NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-3 Practice
Solving Trigonometric Equations
Solve each equation for all values of x.
1. cos x = 3 cos x – 2
2. 2 sin2 x – 1 = 0
3. √cos 𝑥 = 2 cos x – 1
4. 2 sin2 x – 5 sin x + 2 = 0
5. cos x = –1
6. sin3 x – 4 sin x = 0
7. tan2 x = 1
8. 2 sin2 x – cos x = 1
9. sin x cos x –3 cos x = 0
10. cos2 x + sin x + 1 = 0
Find all solutions of each equation on the interval [0, 2π).
11. sec 2 x + tan x = 1
12. 3 tan x – √3 = 0
13. 4 sin2 x – 4 sin x + 1 = 0
14. 4 cos 2 x – 1 = 0
15.
cos3 𝑥
sin 𝑥
= cot x
17. 2 cos x = 1
16. tan x sin2 x = 3 tan x
18. 5 + 2 sin x – 7 = 0
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
19. 4 sin2 x tan x = tan x
20. 2 cos x – √3 = 0
21. cos x = sin x
22. √3 cos x tan x – cos x = 0
23. tan2 x + sec x – 1 = 0
24. 1 + cos x = √3 sin x
25. CIRCLES To find the diameter d of any circle, first inscribe a triangle in the circle. The diameter is then equal to the
ratio of any side of the triangle and the sine of its opposite angle.
a. Suppose the measure of one side of a triangle inscribed in a circle is 20 centimeters. If the measure of the angle in
the triangle opposite this side is 30°, what is the length of the diameter of the circle?
b. Suppose a circle with a diameter of 12.4 inches circumscribes a triangle with one side of the triangle measuring
4.6 inches. What is the measure of the angle in the triangle opposite this side?
26. AVIATION An airplane takes off from the ground and reaches a height of 500 feet after flying 2 miles. Given the
formula H = d tan θ, where H is the height of the plane and d is the distance (along the ground) the plane has flown,
find the angle of ascent θ at which the plane took off.
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27. DARTS Suppose a dart is thrown at a dartboard 18 feet away. The distance to the dartboard r is given by r = 𝑣0 2
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sin 2θ, where 𝑣0 is the initial velocity and θ is the angle of elevation.
a. If the dart is thrown with a velocity of 60 feet per second, neglecting air resistance, find the dart’s minimum angle of
elevation θ.
b. Find θ if the initial velocity remains the same, but the distance to the dartboard is 30 feet.