1.1 radian and degree measure -...

Chapter One Review Guide Trigonometry

1.1 Radian and Degree Measure

1. Determine the quadrant in which the terminal side of an angle of 395° lies.

395° is co-terminal to 35°, so quadrant I.

2. Find an angle that is coterminal to θ = −7π

12.

7 7 242

12 12 12

31 17 or

12 12

π π ππ

π π

− ± = − ±

= −

3. Find an angle that is complementary to θ =π

6.

6 2

2 6

3

6 6

2

6

3

x

x

x

x

x

π π

π π

π π

π

π

+ =

= −

= −

=

=

4. Find an angle that is supplementary to θ =2π

7.

2

7

2

7

7 2

7 7

5

7

x

x

x

x

ππ

ππ

π π

π

+ =

= −

= −

=

5. Convert to radians: 290°

29

290 5.06180 18

π π° = ≈

°i

6. Convert to degrees: 5π

8

5 180 900

112.58 8

π

π

° °= = °i


1.2 Trigonometric Functions: The Unit Circle

7. Find the point (x, y) on the unit circle that corresponds to the real number t = −π

4.

Use your unit circle, find −π

4, and the point is

2 2,

2 2

−

8. Give the exact value (if defined) of cos −π

6

.

Use your unit circle, find 6

π− , and look at the x-coordinate, which is

3

2.

9. Give the exact value (if defined) of tan15π

6

.

15

2.56

ππ= , so that’s co-terminal to

2

π. Tangent is undefined there.

10. Evaluate: csc(5.23).

Calculator: 1/sin(5.23) = -1.1507

11. Evaluate: sec(4.6).

Calculator: 1/cos(4.6) = -8.9164

1.3 Right Triangle Trigonometry

12. A right triangle has an acute angle θ such that sinθ = 7

9. Find tan.

Draw a picture: Use the Pythagorean Theorem to find the 3rd

side:

2 2 2

2

2

7 9

49 81

32

32

4 2

b

b

b

b

b

+ =

+ =

=

=

=

7 7 2

tan84 2

θ = =

13. Evaluate: cot 15°

Calculator: 1/tan 15° = 3.7321

14. Evaluate: sec 1.2

Calculator: 1/cos 1.2 = 2.7597

θ 7

9


15. Find θ such that 0 ≤ θ <π

2 and cscθ = 1.4736.

If cscθ = 1.4736, then sinθ = 1/1.4736.

To find θ, we take the arcsin(1/1.4736) = 0.7459

16. Find x for the triangle shown at the right.

sin 2615

15sin 26

6.5756

x

x

x

° =

° =

≈

1.4 Trigonometric Functions of Any Angle 17. Find the 6 trigonometric functions for the angle whose terminal side contains the

point (8, -7). (EXACT VALUES!)

Draw a triangle: Use the Pythagorean Theorem:

( )22 2

2

2

8 7

64 49

113

113

c

c

c

c

+ − =

+ =

=

=

7 7 113sin

113113

8 8 113cos

113113

7tan

8

θ

θ

θ

−= = −

= =

= −

113csc

7

113sec

8

8cot

7

θ

θ

θ

= −

=

= −

18. Given sinθ = 7

13 and tanθ < 0, find tanθ.

If sin>0 and tan<0, this is in Quadrant II. Draw a picture:

Use the Pythagorean Theorem:

2 2 2

2

2

7 13

49 169

120

120 2 30

a

a

a

a

+ =

+ =

=

= − = −

7 7 30

tan602 30

θ = − = −

x 15

26°

8

-7

13 7


19. Find the reference angle for θ =17π

15.

17 2

15 15

π ππ− =

20. Find the exact value of sec7π

4.

7 2cos

4 2

7 2 2 2sec 2

4 22

π

π

=

= = =

21. Find two values of θ (0 ≤ θ < 360°) that satisfy cotθ = -0.2679.

Calculator: tan-1

(1/(-0.2679)) = -75°

Tangent repeats every 180°, so to get two values between 0° and 360°, add 180°

twice.

-75° + 180° = 105° and 105° + 180° = 285°

1.5 Graphs of Sine and Cosine Functions

22. Given the function f (x) = −3cos1

2x + π

a.) what is the amplitude?

3

b.) what is the period?

2

2 2 41

2

ππ π= =i

c.) is the graph being shifted left or right?

Left 2 21

2

ππ π= =i

d.) is the graph being shifted up or down?

No

e.) graph 2 periods of the function.


23. Given the function f (x) = sin 3x −π

4

a.) what is the amplitude?

1

b.) what is the period?

2

3

π

c.) is the graph being shifted left or right?

Right 14

3 4 3 12

ππ π

= =i

d.) is the graph being shifted up or down?

No

e.) graph 2 periods of the function.

1.6 Graphs of Other Trigonometric Functions

24. Given the function f (x) = − tan 3x( )

a.) what is the period?

3

π

b.) what are the asymptotes?

32

6

x

x

π

π

= ±

= ±

c.) graph 2 periods of the function.

25. Given the equation y = sec (2x)

a.) what is the period?

2

2

ππ=

b.) what are the asymptotes?

3 5

, , ,4 4 4

etcπ π π

c.) graph 2 periods of the function.

1.7 Inverse Trigonometric Functions

26. Evaluate: arctan (-1)

Calculator: tan-1

(-1) = -0.7854 or use your unit circle, tan is –1 at 4

π− .


27. Evaluate: arccos2

2

Calculator: cos-1 2

2= 0.7854 or use your unit circle, cos is

2

2 at

4

π.

28. Evaluate: cos arctan −3

10

Calculator: 0.9578 or draw a picture for an exact answer.

10 10 109

cos109109

θ = =

29. Write an algebraic expression for tan[arcsin(x)].

Draw a picture. Do the Pyth. Thm.

2 2 2

2 2

2

1

1

1

a x

a x

a x

+ =

= −

= −

2

22

1tan

11

x x xx

xx

−= =

−−

1.8 Applications and Models

30. A ladder is leaning against the side of a house. The base of the ladder is 5 feet

from the wall and makes an angle of 39° with the ground. Find the length of the

ladder.

5cos39

cos39 5

5

cos39

6.43

l

l

l

l

° =

° =

=°

=

The ladder is 6.43 feet long

-3 θ

10

109

θ

x 1

21 x−

39°

5 feet

length


31. A ship leaves port and travels 30 nautical miles due north, then changes course to

N 15° E and travels for 10 nautical miles. Find the ship’s bearing from the port of

departure.

Draw a picture like the one at the right.

Solve to find out how long the orange line is.

sin1510 nm

10 nm sin15

2.5882

x

x

x

° =

° =

=

i 2.5882 nautical miles.

Now, use that to solve for the 3rd

side in the yellow triangle.

2 2 2

2

2

2.5882 10

6.6987 100

93.3013

9.6593

b

b

b

b

+ =

+ =

=

=

So, in the large triangle, you have a side length of

39.6593, and another side of 2.5882, and you need

to find x. Use tangent.

2.5882tan

39.6593

2.5882arctan

39.6593

3.7

x

x

x

=

=

= °

So the ship headed 3.7° East of North.

15°

75°

30 nm

θ

10 nm

1.1 radian and degree measure -...

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