Module 2: Practice Problems
Instructions: Work each problem, then reveal the solution to check your work.
1 Unit Circle Coordinates
What are the coordinates of the point on the unit circle at θ = 3π/2?
Solution
At 3π/2 (270°), the point is (0, −1). So cos(3π/2) = 0, sin(3π/2) = −1.
2 Special Angle Values
Find sin(π/3) and cos(π/3).
Solution
sin(π/3) = √3/2, cos(π/3) = 1/2.
3 Reference Angles
Find cos(5π/3).
Solution
5π/3 is in QIV, reference angle = 2π − 5π/3 = π/3. cos(π/3) = 1/2. In QIV cos is positive. cos(5π/3) = 1/2.
4 Pythagorean Identity
If sin θ = 7/25 and θ is in QI, find cos θ.
Solution
cos²θ = 1 − 49/625 = 576/625. cos θ = 24/25 (positive in QI).
5 Finding tan
Find tan(3π/4).
Solution
3π/4 is in QII, ref angle π/4. tan(π/4) = 1. In QII tan is negative. tan(3π/4) = −1.
6 Right Triangle
A right triangle has hypotenuse 13 and one leg 5. Find all six trig functions of the angle opposite the side 5.
Solution
Other leg = √(169−25) = 12. sin=5/13, cos=12/13, tan=5/12, csc=13/5, sec=13/12, cot=12/5.
7 Solving a Right Triangle
Right triangle with angle A = 50° and adjacent side b = 8. Find the opposite side a.
Solution
tan 50° = a/8, a = 8 tan 50° ≈ 8(1.1918) ≈ 9.53.
8 Angle of Elevation
From 80 m away, the angle of elevation to a building top is 35°. Find the height.
Solution
tan 35° = h/80, h = 80 tan 35° ≈ 80(0.7002) ≈ 56.0 m.
9 Cofunction
Express sin 75° as a cosine function.
Solution
sin 75° = cos 15° (since 75 + 15 = 90).
10 Sign Determination
Determine the sign of sin(200°), cos(200°), and tan(200°).
Solution
200° is in QIII. sin: negative, cos: negative, tan: positive (negative/negative).