Sum and Difference Formulas - University of HoustonExercise Set 6.1: Sum and Difference Formulas MATH 1330 Precalculus 553 Simplify each of the following expressions. 1. sin S x 2

SECTION 6.1 Sum and Difference Formulas

MATH 1330 Precalculus 539

Chapter 6 Trigonometric Formulas and Equations

Section 6.1: Sum and Difference Formulas

Sum, Difference, and Reduction Identities

Sum, Difference, and Reduction Identities

Sum and Difference Formulas for Sine and Cosine:

CHAPTER 6 Trigonometric Formulas and Equations

University of Houston Department of Mathematics 540





Example:

Solution:

Two Basic Reduction Formulas:



Example:

Solution:

Formula (1):

Formula (2):



Example:

Solution:

Sum and Difference Formulas for Tangent:



Example:

Solution:





Additional Cofunction Identities:

Example:



Solution:

Part (a):

Part (b):

Additional Example 1:

Solution:


Solution:




Solution:




Solution:




Solution:


Solution:



Exercise Set 6.1: Sum and Difference Formulas


Simplify each of the following expressions.

1. sin x

2. 3

cos2

x

3. cos2

x

4. sin x

5. sin 60 sin 60

6. cos 60 cos 60

7. cos cos4 4

x x

8. sin sin6 6

x x

9. sin 180 sin 180

10. cos 90 cos 90

Answer the following.

11. Given that tan 4 , evaluate 3

tan4

.

12. Given that tan 2 , evaluate tan4

.

13. Given that 2

tan3

x , evaluate 5

tan4

x

.

14. Given that 1

tan5

x , evaluate 7

tan4

x

.

15. Given that tan 5x and tan 6y , evaluate

tan x y .

16. Given that tan 4x and tan 2y ,

evaluate tan x y .

17. Given that tan 3 and 2

tan5

,

evaluate tan .

18. Given that 4

tan3

and 1

tan2

,

evaluate tan .

Simplify each of the following expressions as much as

possible without a calculator.

19. cos 55 cos 10 sin 55 sin 10

20. cos 75 cos 15 sin 75 sin 15

21. sin 45 cos 15 cos 45 sin 15

22. sin 53 cos 7 cos 53 sin 7

23. cos cos sin sin10 9 10 9

24. sin cos cos sin5 7 5 7

25. 13 13

sin cos cos sin12 12 12 12

26. 5 17 5 17

cos cos sin sin12 12 12 12

27. sin 2 cos cos 2 sinA A A A

28. cos 3 cos sin 3 cos

29.

tan 32 tan 2

1 tan 32 tan 2

30.

tan 49 tan 4

1 tan 49 tan 4



31.

5tan tan

12 12

51 tan tan

12 12

32.

13 7tan tan

12 12

13 71 tan tan

12 12

33.

tan tan

1 tan tan

a b b

a b b

34.

tan 2 3 tan

1 tan 2 3 tan

c d d c

c d d c


35. Rewrite each special angle below so that it has a

denominator of 12.

(a) 6

(b)

4

(c)

3

(d) 2

3

(e)

3

4

(f)

5

6

36. Use the answers from Exercise 35 to write each

fraction below as the sum of two special angles.

(Hint: Each of the solutions contains a multiple

of 4

.)

(a) 5

12

(b)

7

12

(c) 13

12

(d)

11

12

37. Use the answers from Exercise 35 to write each

fraction below as the difference of two special

angles. (Hint: Each of the solutions contains a

multiple of 4

.)

(a) 5

12

(b)

7

12

(c) 7

12

(d)

5

12

38. For fractions with larger magnitude than those in

Exercises 36 and 37, it can be helpful to use

larger multiples of 4

. Rewrite each special

angle below so that it has a denominator of 12.

(a) 5

4

(b)

7

4

(c) 9

4

(d)

11

4

39. Use the answers from Exercises 35 and 38 to

write each fraction below as the sum of two

special angles. (Hint: Each of the solutions

contains a multiple of 4

.)

