. Exercises Simplify each expression. 1. cos x (tan x + cot x) 2. sin x + cos x cot x 3.

5 Csc2 1 + tan2 4. (sec x tan x)(csc x + 1) 5. (cot2 x + 1)(sec2 x 1) 6. 1 + tan2 1 + sec 7. csc x sin x + cot2 x 8. cos x (1 + tan2 x ) 9. cos ( 2 )csc 10. cos ( 2 )csc + cos2 x csc x csc x x cot x x sec x x sec x x 1 NAME _____ DATE _____ PERIOD _____ Chapter 5 7 Glencoe Precalculus 5-1 Practice Trigonometric Identities Find the value of each expression using the given information. 1. If cos = 14 and 0 < < 90 , find tan . 2.

6 If sin = 23 and 0 < < 90 , find cos . 3. If tan = 72 and 0 < < 90 , find sin . 4. If tan = 2 and 0 < < 90 , find cot . 5. If sin = 45 and cos > 0, find tan and sec . 6. If cot x = 32 and sec x < 0, find sin x and cos x. 7. If cos = , find sin ( 2). 8. If cot x = , find tan ( 2). Simplify each expression. 9. cos x + sin x tan x 10. cot tan 11. sin2 cos2 cos2 12. csc2 cot2 sin ( ) cot 13. KITE FLYING Brett and Tara are flying a kite. When the string is tied to the ground, the height of the kite can be determined by the formula = csc , where L is the length of the string and is the angle between the string and the level ground.

7 What formula could Brett and Tara use to find the height of the kite if they know the value of sin ? tan = , sec = sin x = , cos x = sec x A sec x H = L sin NAME _____ DATE _____ PERIOD _____ Chapter 5 10 Glencoe Precalculus 5-2 Study Guide and Intervention Verifying Trigonometric Identities Verify Trigonometric Identities To verify an identity means to prove that both sides of the equation are equal for all values of the variable for which both sides are defined.

8 Example: Verify that = x. The left-hand side of this identity is more complicated, so start with that expression first. sec2 1sec2 = (tan2 + 1) 1sec2 Pythagorean Identity = tan2 sec2 Simplify. = (sin2 cos2 )1cos2 Quotient Identity and Reciprocal Identity = sin2 cos2 cos2 x Simplify. = sin2 x Multiply. Notice that the verification ends with the expression on the other side of the identity. Exercises Verify each identity. 1. sec cos = sin tan 2. sec = sin (tan + cot ) 3.

9 Tan csc cos = 1 4. csc2 cot2 1 sin2 = sec2 sec cos = cos = = = sin ( ) = sin tan sin (tan + cot ) = sin ( + ) = sin ( + ) = sin ( )= = sec tan csc cos = ( )( ) cos = 1 = ( + ) = = NAME _____ DATE _____ PERIOD _____ Chapter 5 11 Glencoe Precalculus 5-2 Study Guide and Intervention (continued) Verifying Trigonometric Identities Identifying Identities and Nonidentities You can use a graphing calculator to test whether an equation might be an identity by graphing the functions related to each side of the equation.

10 If the graphs of the related functions do not coincide for all values of x for which both functions are defined, the equation is not an identity. If the graphs appear to coincide, you can verify that the equation is an identity by using trigonometric properties and algebraic techniques. Example: Use a graphing calculator to test whether csc sin = cot cos is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. The equation appears to be an identity because the graphs of the related functions coincide.