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NAME 13-1 DATE PERIOD Study Guide and Intervention Trigonometric Identities Find Trigonometric Values A trigonometric identity is an equation involving trigonometric functions that is true for all values for which every expression in the equation is defined. sin θ tan θ = − cos θ cot θ = − Reciprocal Identities 1 csc θ = − 1 sec θ = − Pythagorean Identities cos2 θ + sin2 θ = 1 tan2 θ + 1 = sec2 θ cos θ sin θ sin θ cos θ 1 cot θ = − tan θ cot2 θ+ 1 = csc2 θ 11 Find the exact value of cot θ if csc θ = - − and 180° < θ < 270°. Example 5 cot2 θ + 1 = csc2 θ 5) ( 11 cot 2 θ + 1 = - − Trigonometric identity 2 121 cot θ + 1 = − 25 96 2 cot θ = − 25 4 √ 6 cot θ = ± − 5 2 11 Substitute - − for csc θ. 5 11 Square - − . 5 Subtract 1 from each side. Take the square root of each side. 4 √6 5 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Since θ is in the third quadrant, cot θ is positive. Thus, cot θ = − . Exercises Find the exact value of each expression if 0° < θ < 90°. √3 2 1. If cot θ = 4, find tan θ. 2. If cos θ = − , find csc θ. 3 3. If sin θ = − , find cos θ. 1 4. If sin θ = − , find sec θ. 4 5. If tan θ = − , find cos θ. 3 6. If sin θ = − , find tan θ. 5 3 3 7 Find the exact value of each expression if 90° < θ < 180°. 7 7. If cos θ = - − , find sec θ. 8 12 8. If csc θ = − , find cot θ. 5 Find the exact value of each expression if 270° < θ < 360°. 6 9. If cos θ = − , find sin θ. 7 Chapter 13 9 10. If csc θ = - − , find sin θ. 4 5 Glencoe Algebra 2 Lesson 13-1 Basic Trigonometric Identities Quotient Identities NAME DATE 13-1 PERIOD Study Guide and Intervention (continued) Trigonometric Identities Simplify Expressions The simplified form of a trigonometric expression is written as a numerical value or in terms of a single trigonometric function, if possible. Any of the trigonometric identities can be used to simplify expressions containing trigonometric functions. Example 1 Simplify (1 - cos2 θ) sec θ cot θ + tan θ sec θ cos2 θ. 1 (1 - cos2 θ) sec θ cot θ + tan θ sec θ cos2 θ = sin 2 θ · − cos θ cos θ sin θ · 1 · − − cos 2 θ +− · cos θ sin θ cos θ = sin θ + sin θ = 2 sin θ Example 2 θ · cot θ csc θ − - − . Simplify sec 1 - sin θ 1 + sin θ cos θ 1 − ·− cos θ sin θ 1 − sec θ · cot θ csc θ sin θ − - − = − - − 1 - sin θ 1 + sin θ 1 + sin θ 1 + sin θ 1 1 − (1 + sin θ) - − (1 - sin θ) θ sin θ −− = sin (1 - sin θ)(1 + sin θ) 1 1 − +1-− +1 θ sin θ −− = sin 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1 - sin θ 2 = − or 2 sec2 θ cos 2 θ Exercises Simplify each expression. θ · csc θ − 1. tan sin θ · cot θ 2. − 2 2 2 θ - cot θ · tan θ −− 3. sin cos θ 4. − θ · cos θ − 5. tan + cot θ · sin θ · tan θ · csc θ 2 θ - cot 2 θ − 6. csc 7. 3 tan θ · cot θ + 4 sin θ · csc θ + 2 cos θ · sec θ 1 - cos θ 8. − sec θ sec θ - tan θ sec θ - tan θ cot θ · sin θ tan θ · cos θ sin θ Chapter 13 6 2 tan θ · sin θ Glencoe Algebra 2