(a) 19

12

(b)

25

12

(c) 35

12

(d)

43

12

40. Use the answers from Exercises 35 and 38 to

write each fraction below as the difference of

two special angles. (Hint: Each of the solutions

contains a multiple of 4

.)

(a) 19

12

(b)

25

12

(c) 23

12

(d)

29

12

41. Use the answers from Exercises 35 and 38, along

with their negatives to write each fraction below

as the difference, x y , of two special angles,

where x is negative and y is positive.

(a) 7

12

(b)

13

12

(c) 29

12

(d)

41

12

42. Use the answers from Exercises 35 and 38, along

with their negatives to write each fraction below

as the difference, x y , of two special angles,

where x is negative and y is positive.

(a) 5

12

(b)

19

12

(c) 31

12

(d)

43

12




43. Use a sum or difference formula to prove that

sin sin .


cos cos .


tan tan .


cos sin2

.

Find the exact value of each of the following.

47. cos 15

48. sin 75

49. sin 195

50. cos 255

51. tan 105

52. tan 165

53. 7

sin12

54. 5

cos12

55. 17

cos12

56. 11

sin12

57. 31

cos12

58. 43

sin12

59. 13

tan12

60. 7

tan12

61. 37

tan12

62. 29

tan12

Answer the following. (Hint: It may help to first draw

right triangles in the appropriate quadrants and label

the side lengths.)

63. Suppose that 4

sin5

and 12

sin13

,

where 02

. Find:

(a) sin

(b) cos

(c) tan

64. Suppose that 8

cos17

and 2

cos3

,

where 02

. Find:

(a) sin

(b) cos

(c) tan

65. Suppose that 5

tan2

and 1

cos2

,

where 3

22

. Find:

(a) sin

(b) cos

(c) tan

66. Suppose that tan 3 and 1

sin4

,

where 3

2

and

2

. Find:

(a) sin

(b) cos

(c) tan



Evaluate the following.

67. 1 11 1cos tan tan

3 2

68. 1 1 1sin tan 4 tan

4

69. 1 13 5tan cos sin

5 13

70. 1 17 4tan tan cos

24 5

Simplify the following.

71. cos sin cot tanA B B A

72. tan cos cos cos cot tanA B A A A B

73. sin sinA B A B

74. cos cosA B A B

Prove the following.

75. sin sin 2sin cosx y x y x y

76. cos cos 2cos cosx y x y x y

77.

cos cos2cot cot

sin sin

x y x yx y

x y

78.

sin sin2 tan

cos cos

x y x yx

x y

79.

sin tan tan

sin tan tan

x y x y

x y x y

80.

cos 1 tan tan

cos 1 tan tan

x y x y

x y x y

Use right triangle ABC below to answer the following

questions regarding cofunctions, in terms of side

lengths a, b, and c.

81. (a) Find sin A .

(b) Find cos B .

(c) Analyze the answers for (a) and (b). What

do you notice?

(d) What is the relationship between angles A

and B? (i.e. If you knew the measure of one

angle, how would you find the other?) Write

answers in terms of degrees.

(e) Complete the following cofunction

relationships by filling in blank. Write your

answers in terms of A.

sin cos ________A

cos sin ________A

82. (a) Find tan A .

(b) Find cot B .


do you notice?






relationships by filling in blank. Write your

answers in terms of A.

tan cot ________A

cot tan ________A

83. (a) Find sec A .

(b) Find csc B .


do you notice?

Continued on the next page…

b c

A

B C a








relationships by filling in the blank. Write

your answers in terms of A.

sec csc _______A

csc sec _______A


sin 90 cos .

Use cofunction relationships to solve the following for

acute angle x.

85. sin 75 cos x

86. 3

cos sin10

x

87. 2

sec csc5

x

88. csc 41 sec x

89. tan 72 cot x

90. cot tan3

x

Simplify the following.

