Solve sin x = c, cos x = c and tan x = c

About this lesson

Precalculus › Trigonometric identities and equations

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Solve sin x = c, cos x = c and tan x = c

Warm-up

Three quick questions from earlier lessons.

Discover

How many angles xx with 0≤x<2π0 \le x < 2\pi have sin⁡x=12\sin x = \dfrac{1}{2}?

  1. Two
  2. One

Key idea

On [0,2π)[0, 2\pi), an equation like sin⁡x=c\sin x = c usually has two solutions.

Find the reference angle α\alpha (from the absolute value of cc). Then use the quadrants where the function has the sign of cc: sine in I, II (positive) or III, IV (negative); cosine in I, IV or II, III; tangent in I, III or II, IV. The solutions are α\alpha, π−α\pi - \alpha, π+α\pi + \alpha or 2π−α2\pi - \alpha, depending on the quadrant.

Worked example

Solve cos⁡x=−22\cos x = -\dfrac{\sqrt{2}}{2} for 0≤x<2π0 \le x < 2\pi.

Reference angle: π4\dfrac{\pi}{4}
cos(π/4) = √2/2.
x=π−π4x = \pi - \dfrac{\pi}{4} or x=π+π4x = \pi + \dfrac{\pi}{4}
Cosine is negative in quadrants II and III.
x=3π4x = \dfrac{3\pi}{4} or x=5π4x = \dfrac{5\pi}{4}
Simplify.

Your turn, guided

Your turn: sin⁡x=−12\sin x = -\dfrac{1}{2}.

Reference angle π6\dfrac{\pi}{6}: x=π+π6x = \pi + \dfrac{\pi}{6} or x=2π−π6x = 2\pi - \dfrac{\pi}{6}
sin(π/6) = 1/2; sine is negative in quadrants III and IV.
x=7π6x = \dfrac{7\pi}{6} or x=11π6x = \dfrac{11\pi}{6}
Type the quadrant III solution.

Practice ★

  1. Solve sin⁡x=12\sin x = \frac{1}{2} for 0≤x<2π0 \le x < 2\pi.
  2. Solve sin⁡x=22\sin x = \frac{\sqrt{2}}{2} for 0≤x<2π0 \le x < 2\pi.
  3. Solve sin⁡x=32\sin x = \frac{\sqrt{3}}{2} for 0≤x<2π0 \le x < 2\pi.

Watch out!

Practice ★★

  1. Solve cos⁡x=−22\cos x = - \frac{\sqrt{2}}{2} for 0≤x<2π0 \le x < 2\pi.
  2. Solve cos⁡x=−12\cos x = - \frac{1}{2} for 0≤x<2π0 \le x < 2\pi.
  1. Solve tan⁡x=3\tan x = \sqrt{3} for 0≤x<2π0 \le x < 2\pi.
  2. Solve tan⁡x=33\tan x = \frac{\sqrt{3}}{3} for 0≤x<2π0 \le x < 2\pi.

Exit ticket

Three questions to check you have got it.

  1. Solve cos⁡x=−22\cos x = - \frac{\sqrt{2}}{2} for 0≤x<2π0 \le x < 2\pi.
  2. Solve cos⁡x=−12\cos x = - \frac{1}{2} for 0≤x<2π0 \le x < 2\pi.
  3. Solve cos⁡x=−32\cos x = - \frac{\sqrt{3}}{2} for 0≤x<2π0 \le x < 2\pi.

Summary card

Solving sin x = c, cos x = c, tan x = c

  1. Reference angle from ∣c∣|c|.
  2. The two quadrants with the right sign.
  3. Write both solutions in [0,2π)[0, 2\pi).

Watch out: there are usually two solutions, not one.

End of the lesson

Next lesson: Solve trigonometric equations by factoring Back to the chapter

Answers: Solve sin x = c, cos x = c and tan x = c

A

  1. π6, 5π6\frac{\pi}{6},\ \frac{5 \pi}{6}
  2. π4, 3π4\frac{\pi}{4},\ \frac{3 \pi}{4}
  3. π3, 2π3\frac{\pi}{3},\ \frac{2 \pi}{3}

B

  1. 3π4, 5π4\frac{3 \pi}{4},\ \frac{5 \pi}{4}
  2. 2π3, 4π3\frac{2 \pi}{3},\ \frac{4 \pi}{3}

C

  1. π3, 4π3\frac{\pi}{3},\ \frac{4 \pi}{3}
  2. π6, 7π6\frac{\pi}{6},\ \frac{7 \pi}{6}

D

  1. 3π4, 5π4\frac{3 \pi}{4},\ \frac{5 \pi}{4}
  2. 2π3, 4π3\frac{2 \pi}{3},\ \frac{4 \pi}{3}
  3. 5π6, 7π6\frac{5 \pi}{6},\ \frac{7 \pi}{6}

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