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

Find all solutions.

11. 7sin 3 2t

Find all solutions on the interval [0, 2 ) .

14. 3sin 15cos sint t t 18. tan sin sin 0x x x

(Section 5.5) Find all solutions.

10. 8cos 62

x

24. 28sin 6sin 1 0x x 30. 26cos 7sin 8 0x x

34. 3cos cost t 38. 2sin cos sin 2cos 1 0x x x x

(Hint: Factor by grouping)

40. 3cos cotx x

7.2 Addition and Subtraction Identities Sum and Difference Identities (Formulas)

Rewrite in terms of sin x and cos x .

10. 3

sin4

x

12. 2

cos3

x

16.

x

4tan

Rewrite as a single function of the form sin( ) A Bx C .

Ex. 2 sin(2𝑥) + 3 cos(2𝑥) 36. sin 5cosx x 38. 3sin 5 4cos 5x x

The Product-to-Sum and Sum-to-Product Identities

Rewrite the product as a sum.

18. 20cos 36 cos 6t t 20. 10cos 5 sin 10x x

Rewrite the sum as a product.

22. cos 6 cos 4u u 24. sin sin 3h h

The Product-to-Sum Identities

)cos()cos(2

1)cos()cos(

)cos()cos(2

1)sin()sin(

)sin()sin(2

1)cos()sin(

The Sum-to-Product Identities

2cos

2sin2sinsin

vuvuvu

2cos

2sin2sinsin

vuvuvu

2cos

2cos2coscos

vuvuvu

2sin

2sin2coscos

vuvuvu

Rewriting a Sum of Sine and Cosine as a Single Sine

To rewrite )cos()sin( BxnBxm as )sin( CBxA , where

222 nmA , A

mC )cos( , and

A

nC )sin(

26. Given 4

sin5

a and 1

cos3

b , with a and b both in the interval 0,2

:

a. Find sin a b b. Find cos a b

Solve each equation for all solutions.

30. 3

cos 5 cos 3 sin 5 sin 32

x x x x 32. sin 5 sin 3x x

Prove the identity.

44.

tan 1tan

4 1 tan

xx

x

52. 2 2cos cos cos sinx y x y x y

7.3 Double Angle Identities Double-angle Identity:

Power Reduction Identity (Formulas for Lowering Powers):

sin2 𝜃 =1 − cos 2𝜃

2

cos2 𝜃 =1 + cos 2𝜃

2

tan2 𝜃 =1 − cos 2𝜃

1 + cos 2𝜃

Half-angle Identity:

Note: Where the + or – sign is determined by the Quadrant of the angle 𝛼

2.

Exercises 1. If 2

cos3

x and x is in quadrant IV, then find exact values for (without solving for x):

a. sin 2x b. cos 2x c. cos2

x

d. tan2

x

cossin22sin

22 sincos2cos

2sin212cos

2tan1

tan22tan

1cos22cos 2

2

cos1

2sin

2

cos1

2cos

cos1

sin

sin

cos1

cos1

cos1

2tan

Use the Half- angle Formulas to find the exact value of each expression.

Ex2. cos 22.5° Ex3. sin 195° Ex4.

Simplify each expression.

Ex5. 22cos 37 1 6. 2 2cos 6 sin (6 )x x 7. 6sin 5 cos(5 )x x

Solve for all solutions on the interval [0, 2 ) .

8. 2sin 2 3cos 0t t 9. cos 2 sint t

Use a double angle, half angle, or power reduction formula to rewrite without exponents.

10. 2cos (6 )x 11. 4sin 3x

Prove the identity.

12. 2

2 4sin 1 cos 2 sinx x x 13.

sin 2tan

1 cos 2

tan9𝜋

8

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.

Ex2. Sketch each triangle, and then solve the triangle using the Law of Sines.

a) 𝑏 = 4, 𝑐 = 3, 𝐵 = 40°

b) 𝑏 = 2, 𝑐 = 3, 𝐵 = 40°

c) 𝑎 = 3, 𝑏 = 7, 𝐴 = 70°

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

Ex3

.

Ex6. Sketch each triangle, then find the area and solve the triangle using the Law of Cosines.

𝑎 = 40, 𝑏 = 12, 𝑐 = 44

43. Three circles with radii 6, 7, and 8 respectively, all touch as shown. Find the shaded

area bounded by the three circles.

(30o East of North) (60o West of North) (70o West of South) (50o East of South)

Ex7

8.2 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 𝑂𝑃̅̅ ̅̅

Plotting Points in Polar Coordinates

Ex. Plot 𝐴 = (1,0), 𝐵 = (3,𝜋

2) , 𝐶 = (5,−

2𝜋

3) , 𝐷 = (6,

5𝜋

6) , 𝐸 = (−6,

5𝜋

6)

Relation Between Polar (𝒓, 𝜽) and Rectangular (𝒙, 𝒚) Coordinates 𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃

} 𝑃𝑜𝑙𝑎𝑟 𝑡𝑜 𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠

𝑟 = √𝑥2 + 𝑦2

tan 𝜃 =𝑦

𝑥, 𝑥 ≠ 0

cos 𝜃 =𝑥

𝑟=

𝑥

√𝑥2 + 𝑦2

sin 𝜃 =𝑦

𝑟=

𝑦

√𝑥2 + 𝑦2}

𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑡𝑜 𝑃𝑜𝑙𝑎𝑟

Convert the Polar coordinate to a Cartesian coordinate

1. 7

7,6

3. 7

4,4

11. (3, 2)

Convert the Cartesian coordinate to a Polar coordinate

13. (4, 2) Ex.(−√6, √2) 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)

21. 3x 23. 24y x 25.

2 2 4x y y 27. 2 2x y x

Convert the Polar equation to a Cartesian equation. (Note, 𝑟 cos 𝜃 = 𝑥, 𝑟 sin 𝜃 = 𝑦 , 𝑎𝑛𝑑 𝑟2 = 𝑥2 + 𝑦2 )

29. 3sinr 31.

4

sin 7cosr

35. cos 2r r

Use calculator to sketch a graph of the polar equation

49. 3cosr 51. 3sin 2r