91. cos 90 cscx x

92. sin sec2

x x

93. sin tan2

x x

94. csc sin2

x x

95. sec 90 cos

96. tan 90 csc 90x x

97. sec 20 sin 70

98. cos csc6 3

99. 2 2 5cot sec

12 12

100. 2 2cos 72 cos 18



Section 6.2: Double-Angle and Half-Angle Formulas

Additional Trigonometric Identities

Additional Trigonometric Identities

The Double-Angle Formulas:

SECTION 6.2 Double-Angle and Half-Angle Formulas


The Half-Angle Formulas:



Example:

Solution:

Part (a):



Part (b):

Example:

Solution:

Part (a):



Part (b):


Solution:

Part (a):



Part (b):

Part (c):



Part (d):

Part (e):


Solution:

Part (a):



Part (b):




Solution:




Solution:

Part (a):

Part (b):




Solution:

Exercise Set 6.2: Double-Angle and Half-Angle Formulas



1. (a) Evaluate sin 26

.

(b) Evaluate 2sin6

.

(c) Is sin 2 2sin6 6

?

(d) Graph sin 2f x x and 2sing x x

on the same set of axes.

(e) Is sin 2 2sinx x ?

2. (a) Evaluate cos 26

.

(b) Evaluate 2cos6

.

(c) Is cos 2 2cos6 6

?

(d) Graph cos 2f x x and

2cosg x x on the same set of axes.

(e) Is cos 2cosx x ?

3. (a) Evaluate tan 26

.

(b) Evaluate 2 tan6

(c) Is tan 2 2 tan6 6

?

(d) Graph tan 2f x x and

2 tang x x on the same set of axes.

(e) Is tan 2 2 tanx x ?

4. Derive the formula for sin 2 by using a sum

formula on sin .

5. Derive the formula for cos 2 by using a sum

formula on cos .

6. Derive the formula for tan 2 by using a sum

formula on tan .

7. The sum formula for cosine yields the equation

2 2cos 2 cos sin . To write

cos 2 strictly in terms of the sine function,

(a) Using the Pythagorean identity

2 2cos sin 1 , solve for 2cos .

(b) Substitute the result from part (a) into the

above equation for cos 2 .

8. The sum formula for cosine yields the equation

2 2cos 2 cos sin . To write

cos 2 strictly in terms of the cosine function,

(a) Using the Pythagorean identity

2 2cos sin 1 , solve for 2sin .

(b) Substitute the result from part (a) into the

above equation for cos 2 .


9. Suppose that 12

cos13

and 3

22

.

Find:

(a) sin 2

(b) cos 2

(c) tan 2

10. Suppose that 3

tan4

and 3

2

.

Find:

(a) sin 2

(b) cos 2

(c) tan 2

11. Suppose that 2

sin5

and 2

.

Find:

(a) sin 2

(b) cos 2

(c) tan 2



12. Suppose that tan 3 and 3

22

.

Find:

(a) sin 2

(b) cos 2

(c) tan 2

Simplify each of the following expressions as much as

possible without a calculator.

13. 2sin 15 cos 15

14. 2 2cos sin2 2

15. 22cos 34 1

16. 2 51 2sin

12

17. 2

2 tan 105

1 tan 105

18. 21 sin x

19. 7 7

2sin cos8 8

20. 2sin 23 cos 23

21. 2 23 3cos sin

8 8

22. 2

112 tan

12

111 tan

12

23. 2 72sin 1

12

24. 2 2sin 67.5 cos 67.5

25. 2

2 tan 41

1 tan 41

26. 21 2cos 22.5

27. 22cos 12

28.

2

2 tan 3

tan 3 1

The formulas for 2

sinx

and 2

cosx

both contain a

sign, meaning that a choice must be made as to

whether or not the sign is positive or negative. For

each of the following examples, first state the quadrant

in which the angle lies. Then state whether the given

expression is positive or negative. (Do not evaluate the

expression.)

29. (a) cos 105

(b) sin 67.5

30. (a) sin 15

(b) cos 112.5

31. (a) 15

cos8

(b) 13

sin12

32. (a) 7

sin8

(b) 19

cos12



In the text, tan2

s

is defined as:

sintan

2 1 cos

ss

s

.

