Download Unit Circle Trigonometry - Assignment | MATH 1330 and more Assignments Pre-Calculus in PDF only on Docsity!
Math 1330, Precalculus
Sketch each of the following angles in standard position. (Do not use a protractor; just draw a quick sketch of each angle.)
1. (a) 30 D^ (b) − 135 D^ (c) 300 D 2. (a) 120 D^ (b) − 60 D^ (c) 210 D 3. (a) 3
π (b) 7 4
π (c) 5 6
π −
4. (a) 4
π − (b) 4 3
π (c) 11 6
π
5. (a) 90 D^ (b) − π (c) 450 D 6. (a) 270 D^ (b) 180 D^ (c) − 4 π 7. (a) − 240 D^ (b) 13 6
π (c) − 510 D
8. (a) − 315 D^ (b) 9 4
π − (c) 1020 D
Find three angles, one negative and two positive, that are coterminal with each angle below.
9. (a) 50 D^ (b) − 200 D 10. (a) 300 D^ (b) − 830 D 11. (a) 2 5
π (b) 7 2
π −
12. (a) 8 3
π (b) 4 9
π −
Answer the following.
13. List four quadrantal angles in degree measure, where each angle θ satisfies the condition 0 D^ ≤ θ< 360 D^. 14. List four quadrantal angles in radian measure, where each angle θ satisfies the condition 5 2 2
π π ≤ θ<.
15. List four quadrantal angles in radian measure, where each angle θ satisfies the condition 5 13 4 4
π π < θ≤.
16. List four quadrantal angles in degree measure, where each angle θ satisfies the condition 800 D^ < θ≤ 1160 D^.
Sketch each of the following angles in standard position and then specify the reference angle or reference number.
17. (a) 240 D^ (b) − 30 D^ (c) 60 D 18. (a) 315 D^ (b) 150 D^ (c) − 120 D 19. (a) 7 6
π (b) 5 3
π (c) 7 4
π −
20. (a) 3
π − (b) 5 4
π (c) 5 6
π −
21. (a) 840 D^ (b) 37 6
π − (c) − 660 D
22. (a) 11 3
π (b) − 780 D^ (c) 11 4
π −
The exercises below are helpful in creating a comprehensive diagram of the unit circle. Answer the following.
23. Using the following unit circle, draw and then label the terminal side of all multiples of 2
π from
0 to 2 π radians. Write all labels in simplest form.
24. Using the following unit circle, draw and then label the terminal side of all multiples of 4
π (^) from
0 to 2 π radians. Write all labels in simplest form.
x
y
-1^1
1
0
x
y
-1^1
1
0
Math 1330, Precalculus
x
y
1
1
25. Using the following unit circle, draw and then label the terminal side of all multiples of 3
π from
0 to 2 π radians. Write all labels in simplest form.
26. Using the following unit circle, draw and then label the terminal side of all multiples of 6
π from
0 to 2 π radians. Write all labels in simplest form.
27. Use the information from numbers 23-26 to label all the special angles on the unit circle in radians. 28. Label all the special angles on the unit circle in degrees.
Name the quadrant in which the given conditions are satisfied.
29. sin ( θ) > 0, cos ( θ)< 0 2
30. sin ( θ) < 0, sec ( θ)> 0 4
31. cot ( θ ) > 0, sec ( θ)< 0 3
32. csc ( θ) > 0, cot ( θ)> 0 1
33. tan ( θ) < 0, csc ( θ)< 0 4
34. csc ( θ) < 0, tan ( θ)> 0 3
Fill in each blank with < , > , or =.
35. sin 40( D^ ) _____ sin 140( D)
36. cos 20( D^ ) _____ cos 160( D)
37. tan 310( D^ ) _____ tan 50( D)
38. sin 195( D^ ) _____ sin 15( D)
39. cos 355( D^ ) _____ cos 185( D)
40. tan 110( D^ ) _____ tan 290( D)
x
y
1
1
x
y
-1^1
1
0
x
y
-1^1
1
0
Math 1330, Precalculus
(a) Find the missing side measures in each of the diagrams above.
