Pre-Class Problems17 for Monday, March 26

Pre-Class Problems17 for Monday, March 26

Earn one bonus point because you checked these problems. Send me an email with the following in the Subject line: PC17005 if you are in Section 005 (10:00 class) or PC17002 if you are in Section 002 (11:30 class).

These are the type of problems that you will be working on in class. These problems are fromLesson 9.

Solution to Problems on the Pre-Exam.

You can go to the solution for each problem by clicking on the problem number or letter.

Objective of the following problems: To use the inverse tangent function to find the reference angle of an angle if given the coordinates of a point on the terminal side of the angle.

1.Find the exact and the approximate angle , which is between and , which passes through the following points. Round your approximate answer to the nearest tenth of a degree.

a. b. c.

d. e.

Objective of the following problems: To find the exact value of any one of the six trigonometric functions of an angle when the angle is given in terms of an inverse trigonometric function of a number.

2.Given the angle , find the exact value of a) and

b) .

3.Given the angle , find the exact value of

a) and b) .

4.Given the angle , find the exact value of a) and b) .

5.Given the angle , find the exact value of a)

and b) .

6.Given the angle , find the exact value of a) andb) .

7.Given the angle , find the exact value of

a) and b) .

8.Given the angle , find the exact value of

a) and b) .

9.Given the angle , find the exact value of

a) and b) .

10.Given the angle , find the exact value of a) and b) .

Additional problems available in the textbook: Page 553 …61–64, 67 – 70. Examples 6 and 7 starting on Page 546.

Solutions:

1a. Back to Problem 1.

NOTE: Since the point is in the second quadrant and lies on the terminal side of the angle , then the terminal side of is in the second quadrant. Thus, is not an acute angle.

To find the reference angle :

since is an acute angle.

NOTE: Since is in the second quadrant, then . Since the exact value of is , then the exact value of is . Since the approximate value of is , then the approximate value of is .

Answer:Exact:

Approximate:

1b.Back to Problem 1.

NOTE: Since the point is in the fourth quadrant and lies on the terminal side of the angle , then the terminal side of is in the fourth quadrant. Thus, is not an acute angle.

To find the reference angle :

since is an acute angle.

NOTE: Since is in the fourth quadrant, then . Since the exact value of is , then the exact value of is . Since the approximate value of is , then the approximate value of is .

Answer:Exact:

Approximate:

1c. Back to Problem 1.

NOTE: Since the point is in the third quadrant and lies on the terminal side of the angle , then the terminal side of is in the third quadrant. Thus, is not an acute angle.

To find the reference angle :

since is an acute angle.

NOTE: Since is in the third quadrant, then . Since the exact value of is , then the exact value of is . Since the approximate value of is , then the approximate value of is .

Answer:Exact:

Approximate:

1d. Back to Problem 1.

NOTE: Since the point is in the first quadrant and lies on the terminal side of the angle , then the terminal side of is in the first quadrant.

NOTE: By the definition of the inverse tangent function, the inverse tangent of a positive number is an angle in the open interval .

is an acute angle.

Answer:Exact:

Approximate:

1e. Back to Problem 1.

NOTE: Since the point is in the fourth quadrant and lies on the terminal side of the angle , then the terminal side of is in the fourth quadrant. Thus, is not an acute angle.

To find the reference angle :

since is an acute angle.

NOTE: Since is in the fourth quadrant, then . Since the exact value of is , then the exact value of is . Since the approximate value of is , then the approximate value of is .

Answer:Exact:

Approximate:

2.Let . Then is an acute angle in the first quadrant and

.

NOTE: These two statements follow from the definition of the inverse sine function.

:

8

3

NOTE:

a. = =

NOTE:

b. = =

NOTE:

NOTE:

OR = =

NOTE:

Answers:a. b.

Back to Problem2.

3.Let . Then is in the fourth quadrant and

.

NOTE: These two statements follow from the definition of the inverse sine function.

NOTE: Since is in the fourth quadrant, then it is NOT an acute angle.

