If we restrict the domain of y = sin x to [ π 2, π 2

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1 Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the domain of f 1. (2) Domain of f = range of f 1, and range of f = domain of f 1. (3) The graph of f and the graph of f 1 are symmetric with respect to the line y = x. (4) If a function y = f(x) has an inverse function, the equation of the inverse function is x = f(y). The solution of this equation is y = f 1 (x). 2. The inverse sine function If we restrict the domain of y = sin x to [ π 2, π 2 ], the restrict function y = sin x, π 2 x π 2 will have an inverse function. We call it the of x. We denote it by : y = sin 1 x means x = sin y, where 1 x 1 and π 2 y π 2 1

2 3. Graph y = sin 1 x. Domain of y = sin 1 x is and range is. 4. Example: Find the exact value of sin 1 ( 1 2 ). 5. Remark: sin 1 (sin x) = x, where π 2 x π 2 sin(sin 1 x) = x, where 1 x 1 6. Example: Find the exact value of (1) sin 1 [sin( π 8 )] (2) sin[sin 1 ( 3 2 )] 2

3 7. The inverse cosine function If we restrict the domain of y = cos x to [0, π], the restrict function y = cos x, 0 x π will have an inverse function. We call it the of x. We denote it by : y = cos 1 x means x = cos y, where 1 x 1 and 0 y π 8. Graph y = cos 1 x. Domain of y = cos 1 x is and range is. 3

4 9. Example: Find the exact value of cos 1 ( 2 2 ). 10. Remark: cos 1 (cos x) = x, where 0 x π cos(cos 1 x) = x, where 1 x Example: Find the exact values of (1) cos 1 [cos( π 12 )] (2) cos[cos 1 ( 0.4)] 12. The inverse tangent function 4

5 If we restrict the domain of y = tan x to ( π 2, π 2 ), the restrict function y = tan x, π 2 < x < π 2 will have an inverse function. We call it the of x. We denote it by : 13. Graph y = tan 1 x. y = tan 1 x means x = tan y, where < x < and π 2 < y < π 2 Domain of y = tan 1 x is is and range is. 14. Example: Find the exact value of tan 1 ( 3). 5

6 15. Remark: tan 1 (tan x) = x, where π 2 < x < π 2 tan(tan 1 x) = x, where < x < 16. Example: Find the exact values of (1) tan 1 [tan( π 8 )] (2) tan[tan 1 (6)] 6

7 3.2 The inverse trigonometric functions (Continued) 1. Example: Find the exact value of sin 1 (sin 5π 4 ). 2. Example: Find the exact value of sin(tan ). 7

8 3. Example: Find the exact value of cos[sin 1 ( 1 3 )]. 4. Example: Find the exact value of tan[cos 1 ( 1 3 )]. 8

9 5. The remaining inverse trigonometric functions: y = sec 1 x means x = sec y, where x 1 and 0 y π, y π 2. y = csc 1 x means x = csc y, where x 1 and π 2 y π 2, y 0. y = cot 1 x means x = cot y, where < x < and 0 < y < π. 6. Example: Find the exact value of csc

10 3.3 Trigonometric identities 1. Basis trigonometric identities: Quotient identies Reciprocal identities Pythagorean identities Even-odd identities 2. Example: Simplify the following expressions: (1) cot θ csc θ (2) cos θ 1+sin θ (3) 1+sin θ sin θ + cot θ cos θ cos θ 10

11 (4) sin 2 θ 1 tan θ sin θ tan θ 3. Example: Establish the following identities: (1) csc θ tan θ = sec θ (2)sin 2 ( θ) + cos 2 ( θ) = 1 (3) sin2 ( θ) cos 2 ( θ) sin( θ) cos( θ) = cos θ sin θ 11

12 (4) 1+tan θ 1+cot θ = tan θ sin θ (5) 1+cos θ + 1+cos θ sin θ = 2 csc θ (6) tan θ cot θ sec θ csc θ = 1 12

13 (7) 1 sin θ cos θ = cos θ 1+sin θ (8) sin(tan 1 ) = v 1+v 2. 13

14 3.4 Sum and difference formulas 1. Sum and difference formulas for cosines cos(α + β) = cos α cos β sin α sin β cos(α β) = cos α cos β + sin α sin β 2. Example: Find the exact values of cos 75 0 and cos π cos( π 2 θ) = sin θ sin( π 2 θ) = cos θ 14

