Math 180 Chapter 6 Lecture Notes. Professor Miguel Ornelas

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1 Math 180 Chapter 6 Lecture Notes Professor Miguel Ornelas 1

2 M. Ornelas Math 180 Lecture Notes Section 6.1 Section 6.1 Verifying Trigonometric Identities Verify the identity. a. sin x + cos x cot x = csc x b. (tan u + cot u)(cos u + sin u) = csc u + sec u c. 1 + csc 3β sec 3β cot 3β = cos 3β d. tan x sec x + 1 = 1 cos x cos x Section 6.1 continued on next page...

3 M. Ornelas Math 180 Lecture Notes Section 6.1 (continued) e. cot θ tan θ sin θ + cos θ = csc θ sec θ f. 1 = sin β cos β tan β + cot β Section 6. Trigonometric Equations Find all solutions of the equation. a. sin x = b. cos θ + 1 = 0 c. cos 1 4 x = d. ( sin θ + 1)( cos θ + 3) = 0 Section 6. continued on next page... 3

4 M. Ornelas Math 180 Lecture Notes Section 6. (continued) Find all solutions of the equation that are in the interval [0, π). ( a. cos x π ) = 0 4 b. cos t + 3 cos t = 1 c. cos γ + cos γ = 0 d. tan t sec t = 0 4

5 M. Ornelas Math 180 Lecture Notes Section 6.3 Section 6.3 The Addition and Subtraction Formulas Cofunction Formulas ( π ) 1. cos u = sin u ( π ). sin u = cos u ( π ) 3. tan u = cot u ( π ) 4. cot u = tan u ( π ) 5. sec u = csc u ( π ) 6. csc u = sec u Addition and Subtraction Formulas for Cosine, Sine and Tangent 1. cos(u v) = cos u cos v + sin u sin v. cos(u + v) = cos u cos v sin u sin v 3. sin(u + v) = sin u cos v + cos u sin v 4. sin(u v) = sin u cos v cos u sin v 5. tan(u + v) = tan u + tan v 1 tan u tan v 6. tan(u v) = tan u tan v 1 + tan u tan v Express as a cofunction of a complementary angle. a. cos π 3 b. tan 1 Section 6.3 continued on next page... 5

6 M. Ornelas Math 180 Lecture Notes Section 6.3 (continued) Find the exact values. a. cos π 4 + cos π 6 b. tan 60 + tan 5 Express as a trigonometric function of one angle. a. cos 61 sin 8 sin 61 cos 8 b. sin( 5) cos + cos 5 sin( ) If sin α = 5 (α 13 and tan α > 0, find the exact value of sin π ) 3 Section 6.3 continued on next page... 6

7 M. Ornelas Math 180 Lecture Notes Section 6.3 (continued) If α and β are acute angles such that csc α = 13 1 and cot β = 4 3, find a. sin(α + β) b. tan(α + β) c. the quadrant containing α + β Verify the identity. ( a. tan u + π ) = 1 + tan u 4 1 tan u b. sin(u + v) sin(u v) = sin u sin v 7

8 M. Ornelas Math 180 Lecture Notes Section 6.4 Section 6.4 Multiple-Angle Formulas Double-Angle Formulas 1. sin u = sin u cos u. cos u = cos u sin u 3. cos u = 1 sin u 4. cos u = cos u 1 5. tan u = tan u 1 tan u If sin α = 4 and α is an acute angle, find the exact values of sin α and cos α 5 Express cos 3θ in terms of cos θ Section 6.4 continued on next page... 8

9 M. Ornelas Math 180 Lecture Notes Section 6.4 (continued) Half-Angle Identities 1. sin u = 1 cos u. cos u = 1 + cos u 3. tan u = 1 cos u 1 + cos u Verify the identity sin x cos x = 1 (1 cos 4x). 8 Express cos 4 t in terms of values of the cosine function with exponent 1. Section 6.4 continued on next page... 9

10 M. Ornelas Math 180 Lecture Notes Section 6.4 (continued) Half-Angle Formulas 1. sin v = ± 1 cos v. cos v = ± 1 + cos v 3. tan v = ± 1 cos v 1 + cos v Find exact values for a. sin.5 b. cos 11.5 Half-Angle Formulas for the Tangent 1. tan v = 1 cos v sin v. tan v = sin v 1 + cos v If tan α = 4 3 and α is in quadrant IV, find tan α. 10

11 M. Ornelas Math 180 Lecture Notes Section 6.5 Section 6.5 Product-to-Sum and Sum-to-Product Formulas Product-to-Sum Formulas 1. sin u cos v = 1 [sin(u + v) + sin(u v)]. cos u sin v = 1 [sin(u + v) sin(u v)] 3. cos u cos v = 1 [cos(u + v) + cos(u v)] 4. sin u sin v = 1 [cos(u v) cos(u + v)] Express as a sum or difference. a. sin 7t sin t b. 3 cos x sin x Sum-to-Product Formulas 1. sin a + sin b = sin a + b cos a b. sin a sin b = cos a + b sin a b 3. cos a + cos b = cos a + b cos a b 4. cos a cos b = sin a + b sin a b Section 6.5 continued on next page... 11

12 M. Ornelas Math 180 Lecture Notes Section 6.5 (continued) Express as a product. a. sin θ sin 8θ b. cos x + cos x Verify the identity. a. sin 4t + sin 6t cos 4t cos 6t = cot t b. sin u sin v sin u + sin v = tan 1 (u v) tan 1 (u + v) Use sum-to-product formulas to find the solutions of the equation. a. sin 5t + sin 3t = 0 b. cos 3x = cos 6x 1

13 M. Ornelas Math 180 Lecture Notes Section 6.6 Section 6.6 The Inverse Trigonometric Functions Definition of the Inverse Sine Function The inverse sine function, denoted by sin 1, is defined by y = sin 1 x if and only if x = sin y for 1 x 1 and π y π. Definition of the Inverse Cosine Function The inverse cosine function, denoted by cos 1, is defined by y = cos 1 x if and only if x = cos y for 1 x 1 and 0 y π. Definition of the Inverse Tangent Function The inverse tangent function, denoted by tan 1, is defined by y = tan 1 x if and only if x = tan y for any real number x and for π < y < π. Find the exact value of the expression whenever it is defined. a. sin 1 b. cos 1 π c. tan 1 ( 3 ) d. arccos π 3 Section 6.6 continued on next page... 13

14 M. Ornelas Math 180 Lecture Notes Section 6.6 (continued) [ ( e. sin arcsin 3 )] 10 f. cos 1 [cos ( )] 5π 6 ( g. arctan tan 7π ) 4 h. tan ( cos 1 0 ) [ ( i. cos arctan 3 ) arcsin 4 ] 4 5 ( j. tan tan 1 3 ) 4 Section 6.6 continued on next page... 14

15 M. Ornelas Math 180 Lecture Notes Section 6.6 (continued) Write the expression as an algebraic expression in x for x > 0. a. sin ( tan 1 x ) ( b. csc tan 1 x ) Use inverse trigonometric functions to find the solutions of the equation cos x + cos x 1 = 0 that are in the interval [0, π). 15