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General Solutions

A free Trigonometry lesson from the “Inverse Trig and Equations” unit, with a worked example and practice problems including step-by-step solutions.

A single trig equation has infinitely many solutions because sine and cosine repeat every 2pi and tangent repeats every pi. The general solution captures every one of them by writing a base angle plus a period term: add 2pi*k for sin(x)=k or cos(x)=k, and add pi*k for tan(x)=k, where k is any integer.

What you'll learn

Why it matters: Oscillations like AC voltage, sound pressure, and tides return to the same value over and over, so engineers need the full general solution, not just one angle, to describe every moment it occurs.

Worked example

Problem. Write the general solution to sin(x) = 1/2.

  1. The base angles in [0, 2pi) are pi/6 and 5pi/6.
  2. Sine has period 2pi, so add 2pi*k to each.
  3. x = pi/6 + 2pi*k or x = 5pi/6 + 2pi*k.

Answer: x = pi/6 + 2pi*k or x = 5pi/6 + 2pi*k

Practice problems

1. Which period term is added to every solution of a tangent equation tan(x) = k?

Choices: + pi*k · + 2pi*k · + pi/2*k · + k

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which period term is added to every solution of a tangent equation tan(x) = k?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Tangent repeats every pi.
  4. So the period term is pi*k.
  5. Sine and cosine instead use 2pi*k.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: + pi*k

2. In the general solution x = pi/3 + 2pi*k, what does the letter k represent?

Choices: any integer · a positive integer only · the angle in degrees · the period of cosine

Show solution
  1. Warm-up: First identify exactly what the question is asking: In the general solution x = pi/3 + 2pi*k, what does the letter k represent?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. k indexes each full revolution.
  4. It can be negative, zero, or positive.
  5. So k is any integer.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: any integer

3. Write the general solution to tan(x) = 1.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Write the general solution to tan(x) = 1.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The base angle is arctan(1) = pi/4.
  4. Tangent has period pi.
  5. x = pi/4 + pi*k.
  6. Check the result by substituting or estimating: the response should match x = pi/4 + pi*k and make sense in the original problem.

Answer: x = pi/4 + pi*k

4. Write the general solution to cos(x) = 1/2 using a plus/minus base angle.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Write the general solution to cos(x) = 1/2 using a plus/minus base angle.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. arccos(1/2) = pi/3.
  4. Cosine is even, so both +pi/3 and -pi/3 work.
  5. x = +/-pi/3 + 2pi*k.
  6. Check the result by substituting or estimating: the response should match x = +/-pi/3 + 2pi*k and make sense in the original problem.

Answer: x = +/-pi/3 + 2pi*k

5. Write the general solution to sin(x) = sqrt(3)/2.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Write the general solution to sin(x) = sqrt(3)/2.
  2. For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
  3. The base angles are pi/3 and 2pi/3.
  4. Sine has period 2pi, so add 2pi*k to each.
  5. x = pi/3 + 2pi*k or x = 2pi/3 + 2pi*k.
  6. Check the result by substituting or estimating: the response should match x = pi/3 + 2pi*k or x = 2pi/3 + 2pi*k and make sense in the original problem.

Answer: x = pi/3 + 2pi*k or x = 2pi/3 + 2pi*k

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