15 4. Sum and difference formulas for cosines sin(α + β) = sin α cos β + cos α sin β sin(α β) = sin α cos β cos α sin β 5. Example: Find the exact values of sin 7π 12 and sin 800 cos 20 0 cos 80 0 sin Example: If it is known that sin α = 4 5, π 2 < α < π, and that sin β = 2 5 = 2 5 5, π < β < 3π 2, find the exact value of (1) cos α (2) cos β 15

16 (3) cos(α + β) (4) sin(α + β). 7. Example: Establish the identity cos(α β) sin α sin β = cot α cot β Sum and difference formulas for tangents tan(α + β) = tan(α β) = tan α+tan β 1 tan α tan β tan α tan β 1+tan α tan β 16

17 9. Example: Prove the identity tan(θ + π) = tan θ. 10. Example: Prove the identity tan(θ + π 2 ) = cot θ. 11. Example: Find the exact value of sin(cos sin ). 17

18 12. Example: Write sin(sin 1 u + cos 1 v) as an algebraic expression containing u and v (that is, without any trigonometric functions). 18

19 3.5 Double-angle and half-angle formulas 1. Double-angle formulas sin(2θ) = 2 sin θ cos θ cos(2θ) = cos 2 θ sin 2 θ cos(2θ) = 1 2 sin 2 θ cos(2θ) = 2 cos 2 θ 1 2. Example: If sin θ = 3 5, π 2 sin(2θ) and cos(2θ). < θ < π, find the exact values of 3. Example: (1) Develop a formula for tan(2θ) in terms of tan θ. 19

20 (2) Develop a formula for sin(3θ) in terms of sin θ and cos θ. 4. Other variations of the double-angle formulas sin 2 θ = 1 cos(2θ) 2, cos 2 θ = 1+cos(2θ) 2, tan 2 θ = 1 cos(2θ) 1+cos(2θ) 5. Example: Write an equivalent expression for cos 4 θ that does not involve any powers of sine or cosine greater that 1. 20

21 6. Half-angle formulas 1 cos sin α 2 = ± α 2, cos α 2 = ± 1+cos α 2, tan α 2 = ± 1 cos α 1+cos α where the + and sign is determined by the quadrant of the angle α Example: Use half-angle formulas find the exact values of cos 15 0 and sin( 15 0 ). 8. Example: If cos α = 3 5, π < α < 3π 2 of, find the exact value (1) sin α 2 21

22 (2) cos α 2 (3) tan α Half-angle for tan α 2 tan α 2 = 1 cos α sin α = sin α 1+cos α. 22

23 3.6 Product-to-sum and sum-to-product formulas 1. Product-to-sum formulas sin α sin β = 1 2 [cos(α β) cos(α + β)] cos α cos β = 1 2 [cos(α β) + cos(α + β)] sin α cos β = 1 2 [sin(α + β) + sin(α β)] 2. Example: Express each of the following products as a sum containing only sines or cosines: (1) sin(6θ) sin(4θ) (2) cos(3θ) cos θ (3) sin(3θ) cos(5θ) 23

24 3. Sum-to-product formulas sin α + sin β = 2 sin α+β cos α + cos β = 2 cos α+β 2 cos α β 2 2 cos α β 2 α β, sin α sin β = 2 sin 2 cos α+β 2 α+β, cos α cos β = 2 sin 2 sin α β 2 4. Example: Express each sum or difference as a product of sines and/or cosines: (1) sin(5θ) sin(3θ) (2) cos(3θ) + cos(2θ) 24

25 3.7 Trigonometric equations (I) 1. Example: Determine whether θ = π 4 equation sin θ = 1 2. Is θ = π 6 a solution? is a solution of the 2. Example: Slove the equation cos θ = 1 2. Give a general formula for all soultions. List six solutions 25

26 3. Example: Slove the equation 2 sin θ + 3 = 0, 0 θ < 2π. 4. Example: Slove the equation sin(2θ) = 1 2, 0 θ < 2π. 5. Example: Slove the equation tan(θ π 2 ) = 1, 0 θ < 2π. 26

27 3.8 Trigonometric equations (II) 1. Example: Solve the equation 2 sin 2 θ 3 sin θ + 1 = 0, 0 θ < 2π. 2. Example: Solve the equation 3 cos θ+3 = 2 sin 2 θ, 0 θ < 2π. 27

28 3. Example: Solve the equation cos(2θ)+3 = 5 cos θ, 0 θ < 2π. 4. Example: Solve the equation cos 2 θ + sin θ = 2, 0 θ < 2π. 28

29 5. Example: Solve the equation sin θ cos θ = 1 2, 0 θ < 2π. 6. Example: Solve the equation sin θ + cos θ = 1, 0 θ < 2π. 29