:

9

NOTE:

8

a. = =

NOTE: Cosine is positive in the fourth quadrant and .

b. = =

NOTE: Tangent is negative in the fourth quadrant and .

NOTE:

OR = =

NOTE: Cotangent is negative in the fourth quadrant and .

Answers:a.

Back to Problem 3.

b.

4.Let . Then is an acute angle in the first quadrant and

.

NOTE: These two statements follow from the definition of the inverse cosine function.

:

7

NOTE:

4

a. = =

NOTE:

b. = =

NOTE:

NOTE:

OR = =

NOTE:

Answers:a. b.

Back to Problem 4.

5.Let . Then is in the second quadrant and.

NOTE: These two statements follow from the definition of the inverse cosine function.

NOTE: Since is in the second quadrant, then it is NOT an acute angle.

:

5

NOTE:

2

a. = =

NOTE: Tangent is negative in the second quadrant and .

b. = =

NOTE: Sine is positive in the second quadrant and .

NOTE:

OR = =

NOTE: Cosecant is positive in the second quadrant and .

Answers:a.

Back to Problem 5.

b.

6.Let . Then is an acute angle in the first quadrant and

.

NOTE: These two statements follow from the definition of the inverse tangent function.

:

NOTE:

7

a. = =

NOTE:

b. = =

NOTE:

NOTE:

OR = =

NOTE:

Answers:a. b.

Back to Problem 6.

7.Let . Then is in the fourth quadrant and

.

NOTE: These two statements follow from the definition of the inverse tangent function.

NOTE: Since is in the fourth quadrant, then it is NOT an acute angle.

:

7 5

NOTE:

a. = =

NOTE: Sine is negative in the fourth quadrant and .

b. = =

NOTE: Cosine is positive in the fourth quadrant and .

NOTE:

OR = =

NOTE: Secant is positive in the fourth quadrant and .

Answers:a.

Back to Problem 7.

b.

8.Let . Then is in the fourth quadrant and.

NOTE: These two statements follow from the definition of the inverse sine function.

NOTE: Since is in the fourth quadrant, then it is NOT an acute angle.

:

3

2

NOTE:

a. = =

NOTE: Tangent is negative in the fourth quadrant and .

b. = =

NOTE: Cosine is positive in the fourth quadrant and .

Answers:a.

Back to Problem 8.

b.

9.Let . Then is in the second quadrant and.

NOTE: These two statements follow from the definition of the inverse cosine function.

NOTE: Since is in the second quadrant, then it is NOT an acute angle.

:

8

NOTE:

a. = =

NOTE: Sine is positive in the second quadrant and .

b. = =

NOTE: Tangent is negative in the second quadrant and .

NOTE:

OR = =

NOTE: Cotangent is negative in the second quadrant and .

Answers:a.

Back to Problem 9.

b.

10.Let . Then is in the fourth quadrant and .

NOTE: These two statements follow from the definition of the inverse tangent function.

NOTE: Since is in the fourth quadrant, then it is NOT an acute angle.

:

NOTE: 4

1

a. = =

NOTE: Cosine is positive in the fourth quadrant and .

b. = =

NOTE: Sine is negative in the fourth quadrant and .

NOTE:

OR = =

NOTE: Cosecant is negative in the fourth quadrant and .

Answers:a.

Back to Problem 10.

b.

Solution to Problems on the Pre-Exam:Back to Page 1.

14.Find the exact angle , where, if the terminal side of passes through the point. (5 pts.)

NOTE: Since the point is in the third quadrant and lies on the terminal side of the angle , then the terminal side of is in the third quadrant. Thus, is not an acute angle.

To find the reference angle :

since is an acute angle.

NOTE: Since is in the third quadrant, then . Since the exact value of is , then the exact value of is .

Answer:

19.Find the exact value of (8 pts.)

Let . Then is in the fourth quadrant and .

NOTE: These two statements follow from the definition of the inverse sine function.

:

8

3

NOTE:

= =

NOTE: Cosine is positive in the fourth quadrant and .

NOTE:

OR = =

NOTE: Secant is positive in the fourth quadrant and .

is in IV quadrant =