Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Trigonometry - Solving equations.
Test Yourself 1.


 

Make sure you have read the page titled Trigonometric Equations - summary of strategies.

The questions below focus on the formats:
1. Trig ratio and a constant.
2. Squared trig ratio and a constant.
3. Sin and cos terms but no constant.
4. Trig ratios in a quadratic format.
5. Using angles of any magnitude.
6. Number of solutions.

 

=
1. Trig ratio and a constant.

(solve all equations for
0 ≤ angle ≤ 360°) unless otherwise inducted.

 1. .
Answer.150°, 330°.		 2. sec x = 2.
Answer.60°, 300°.
3. Solve .
Answer.240°, 300°.		 4. Solve sin 2x = 1.
Answer.45°, 225°.
  5. Solve
for 0 ≤ x ≤ 2π.
Answer.5π/12, 7π/12, 17π/12, 19π/12.		 6. sin (x - 25°) = 0.5.
Answer.55°, 175°.
  7. .
Answer.60°, 120°.		 8. 2cos θ = -1.
Answer.120°, 240°.
  9. tan θ = -0.8391
for -180° ≤ θ ≤ 180°.
Answer.140°, -40°.		 10. cos (x - 35°) = -0.5
for -180° ≤ θ ≤ 180°.
Answer.155°, -85°.
2. Squared trig ratio and a constant. 11. 3tan2 β - 1 = 0
Answer.β = 30°, 150°, 210°, 330°.		 12. 4cos2 x - 1 = 0
for -180° ≤ θ ≤ 180°.
Answer.x = -120°, -60°, 60°, 120°.
  13. .
Answer.x = 15°, 165°, 195°, 330°
195°, 255°, 285°, 345°.		 14. sin2 x = 1
Answer.x = 90°, 270°.
  15. .
Answer.x = 30°, 60°, 120°, 150°
210°, 240°, 300°, 330°.		 16. 4cot2 (x + 30°) - 12 = 0.
Answer.x = 0°, 120°, 180°, 270°, 360°.
3. Sin and cos terms but no constant. 17. .
Answer.α = 30°, 210°.		 18.
Answer.x = 0°, 120°, 180°, 300°.
  19. .
Answer.26° 34', 206° 34.		 20. 2sinθ cosθ = sin θ.
Answer.0°, 63° 26', 180°, 246° 26', 360°.
4. Trig ratios in a quadratic format. 21. Find the exact solutions for θ given the equation

2sin2 θ - sin θ - 1 = 0

(0 ≤ θ ≤ 360°).
Answer.180°, 210°, 330°.		 22. Solve for θ:

2cos2θ + 3sinθ cosθ + sin2θ = 0

(0 ≤ θ ≤ 360°).
Answer.135°, 315°, 153°26', 333°26'.
  23. (i) Show that

(cosec2 A - 1)sin2 A = cos2 A

(ii) Hence or otherwise solve

(cosec2A - 1)sin2A = 0.75

for -π ≤ A ≤ π

Answer.π/6, 5π/6, 7π/6, 11π/6.		 24. Solve for θ:

8cos2x = 2sinx + 7

(0 ≤ θ ≤ 360°)
(Answer if necessary to the nearest minute).


Answer.210°, 330°, 14°29' or 165°31'.
  25. Solve the equation

cos2x + sinx + 1 = 0

(0 ≤ θ ≤ 360°).
Answer.0°, 90°, 180°.		 26. Solve the equation

2 cos2J - 5sin J + 1 = 0

for 0 ≤ J ≤ 360°.
Answer.270°.
5. Complementary or supplementary angles.

(0 ≤ θ ≤ 360°).

 27. Solve:

Answer.θ = 40°.		 28. cot (90° - 5x) = tan (3x + 36°)
Answer.x = 18°.
  29.
sin (180° - 3x) = cos (5x-22°).
Answer.x = 14°.		 30.
Answer.x = 45°, 224°, 90°, 270°.
6. Number of solutions. 31. How many solutions are there for the equation

(2cos 2x - 1)(sin2 x + 1) = 0
0 ≤ x ≤ 2π.

Answer.4 solutions. 		32. How many solutions are there between 0 and 2π for the equation (sin x - 1)(tan x + 2) = 0
Answer.2 solutions.
  33. How many solutions are there for the equation

(cos θ + 1)(cosec θ - 1) = 0.
0 ≤ x ≤ 2π.

Answer.1 solution
(θ = π/2 but ≠ π).