The following exercises can be used to derive this

formula along with two additional formulas for

tan2

s

.

33. (a) Write the formula for sin2

s

.

(b) Write the formula for cos2

s

.

(c) Derive a new formula for tan `2

s

using the

identity

sintan

cos

, where

2

s .

Leave both the numerator and denominator

in radical form. Show all work.

34. (a) This exercise will outline the derivation for:

sintan

2 1 cos

ss

s

. In exercise 33, it was

discovered that

1 costan

2 1 cos

ss

s

.

Rationalize the denominator by multiplying

both the numerator and denominator by

1 cos s . Simplify the expression and

write the result for tan2

s

.

(b) A detailed analysis of the signs of the

trigonometric functions of s and 2s in

various quadrants reveals that the symbol

in part (a) is unnecessary. (This analysis is

lengthy and will not be shown here.) Given

this fact, rewrite the formula from part (a)

without the symbol.

(c) How does this result from part (b) compare

with the formula given in the text

for tan2

s

?

35. (a) In exercise 33, it was discovered that

1 costan

2 1 cos

ss

s

.

Rationalize the numerator by multiplying

both the numerator and denominator by

1 cos s . Simplify the expression and

write the new result for tan2

s

.

(b) A detailed analysis of the signs of the

trigonometric functions of s and 2s in

various quadrants reveals that the symbol

in part (a) is unnecessary. (This analysis is

lengthy and will not be shown here.) Given

this fact, rewrite the formula from part (a)

without the symbol. This gives yet

another formula which can be used for

tan2

s

.

(c) Use the results from Exercises 33-35 to

write the three formulas for tan2

s

. Which

formula seems easiest to use and why?

Which formula seems hardest to use and

why?

36. (a) In the text, tan2

s

is defined as:

sintan

2 1 cos

ss

s

.

Multiply both the numerator and

denominator of the right-hand side of the

equation by 1 cos s . Then simplify to

obtain a formula for tan2

s

. Show all

work.

(b) How does the result from part (a) compare

to the identity obtained in part (b) of

Exercise 39?



Answer the following. SHOW ALL WORK. Do not

leave any radicals in the denominator, i.e. rationalize

the denominator whenever appropriate.

37. (a) Find cos 75 by using a sum or difference

formula.

(b) Find cos 75 by using a half-angle

formula.

(c) Enter the results from parts (a) and (b) into a

calculator and round each one to the nearest

hundredth. Are they the same?

38. (a) Find sin 165 by using a sum or difference

formula.

(b) Find sin 165 by using a half-angle

formula.

(c) Enter the results from parts (a) and (b) into a

calculator and round each one to the nearest

hundredth. Are they the same?

39. (a) Find sin 112.5 by using a half-angle

formula.

(b) Find cos 112.5 by using a half-angle

formula.

(c) Find tan 112.5 by computing

sin 112.5

cos 112.5.

(d) Find tan 112.5 by using a half-angle

formula.

40. (a) Find sin 105 by using a half-angle

formula.

(b) Find cos 105 by using a half-angle

formula.

(c) Find tan 105 by computing

sin 105

cos 105.

(d) Find tan 105 by using a half-angle

formula.

Find the exact value of each of the following by using a

half-angle formula. Do not leave any radicals in the

denominator, i.e. rationalize the denominator

whenever appropriate.

41. (a) 13

sin12

(b) 13

cos12

(c) 13

tan12

42. (a) 3

sin8

(b) 3

cos8

(c) 3

tan8

43. (a) 11

sin8

(b) 11

cos8

(c) 11

tan8

44. (a) sin 157.5

(b) cos 157.5

(c) tan 157.5

45. (a) sin 285

(b) cos 285

(c) tan 285

46. (a) 17

sin12

(b) 17

cos12

(c) 17

tan12




47. If 5

cos9

and 3

22

,

(a) Determine the quadrant of the terminal side

of .

