Trigonometric identities and equations

Practice sheet

Fresh questions from the whole chapter, to do on paper.

New questions

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  1. sin⁡θ=15\sin\theta = \frac{1}{5} and θ\theta is in quadrant II. Find the exact value of cos⁡θ\cos\theta.
  2. Which expression is equal to 1−cos⁡2xsin⁡x\dfrac{1 - \cos^2 x}{\sin x} (where sin⁡x≠0\sin x \ne 0)?
    1. sin⁡x\sin x
    2. cos⁡x\cos x
    3. 1−cos⁡x1 - \cos x
  3. Which equation is true for every xx?
    1. sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
    2. sin⁡x+cos⁡x=1\sin x + \cos x = 1
    3. sin⁡2x−cos⁡2x=1\sin^2 x - \cos^2 x = 1
    4. tan⁡x=cos⁡xsin⁡x\tan x = \dfrac{\cos x}{\sin x}
  4. Solve sin⁡x=12\sin x = \frac{1}{2} for 0≤x<2π0 \le x < 2\pi.
  5. Solve cos⁡x=−22\cos x = - \frac{\sqrt{2}}{2} for 0≤x<2π0 \le x < 2\pi.
  6. Solve tan⁡x=3\tan x = \sqrt{3} for 0≤x<2π0 \le x < 2\pi.
  7. Solve sin⁡x(sin⁡x−32)=0\sin x\left(\sin x - \frac{\sqrt{3}}{2}\right) = 0 for 0≤x<2π0 \le x < 2\pi.
  8. Solve 3cos⁡(x)+cos⁡(2x)+2=03 \cos{\left(x \right)} + \cos{\left(2 x \right)} + 2 = 0 for 0≤x<2π0 \le x < 2\pi.
  9. Put the steps for solving 2sin⁡2x+sin⁡x−1=02\sin^2 x + \sin x - 1 = 0 in order.
    • Let u=sin⁡xu = \sin x: 2u2+u−1=02u^2 + u - 1 = 0
    • Factor: (2u−1)(u+1)=0(2u - 1)(u + 1) = 0
    • sin⁡x=12\sin x = \dfrac{1}{2} or sin⁡x=−1\sin x = -1
    • List every xx in [0,2π)[0, 2\pi): π6,5π6,3π2\dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}
  10. Solve sin⁡2x=12\sin 2x = \frac{1}{2} for 0≤x<2π0 \le x < 2\pi.
  11. How many solutions does sin⁡(5x)=12\sin(5x) = \dfrac{1}{2} have for 0≤x<2π0 \le x < 2\pi?
  12. Solve cos⁡3x=−1\cos 3x = -1 for 0≤x<2π0 \le x < 2\pi.

Answers

  1. −265- \frac{2 \sqrt{6}}{5}
  2. sin⁡x\sin x
  3. sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1
  4. π6, 5π6\frac{\pi}{6},\ \frac{5 \pi}{6}
  5. 3π4, 5π4\frac{3 \pi}{4},\ \frac{5 \pi}{4}
  6. π3, 4π3\frac{\pi}{3},\ \frac{4 \pi}{3}
  7. 0, π, π3, 2π30,\ \pi,\ \frac{\pi}{3},\ \frac{2 \pi}{3}
  8. π, 2π3, 4π3\pi,\ \frac{2 \pi}{3},\ \frac{4 \pi}{3}
  9. 1. Let u=sin⁡xu = \sin x: 2u2+u−1=02u^2 + u - 1 = 0 2. Factor: (2u−1)(u+1)=0(2u - 1)(u + 1) = 0 3. sin⁡x=12\sin x = \dfrac{1}{2} or sin⁡x=−1\sin x = -1 4. List every xx in [0,2π)[0, 2\pi): π6,5π6,3π2\dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}
  10. π12, 5π12, 13π12, 17π12\frac{\pi}{12},\ \frac{5 \pi}{12},\ \frac{13 \pi}{12},\ \frac{17 \pi}{12}
  11. 1010
  12. π3, π, 5π3\frac{\pi}{3},\ \pi,\ \frac{5 \pi}{3}