Pre-Class Problems 22 for Friday, April 6
These are the type of problems that you will be working on in class. These problems are from Lesson 12.
Solution to Problems on the Pre-Exam.
You can go to the solution for each problem by clicking on the problem letter.
Objective of the following problems: To use a reference angle to the exact value of the six trigonometric functions of the following angles. You will need to use either the addition or subtraction for cosine, sine, or tangent in order to find the value of the reference angle, which will either be or . The cosine, sine, and tangent of . The cosine, sine, and tangent of .
1. Find the exact value of the six trigonometric functions of the given angle using the reference angle of the angle.
a. b. c.
d. e. f.
g. h. i.
j. k. l.
m. n.
Objective of the following problems: To find the exact value any one of the six trigonometric functions of the sum or difference of two angles when the angles are given in terms of an inverse trigonometric function of a number.
2. Find the exact value of the following.
a.
b.
c.
Additional problems available in the textbook: Page 587 … 7 - 40. Examples 1 – 6 starting on page 580.
Solutions:
The cosine of or : Back to Problem 1.
NOTE: Since , then we will need the formula
from our formula sheet.
=
= = …… (a)
The sine of or : Back to Problem 1.
NOTE: Since , then we will need the formula
from our formula sheet.
=
= = …… (b)
The tangent of or : Back to Problem 1.
NOTE: Since , then we will need the formula
from our formula sheet.
= =
= = = =
= = = =
…… (c)
NOTE: . Thus, =
.
NOTE: . Thus, .
The cosine of or : Back to Problem 1.
NOTE: Since , then we will need the formula
from our formula sheet.
=
= = …… (d)
The sine of or : Back to Problem 1.
NOTE: Since , then we will need the formula
from our formula sheet.
=
= = …… (e)
The tangent of or : Back to Problem 1.
NOTE: Since , then we will need the formula
from our formula sheet.
= =
= = = =
= = = =
= …… (f)
NOTE: . Thus, =
.
NOTE: . Thus, .
1a. Back to Problem 1.
NOTE: The angle of is the same as the angle of radians. The terminal side of this angle is in the III quadrant.
NOTE: In the third quadrant, cosine is negative, sine is negative, and tangent is positive.
NOTE: The reference angle of is .
=
NOTE: by (a) above.
=
=
= =
NOTE: by (b) above.
=
=
=
NOTE: by (c) above.
=
Answer: ,
,
,
1b. Back to Problem 1.
NOTE: The angle of radians is the same as the angle of . The terminal side of this angle is in the IV quadrant.
NOTE: In the fourth quadrant, cosine is positive, sine is negative, and tangent is negative.
NOTE: The reference angle of is .
=
NOTE: by (d) above.
=
=
=
NOTE: by (e) above.
=
=
=
NOTE: by (f) above.
=
Answer: ,
,
,
1c. Back to Problem 1.
NOTE: The angle of radians is the same as the angle of . The terminal side of this angle is in the III quadrant.
NOTE: In the third quadrant, cosine is negative, sine is negative, and tangent is positive.
NOTE: The reference angle of is .
= =
NOTE: by (d) above.
=
=
=
NOTE: by (e) above.
=
=
=
NOTE: by (f) above.
=
Answer: ,
,
,
1d. Back to Problem 1.
NOTE: The angle of radians is the same as the angle of . The terminal side of this angle is in the IV quadrant.
NOTE: In the fourth quadrant, cosine is positive, sine is negative, and tangent is negative.
NOTE: The reference angle of is .
=
NOTE: by (a) above.
=
=
= =
NOTE: by (b) above.
=
=
=
NOTE: by (c) above.
=
Answer: ,
,
,
1e. Back to Problem 1.
NOTE: The angle of is the same as the angle of radians. The terminal side of this angle is in the I quadrant.
NOTE: In the first quadrant, cosine is positive, sine is positive, and tangent is positive.
NOTE: The reference angle of is .
=
NOTE: by (d) above.
=
=
=
NOTE: by (e) above.
=
=
=
NOTE: by (f) above.
=
Answer: ,
,
,
1f. Back to Problem 1.
NOTE: The angle of is the same as the angle of radians. The terminal side of this angle is in the III quadrant.
NOTE: In the third quadrant, cosine is negative, sine is negative, and tangent is positive.
NOTE: The reference angle of is .
=
NOTE: by (a) above.
=
=
= =
NOTE: by (b) above.
=
=
=
NOTE: by (c) above.
=
Answer: ,
,
,
1g. Back to Problem 1.
NOTE: The angle of radians is the same as the angle of . The terminal side of this angle is in the II quadrant.
NOTE: In the second quadrant, cosine is negative, sine is positive, and tangent is negative.
NOTE: The reference angle of is .
=
NOTE: by (a) above.
=
=
=
NOTE: by (b) above.
=
=
=
NOTE: by (c) above.
=
Answer: ,
,
,
1h. Back to Problem 1.
NOTE: The angle of radians is the same as the angle of . The terminal side of this angle is in the III quadrant.
NOTE: In the third quadrant, cosine is negative, sine is negative, and tangent is positive.
NOTE: The reference angle of is .
= =
NOTE: by (d) above.
=
=
=
NOTE: by (e) above.
=
=
=
NOTE: by (f) above.
=
Answer: ,
,
,
1i. Back to Problem 1.
NOTE: The angle of is the same as the angle of radians. The terminal side of this angle is in the II quadrant.
NOTE: In the second quadrant, cosine is negative, sine is positive, and tangent is negative.
NOTE: The reference angle of is .
= =
NOTE: by (d) above.
=
=
=
NOTE: by (e) above.
=
=
=
NOTE: by (f) above.
=
= =
Answer: ,
,
,
1j. Back to Problem 1.
NOTE: The angle of radians is the same as the angle of . The terminal side of this angle is in the IV quadrant.
NOTE: In the fourth quadrant, cosine is positive, sine is negative, and tangent is negative.
NOTE: The reference angle of is .
=
NOTE: by (a) above.
=
=
= =
NOTE: by (b) above.
=
=
=
NOTE: by (c) above.
=
Answer: ,
,
,
1k. Back to Problem 1.
NOTE: The angle of is the same as the angle of radians. The terminal side of this angle is in the II quadrant.
NOTE: In the second quadrant, cosine is negative, sine is positive, and tangent is negative.
NOTE: The reference angle of is .
= =
NOTE: by (d) above.
=
=
=
NOTE: by (e) above.
=
=
=
NOTE: by (f) above.
=
Answer: ,
,
,
1l. Back to Problem 1.
NOTE: The angle of radians is the same as the angle of . The terminal side of this angle is in the IV quadrant.
NOTE: In the fourth quadrant, cosine is positive, sine is negative, and tangent is negative.
NOTE: The reference angle of is .
=
NOTE: by (d) above.
=
=
=
NOTE: by (e) above.
=
=
=
NOTE: by (f) above.
=
Answer: ,
,
,
1m. Back to Problem 1.
NOTE: The angle of is the same as the angle of radians. The terminal side of this angle is in the I quadrant.
NOTE: In the first quadrant, cosine is positive, sine is positive, and tangent is positive.
NOTE: The reference angle of is .
=
NOTE: by (a) above.
=
=
=
NOTE: by (b) above.
=
=
=
NOTE: by (c) above.
=
Answer: ,
,
,
1n. Back to Problem 1.
NOTE: The angle of radians is the same as the angle of . The terminal side of this angle is in the II quadrant.
NOTE: In the second quadrant, cosine is negative, sine is positive, and tangent is negative.
NOTE: The reference angle of is .
=
NOTE: by (a) above.
=
=
=
NOTE: by (b) above.
=
=
=
NOTE: by (c) above.
=
Answer: ,
,
,
2a. Back to Problem 2.
Let and let
is an acute angle and is in II and
NOTE: These two statements follow NOTE: These two statements
from the definition of the inverse follow from the definition of the
tangent function. inverse cosine function.
5 4 3
8
NOTE: In the second quadrant, sine is positive.
= =
= =
= Answer:
2b. Back to Problem 2.
Let and let
is an acute angle and is in IV and
NOTE: These two statements follow NOTE: These two statements
from the definition of the inverse follow from the definition of the
sine function. inverse tangent function.
7 3 9 8
NOTE: In the fourth quadrant, cosine is positive and sine is negative.
= =
= =
=
Answer:
2c. Back to Problem 2.
Let and let
is in II and is in IV and
NOTE: These two statements follow NOTE: These two statements
from the definition of the inverse follow from the definition of the
cosine function. inverse sine function.
3 6
2 5
NOTE: In the second quadrant, NOTE: In the fourth quadrant,
tangent is negative. tangent is negative.
= =
= = =
= =
Answer:
Solution to Problems on the Pre-Exam: Back to Page 1.
22. Find the exact value of . (5 pts.) Put a box around your answer.
NOTE: The terminal side of is in the III quadrant.
NOTE: Sine is negative in the third quadrant.
NOTE: The reference angle of is .
=
NOTE: Since , then we will need the formula
from our formula sheet.
=
= =
= = =