(b) Complete the following: ___ ___2

(c) Determine the quadrant of the terminal side

of 2

.

(d) Based on the answer to part (c), determine

the sign of sin2

.

(e) Based on the answer to part (c), determine

the sign of cos2

.

(f) Find the exact value of sin2

.

(g) Find the exact value of cos2

.

(h) Find the exact value of tan2

.

48. If 21

sin5

and 3

2

,

(a) Determine the quadrant of the terminal side

of .

(b) Complete the following:

_____ _____2

(c) Determine the quadrant of the terminal side

of 2

.

(d) Based on the answer to part (c), determine

the sign of sin2

.

(e) Based on the answer to part (c), determine

the sign of cos2

.

(f) Find the exact value of sin2

.

(g) Find the exact value of cos2

.

(h) Find the exact value of tan2

.

49. If 7

tan3

and 2

,

(a) Find the exact value of sin2

.

(b) Find the exact value of cos2

.

(c) Find the exact value of tan2

.

50. If 5

cos6

and 3

22

,

(a) Find the exact value of sin2

.

(b) Find the exact value of cos2

.

(c) Find the exact value of tan2

.

Prove the following.

51.

1 cos 2tan

sin 2

xx

x

52.

cos 3 sin 32

cos sin

x x

x x

53. 1 2 sin 1 2 sin cos 2x x x

54. 4 4cos sin cos 2x x x

55. csc cot tan2

xx x

56.

2

2

1 tan2

sec

1 tan2

x

x



Section 6.3: Solving Trigonometric Equations

Solving Equations Involving Trigonometric Functions

Solving Equations Involving Trigonometric Functions

SECTION 6.3 Solving Trigonometric Equations


Example:

Solution:



Example:

Solution:


Solution:




Solution:




Solution:




Solution:




Solution:




Solution:



Exercise Set 6.3: Solving Trigonometric Equations


Does the given x-value represent a solution to the

given trigonometric equation? Answer yes or no.

1. 6

x

; 22cos 7cos 4x x

2. 5

4x

; 23tan 8tan 5x x

3. 3

2x

; 3 2sin 4sin 2sin 3x x x

4. 3

x

; 26cos 7cos 2 0x x

For each of the following equations,

(a) Solve the equation for 0 360x

(b) Solve the equation for 0 2x .

(c) Find all solutions to the equation, in radians.

5. 1

cos2

x

6. 3

sin2

x

7. tan 1x

8. cos 0x

9. sin 1x

10. 3 cot 1x

11. 2 3

sec3

x

12. csc 2x

13. 2cos 2x

14. sin 2x

15. 22sin 5sin 3 0x x

16. 22cos cos 1x x

17. 2cos 2cos 1x x

18. 22sin 7sin 4x x

When solving an equation such as 1

2sin x

for 0 360x , a typical thought process is to first

compute 1 1

2sin

to find the reference angle (in this

case, 30 ) and then use that reference angle to find

solutions in quadrants where the sine value is negative

(in this case, 210 and 330 . In each of the the

following examples of the form sin x C ,

(a) Use a calculator to find 1sin C

to the nearest

tenth of a degree. This represents the

reference angle.

(b) Use the reference angle from part (a) to find

the solutions to the equation for

0 360x .

19. 4

sin5

x

20. sin 0.3x

21. sin 0.46x

22. 2

sin5

x

The method in Exercises 19-22 can be used for other

trigonometric functions as well. Use a calculator to

solve each of the following equations for 0 360x .

Round answers to the nearest tenth of a degree. If no

solution exists, state “No solution.”

Note: Since calculators do not contain inverse keys for

cosecant, secant, and cotangent, use reciprocal

relationships to rewrite equations in terms of sine, cosine

and tangent. For example, to solve the equation

sec 4x , first rewrite the equation as 14

cos x and

then proceed to solve the equation.

