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Skill 30: Finding Values of Trigonometric Ratios
For each of the angles, θ, described below, draw the angle in standard position, find the measure of the
reference angle, a positive coterminal angle, and a negative coterminal angle.
θ = –473°
θ=
13
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For each of the angles, θ, described below, determine the quadrant or axis in/on which the terminal side of the
angle lies. If no such angle exists, state why. Completely explain your reasoning.
cos θ < 0 and tan θ < 0
cot θ is undefined and cos θ < 0
The terminal side of an angle, α, passes through the points given below. Find the exact value of each of the six
trigonometric ratios of α.
α passes through (5, –3)
α passes through (–3, –2)
Suppose that sin    3 and tan   0 . Use this information to answer the following questions.
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a. In which quadrant does the angle α terminate? Explain your reasoning.
b. Draw and label the reference triangle for α and find the exact value of the other 5 trigonometric
ratios of α.
Suppose that sec    3 and tan   0 . Use this information to answer the following questions.
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a. In which quadrant does the angle θ terminate? Explain your reasoning.
b. Draw and label the reference triangle for θ and find the exact value of the other 5 trigonometric
ratios of θ.
Suppose that csc   7 and cot   0 . Use this information to answer the following questions.
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a. In which quadrant does the angle α terminate? Explain your reasoning.
b. Draw and label the reference triangle for α and find the exact value of the other 5 trigonometric
ratios of α.
The terminal side of an angle β lies in quadrant II along the line 2x + 5y = 0. Order the values sin  , tan  ,
and csc  from least to greatest.
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Suppose tan    and sec  0 . Find the values of csc θ and cos θ.
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Pre-AP Calculus Skill 30 Independent Practice
The terminal side of an angle passes through the given ordered pairs below. For each angle, draw and label the
reference triangle and find the values of the six trigonometric ratios. Show your work.
1. (–12, –5)
2. (7, –24)
3.  3 , 1


Given the constraints on the angles below, determine in which quadrant or on which axis the terminal side of
the angle lies. If no such angle exists, state so. Completely explain your reasoning.
4. sin θ > 0 and tan θ < 0
5. sec θ > 0 and cot θ < 0
6. sin θ < 0 and cos θ > 0
7. cos θ > 0 and sec θ is undefined
8. If tan    15 and sin θ < 0, find the exact value of the six trigonometric ratios.
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9. If sec  2 and sin θ > 0, find the exact value of the six trigonometric ratios.
10. If sin   3 and sec θ > 0, find the exact value of the six trigonometric ratios.
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11. If the terminal side of θ is in Quadrant IV and lies on the line 4x + 3y = 0, find the exact value of the
six trigonometric ratios.
12. If cos    4 and θ terminates in Quadrant IV, find the exact value of the six trigonometric ratios.
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