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Basic Sine and Cosine Equations

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

A basic sine or cosine equation like sin(x)=k or cos(x)=k usually has TWO solutions in the interval [0, 2pi), because each horizontal line crosses one full period of the curve twice. To find them, identify the reference angle from the standard value of k, then use the sign of k to choose the two quadrants where the function takes that value. The special cases k = 0, 1, and -1 land at the axis points and may give one or two solutions.

What you'll learn

Why it matters: Anything that cycles — daylight hours through the year, the height of a Ferris-wheel seat, alternating current voltage — is modeled by sine or cosine, so asking "when does the value equal k?" is exactly solving sin(x)=k or cos(x)=k.

Worked example

Problem. Solve cos(x) = -1/2 for all x in [0, 2pi).

  1. The reference angle for cos = 1/2 is pi/3, since cos(pi/3) = 1/2.
  2. Cosine is negative in Quadrants II and III, so use pi - pi/3 and pi + pi/3.
  3. x = 2pi/3 and x = 4pi/3.

Answer: x = 2pi/3, 4pi/3

Practice problems

1. Solve sin(x) = 1 for all x in [0, 2pi).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve sin(x) = 1 for all x in [0, 2pi).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Sine reaches its maximum value of 1 only at the top of the circle.
  4. That occurs at x = pi/2, and nowhere else in [0, 2pi).
  5. Check the result by substituting or estimating: the response should match x = pi/2 and make sense in the original problem.

Answer: x = pi/2

2. Solve cos(x) = 1 for all x in [0, 2pi).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve cos(x) = 1 for all x in [0, 2pi).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Cosine equals its maximum 1 at the rightmost point of the circle.
  4. In [0, 2pi) that happens only at x = 0.
  5. Check the result by substituting or estimating: the response should match x = 0 and make sense in the original problem.

Answer: x = 0

3. Solve sin(x) = 0 for all x in [0, 2pi).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve sin(x) = 0 for all x in [0, 2pi).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Sine is zero where the curve crosses the x-axis.
  4. On [0, 2pi) that occurs at x = 0 and x = pi.
  5. Check the result by substituting or estimating: the response should match x = 0, pi and make sense in the original problem.

Answer: x = 0, pi

4. Solve sin(x) = 1/2 for all x in [0, 2pi).

Show solution
  1. The reference angle for sin = 1/2 is pi/6.
  2. Sine is positive in Quadrants I and II, so use pi/6 and pi - pi/6.
  3. x = pi/6 and x = 5pi/6.

Answer: x = pi/6, 5pi/6

5. Solve cos(x) = sqrt(2)/2 for all x in [0, 2pi).

Show solution
  1. The reference angle for cos = sqrt(2)/2 is pi/4.
  2. Cosine is positive in Quadrants I and IV, so use pi/4 and 2pi - pi/4.
  3. x = pi/4 and x = 7pi/4.

Answer: x = pi/4, 7pi/4

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