7.1 Solving Trigonometric Equations with Identities

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. 8 cos  x   6 2 

Find all solutions. 11. 7sin  3t   2

Find all solutions on the interval [0, 2 ) . 14. 3sin  t   15cos  t  sin  t 

18. tan  x  sin  x   sin  x   0

24. 8sin 2 x  6sin  x   1  0

30. 6cos 2  x   7sin  x   8  0

34. cos3  t    cos  t 

38. 2sin  x  cos  x   sin  x   2cos  x   1  0 (Hint: Factor by grouping)

40. 3cos  x   cot  x 

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

Rewrite in terms of sin  x  and cos  x  . 3   10. sin  x   4  

2   12. cos  x   3  

  16. tan   x  4 

Rewriting a Sum of Sine and Cosine as a Single Sine To rewrite m sin( Bx )  n cos(Bx ) as A sin( Bx  C ) , where m n A 2  m 2  n 2 , cos(C )  , and sin(C )  A A Rewrite as a single function of the form A sin( Bx  C ) . Ex. 2 sin(2𝑥) + 3 cos(2𝑥) 36.  sin  x   5cos  x 

38. 3sin  5 x   4cos  5 x 

The Product-to-Sum and Sum-to-Product Identities The Product-to-Sum Identities 1 sin( ) cos( )  sin(   )  sin(   )  2 1 sin( ) sin( )  cos(   )  cos(   )  2 1 cos( ) cos( )  cos(   )  cos(   )  2

The Sum-to-Product Identities u v u v sinu   sinv   2 sin  cos   2   2  u v u v sinu   sinv   2 sin  cos   2   2  u v u v cosu   cosv   2 cos  cos   2   2  u v u v cosu   cosv   2 sin  sin   2   2 

Rewrite the product as a sum. 18. 20cos  36t  cos  6t 

20. 10cos  5 x  sin 10 x 

Rewrite the sum as a product. 22. cos  6u   cos  4u 

24. sin  h   sin  3h 

4 1   and cos  b   , with a and b both in the interval  0,  : 5 3  2 a. Find sin  a  b  b. Find cos  a  b 

26. Given sin  a  

Solve each equation for all solutions. 30. cos  5 x  cos  3 x   sin  5 x  sin  3 x  

Prove the identity.   tan  x   1  44. tan  x    4  1  tan  x  

3 2

32. sin  5 x   sin  3x 

52. cos  x  y  cos  x  y   cos 2 x  sin 2 y

7.3 Double Angle Identities Double-angle Identity:

sin2   2 sin cos

cos2   cos2   sin 2 

cos2   2 cos2   1 cos2   1  2 sin 2  2 tan  tan 2   1  tan 2 

Power Reduction Identity (Formulas for Lowering Powers): 1 − cos 2𝜃 sin2 𝜃 = 2 1 + cos 2𝜃 cos 2 𝜃 = 2 1 − cos 2𝜃 tan2 𝜃 = 1 + cos 2𝜃 Half-angle Identity: 1  cos   sin    2 2 1  cos   cos    2 2 1  cos 1  cos sin    tan       1  cos sin  1  cos 2

Note: Where the + or – sign is determined by the Quadrant of the angle Exercises 1. If cos  x   a. sin  2x 

𝛼 2

.

2 and x is in quadrant IV, then find exact values for (without solving for x): 3

b. cos  2x 

x c. cos   2

x d. tan   2

Use the Half- angle Formulas to find the exact value of each expression. Ex2. cos 22.5° Ex3. sin 195°

Ex4. tan

Simplify each expression. Ex5. 2cos2  37  1

7. 6sin  5 x  cos(5 x)

6. cos2  6 x   sin 2 (6 x)

Solve for all solutions on the interval [0, 2 ) . 8. 2sin  2t   3cos  t   0

9𝜋 8

9. cos  2t   sin  t 

Use a double angle, half angle, or power reduction formula to rewrite without exponents. 10. cos 2 (6 x) 11. sin 4  3x 

Prove the identity. 12.  sin 2 x  1  cos  2 x   sin 4 x 2

13.

sin  2 

1  cos  2 

 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.

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°

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 = 12, 𝑐 = 44

(30o East of North)

(60o West of North)

(70o West of South)

(50o East of South)

Ex7

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

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 𝑂𝑃

Plotting Points in Polar Coordinates 𝜋

Ex. Plot 𝐴 = (1,0), 𝐵 = (3, 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 } 𝑃𝑜𝑙𝑎𝑟 𝑡𝑜 𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝐶𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑦 = 𝑟 sin 𝜃 𝑟 = √𝑥 2 + 𝑦 2 𝑦 tan 𝜃 = , 𝑥 ≠ 0 𝑥 𝑥 𝑥 𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑡𝑜 𝑃𝑜𝑙𝑎𝑟 cos 𝜃 = = 𝑟 √𝑥 2 + 𝑦 2 𝑦 𝑦 sin 𝜃 = = 𝑟 √𝑥 2 + 𝑦 2 }

Convert the Polar coordinate to a Cartesian coordinate  7  1.  7,   6 

 7  3.  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. x  3 23. y  4 x 2 25. x 2  y 2  4 y 27. x 2  y 2  x

Convert the Polar equation to a Cartesian equation. (Note, 𝑟 cos 𝜃 = 𝑥, 𝑟 sin 𝜃 = 𝑦 , 𝑎𝑛𝑑 𝑟 2 = 𝑥 2 + 𝑦 2 ) 4 29. r  3sin   31. r  35. r  r cos    2 sin    7 cos  

Use calculator to sketch a graph of the polar equation 49. r  3cos   51. r  3sin  2 