(b) Use right triangle trigonometric ratios to find the following, using Diagram 1:
sin 45 ( D^ )=_____ csc 45( D)=_____
cos 45 ( D^ )=_____ sec 60( D)=_____
tan 45 ( D^ )=_____ cot 60( D)=_____
(c) Repeat part (b), using Diagram 2.
(d) Use the unit circle to find the trigonometric ratios listed in part (b).
(e) Examine the answers in parts (b) through (d). What do you notice?
The following two diagrams can be used to quickly evaluate the trigonometric functions of any angle having a reference angle of 30 o, 45 o, or 60o. Use right trigonometric ratios along with the concept of reference angles, to evaluate the following. (Remember that when converting degrees to radians,
30 , 45 , 60 6 4 3
π π π D (^) = D (^) = D (^) = .)
67. (a) sin 240( D^ ) (b) tan 135( D)
68. (a) cos 330( D^ ) (b) csc ( − 225 D)
69. (a) cos 4
⎛ π⎞ ⎜ − ⎟ ⎝ ⎠
(b) 11 sec 6
⎛ π⎞ ⎜ ⎟ ⎝ ⎠
70. (a) 2 sin 3
⎛ π⎞ ⎜ ⎟ ⎝ ⎠
(b) 7 cot 6
⎛ π⎞ ⎜ − ⎟ ⎝ ⎠
Use either the unit circle or the right triangle method from numbers 65-70 to evaluate the following. (Note: The right triangle method can not be used for quadrantal angles.) If a value is undefined, state “Undefined.”
71. (a) tan 30( D^ ) (b) sin ( − 135 D)
72. (a) cos 180( D^ ) (b) csc ( − 60 D)
73. (a) csc ( − 150 D^ ) (b) sin 270( D)
74. (a) sec 225( D^ ) (b) tan ( − 240 D)
75. (a) cot ( − 450 D^ ) (b) cos 495( D)
76. (a) sin ( − 210 D^ ) (b) cot ( − 420 D)
77. (a) csc( π ) (b) cos 2
3
⎛ π⎞ ⎜⎝ ⎟⎠
78. (a) sin 3
⎛ (^) − π⎞ ⎜⎝ ⎟⎠ (b)^ cot 5 4
⎛ π⎞ ⎜⎝ ⎟⎠
79. (a) sec 6
⎛ (^) − π⎞ ⎜⎝ ⎟⎠ (b)^ tan 3 4
⎛ π⎞ ⎜⎝ ⎟⎠
80. (a) csc 11 4
⎛ π⎞ ⎜⎝ ⎟⎠ (b)^ sec 3 2
⎛ (^) − π⎞ ⎜⎝ ⎟⎠
81. (a) cot 10 3
⎛ (^) − π⎞ ⎜⎝ ⎟⎠ (b)^ sec 7 4
⎛ π⎞ ⎜⎝ ⎟⎠
82. (a) tan ( − 5 π) (b) cos 11
2
⎛ (^) − π⎞ ⎜⎝ ⎟⎠
Use a calculator to evaluate the following to the nearest ten-thousandth. Make sure that your calculator is in the appropriate mode (degrees or radians). Note: Be careful when evaluating the reciprocal trigonometric functions.
For example, when evaluating csc( θ ) on your calculator, use the
identity ( )
1 csc θ = (^) sin θ. Do NOT use the calculator key labeled
sin−^1 ( θ) ; this represents the inverse sine function, which will be discussed in Section 5.4.
83. (a) sin 37( D^ ) (b) tan ( − 218 D)
84. (a) tan 350( D^ ) (b) cos ( − 84 D)
85. (a) csc 191( D^ ) (b) cot 21( D)
60 o
30 o
45 o
45 o
2
Diagram 1 (^) Diagram 2
45 o
45 o
45 o
45 o
Math 1330, Precalculus
86. (a) cot 310( D^ ) (b) sec 73( D)
87. (a) cos 5
(b)
csc 7
88. (a)
sin 9
(b) cot 4.7( π)
89. (a) tan ( −4.5 ) (b) sec 3( )
90. (a) csc ( −0.457 ) (b) tan 9.4( )