23. (a) 3

cos7

x (b) 7

csc5

x

24. (a) tan 6x (b) sec 7x

25. (a) cot 2.9x (b) sin 5.6x

26. (a) 1

csc4

x (b) tan 2.5x




27. The following example is designed to

demonstrate a common error in solving

trigonometric equations. Consider the equation

22cos 3 cosx x for 0 2x .

(a) Divide both sides of the equation by cos x

and then solve for x.

(b) Move all terms to the left side of the

equation, and then solve for x.

(c) Are the answers in parts (a) and (b) the

same?

(d) Which method is correct, part (a) or (b)?

Why is the other method incorrect?

28. The following example is designed to

demonstrate a common error in solving

trigonometric equations. Consider the equation

2tan tanx x for 0 360x .

(a) Divide both sides of the equation by tan x

and then solve for x.

(b) Move all terms to the left side of the

equation, and then solve for x.

(c) Are the answers in parts (a) and (b) the

same?

(d) Which method is correct, part (a) or (b)?

Why is the other method incorrect?

Solve the following equations for 0 360x . If no

solution exists, state “No solution.”

29. 22 sin sinx x

30. 2cos cosx x

31. 23 tan tanx x

32. 22sin sinx x

33. 24sin cos cos 0x x x

34. 2sin tan sinx x x

35. 3 22cos 3cos cosx x x

36. 3 22sin 9sin 5sinx x x

The following exercises show a method of solving an

equation of the form:

sin Ax B C , for 0 2x .

(The same method can be used for the other five

trigonometric functions as well, and can similarly be

applied to intervals other than 0 2x .) Answer

the following, using the method described below.

(a) Write the new interval obtained by

multiplying each term in the solution interval

(in this case, 0 2x ) by A and then

adding B.

(b) Let u Ax B . Find all solutions to

sin u C within the interval obtained in

part (a).

(c) For each solution u from part (b), set up and

solve u Ax B for x. These x-values

represent all solutions to the initial equation.

37. 2

sin 22

x

38. 1

sin 32

x

39. 3

sin2 2

x

40. 2

sin4 2

x

41. sin 3 0x

42. 1

sin2 3 2

x

Solve the following, using either the method above or

the method described in the text. If no solution exists,

state “No solution.”

43. 2cos 2 2x , for 0 2x

44. 3 tan 2 1 0x , for 0 360x

45. csc 2 2x , for 0 360x

46. sec 2 2x , for 0 2x



47. 2sin 12

x

, for 0 2x

48. tan 3 3x , for 0 360x

49. 2sin 3 3 0x , for 0 360x

50. 2cos 32

x

, for 0 360x

51. 2cos 4 3x , for 0 2x

52. csc 5 1x , for 0 2x

53. tan 90 1x , for 0 360x

54. 2cos 45 1x , for 0 360x

55. 5

3 sec 2 06

x

, for 0 2x

56. cot 0x , for 0 2x

57. sin 2 270 1x , for 0 360x

58. tan 3 2 0x , for 0 2x .

59. tan 60 12

x

, for 0 360x

60. cos 1 03 3

x

, for 0 2x .

61. 2cos 45 2 3x , for 0 360x

62. 2sin 3 24

x

, for 0 2x .

Use a calculator to solve each of the following

equations for 0 360x . Round answers to the

nearest tenth of a degree. If no solution exists, state

“No solution.”

63. 215sin 6sinx x

64. 26tan tan 1x x

65. 23cot 5cot 2x x

66. 24csc 8csc 3x x

67. 2sec 16x

68. 23cos 2cosx x

Solve the following for 0 2x . If no solution

exists, state “No solution.”

69. 22sin cos 1x x

70. 26cos 13sin 11 0x x

71. 2cot csc 1 0x x

72. 2sec 2 tan 0x x

73. tan 3x

74. sec 1x

75. 2log cos 1x

76. 3

1log cot

2x

77. 3log tan 0x

78. 2

log csc 1x

Documents

Sum and Difference Formulas - University of HoustonExercise Set 6.1: Sum and Difference Formulas MATH 1330 Precalculus 553 Simplify each of the following expressions. 1. sin S x 2