Trigonometry

Trigonometric equations

Trigonometric equations usually have more than one solution in an interval. München names them for the T-Kurs entrance test, so learn to find every solution, not just the first.

München Named for the T-Kurs maths test at Studienkolleg München. Source

The rules

• sin x = a: if x₁ is one solution, 180° − x₁ is the second in [0°; 360°).
• cos x = a: if x₁ is one solution, 360° − x₁ is the second.
• tan x = a: solutions repeat every 180°.
• Add multiples of 360° (or 180° for tan) to find solutions in other intervals.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Solve cos(x) = √2/2 for the smallest solution with 0° ≤ x < 360°.

   • a45
   • b315
   • c135
   • d15
   Show solution

   Answer: 45

   Use the standard-angle values of cos to find where cos(x) = √2/2.

   x = 45° or x = 315°; the smallest is 45°.

2. 2. Solve cos(x) = 1 for the smallest solution with 0° ≤ x < 360°.

   • a-30
   • b90
   • c0
   • d180
   Show solution

   Answer: 0

   Use the standard-angle values of cos to find where cos(x) = 1.

   x = 0°; the smallest is 0°.

3. 3. How many solutions does cos(2x) = 0 have for 0° ≤ x < 360°?

   • a2
   • b4
   • c3
   • d5
   Show solution

   Answer: 4

   Factor (never divide by sin x or cos x, you lose solutions), or substitute u = 2x and run u over twice the interval.

   2x = 90°, 270°, 450°, 630° ⇒ x = 45°, 135°, 225°, 315°.

4. 4. Find the sum of all solutions to sin(x) = −√2/2 for 0° ≤ x < 360°.

   • a630
   • b540
   • c315
   • d225
   Show solution

   Answer: 540

   sin(x) = k has, in general, two solutions in one full turn (except at the extremes k = ±1).

   Solutions: x = 225°, 315°. Sum = 540°.

5. 5. Solve sin(x) = 0 for the smallest solution with 0° ≤ x < 360°.

   • a0
   • b90
   • c-30
   • d180
   Show solution

   Answer: 0

   Use the standard-angle values of sin to find where sin(x) = 0.

   x = 0° or x = 180°; the smallest is 0°.

6. 6. How many solutions does sin(2x) = 1/2 have for 0° ≤ x < 360°?

   • a4
   • b3
   • c2
   • d5
   Show solution

   Answer: 4

   Factor (never divide by sin x or cos x, you lose solutions), or substitute u = 2x and run u over twice the interval.

   2x = 30°, 150°, 390°, 510° ⇒ x = 15°, 75°, 195°, 255°.

Common mistakes

• Giving only the calculator solution and missing the second one.
• Using the sine symmetry for cosine.
• Answering in radians when the interval is given in degrees.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs