7.1 Solving Trigonometric Equations with Identities In this section, we explore the techniques needed to solve more complex trig equations: By Factoring Using the Quadratic Formula Utilizing Trig Identities to simplify (Section 5.5) Find all solutions. 10. 8cos 6 x Find all solutions. 7sin 3t 11. Find all solutions on the interval [0, ). 14. 3sin t 15cos tsin t 18. x x x tan sin sin 0
4. 8sin x 6sin x 1 0 30. x x 6cos 7sin 8 0 3 34. cos t cos t 38. x x x x sin cos sin cos 1 0 (Hint: Factor by grouping) 40. 3cos x cot x
7. Addition and Subtraction Identities Sum and Difference Identities (Formulas) Rewrite in terms of 3 10. sin x 4 sin x and cos x. 1. cosx 3 16. tan x 4
Rewriting a Sum of Sine and Cosine as a Single Sine To rewrite msin( Bx) ncos( Bx) as Asin( Bx C), where m n A m n, cos( C), and sin( C) A A Rewrite as a single function of the form Asin( Bx C). Ex. sin(x) + 3 cos(x) 36. sin x 5cos x 38. 3sin 5x 4cos 5x The Product-to-Sum and Sum-to-Product Identities The Product-to-Sum Identities 1 sin( )cos( ) sin( ) sin( ) 1 sin( )sin( ) cos( ) cos( ) 1 cos( )cos( ) cos( ) cos( ) Rewrite the product as a sum. 0cos 36 cos 6 18. t t 0. 10cos 5xsin 10x The Sum-to-Product Identities u v u v sinu sinv sin cos u v u v sinu sinv sin cos cos u cos v u v u v cos cos cos u cos v u v u v sin sin Rewrite the sum as a product. cos 6u cos 4u. 4. sin h sin 3h
4 5 6. Given sin a and cos 1 b, with a and b both in the interval 0, 3 : a. Find sin a b b. Find cos a b Solve each equation for all solutions. 30. cos5xcos3x sin 5xsin 3x 3. sin 5x sin 3x 3 Prove the identity. tan x 1 44. tan x 4 1 tan x 5. cos x y cos x y cos x sin y
7.3 Double Angle Identities Double-angle Identity: sin sin cos cos cos sin cos cos cos 1 1 sin tan Power Reduction Identity (Formulas for Lowering Powers): sin 1 cos θ θ = cos 1 + cos θ θ = tan 1 cos θ θ = 1 + cos θ Half-angle Identity: sin cos 1 cos 1 cos sin tan 1 cos sin 1 cos Note: Where the + or sign is determined by the Quadrant of the angle α. Exercises 1. If tan 1 tan 1 cos 1 cos cos x and x is in quadrant IV, then find exact values for (without solving for x): 3 a. sin x b. cos x x c. cos x d. tan
Use the Half- angle Formulas to find the exact value of each expression. Ex. cos.5 Ex3. sin 195 Ex4. tan 9π 8 Simplify each expression. cos 37 1 Ex5. 6. cos 6x sin (6 x) 7. x 6sin 5 cos(5 x ) Solve for all solutions on the interval [0, ). 8. sin t 3cos t 0 9. cos t sin t Use a double angle, half angle, or power reduction formula to rewrite without exponents. 4 10. cos (6 ) sin 3x x 11. Prove the identity. 1. 4 sin x 1 cos x sin x 13. sin 1 cos tan
8.1 Non-right Triangles: Law of Sines and Cosines Find the area of a Triangle The Law of Sines Note: The law of Sines is useful when we know a side and the angle opposite it. Case c (SSA) is referred to as the ambiguous case because the known information may result in two triangles, one triangle, or no triangle at all. Ex1. Solve the Triangle using the Law of Sines, and find the area of the Triangle.
Ex. Sketch each triangle, and then solve the triangle using the Law of Sines. a) b = 4, c = 3, B = 40 b) b =, c = 3, B = 40 c) a = 3, b = 7, A = 70
Ex3. The Law of Cosines Note: The law of Cosines is use to solve triangles like SAS and SSS. Ex4. Solve x using the Law of Cosines Ex5. Solve θ using the Law of Cosines
Ex6. Sketch each triangle, then find the area and solve the triangle using the Law of Cosines. a = 40, b = 1, c = 44 Ex7 (30 o East of North) (60 o West of North) (70 o West of South) (50 o East of South) 43. Three circles with radii 6, 7, and 8 respectively, all touch as shown. Find the shaded area bounded by the three circles.
8. Polar Coordinates Definition of Polar Coordinates: The polar coordinate system, (r, θ), use distances and directions to specify the location of a point in the plane. r is the distance from O to P θ is the angle between the polar axis and the segment OP Plotting Points in Polar Coordinates Ex. Plot A = (1,0), B = (3, π ), C = (5, π 3 ), D = (6, 5π 6 ), E = ( 6, 5π 6 ) Relation Between Polar (r, θ) and Rectangular (x, y) Coordinates x = r cos θ } Polar to Rectangular Coordinates y = r sin θ r = x + y tan θ = y x, x 0 cos θ = x r = x Rectangular to Polar x + y sin θ = y r = y x + y }
Convert the Polar coordinate to a Cartesian coordinate 1. 7 7, 6 3. 7 4, 4 11. (3, ) Convert the Cartesian coordinate to a Polar coordinate 13. (4, ) Ex.( 6, ) Ex. (-5, -5) Convert the Cartesian equation to a Polar equation. Express your answer as r = f(θ) (Note, x = rcos θ, and y = r sin θ) 1. x 3 3. y 4x 5. x y 4y 7. x y x Convert the Polar equation to a Cartesian equation. (Note, r cos θ = x, r sin θ = y, and r = x + y ) 4 9. r 3sin 31. r 35. r rcos sin 7cos Use calculator to sketch a graph of the polar equation r 3cos r 3sin 49. 51.