Question 1

Solve for $$x in $$the interval $$[0, 2\pi)$$: $$2\cos x = -2.$$

Accepted Answer

$$x = \dfrac{3\pi}{4},\ \dfrac{5\pi}{4}$$

Question 2

Find all solutions for $$\sin x = -\dfrac{1}{2} in$$ $$[0, 2\pi).$$

Accepted Answer

$$x = \dfrac{7\pi}{6},\ \dfrac{11\pi}{6}$$

Question 3

Solve for $$x in$$ $$[0, 2\pi)$$: $$\cos^2$ x = \dfrac{3}{4}.$$

Accepted Answer

$$x = \dfrac{\pi}{6},\ \dfrac{5\pi}{6},\ \dfrac{7\pi}{6},\ \dfrac{11\pi}{6}$$

Question 4

Solve for $$x in$$ $$[0, 2\pi)$$: $$\sin 2x = \cos x.$$

Accepted Answer

$$x = 0,\ \dfrac{\pi}{3},\ \dfrac{5\pi}{3}$$

Question 5

Solve for $$x in$$ $$[0, 2\pi)$$: $$\sin x (2\cos x - 1) = 0.$$

Accepted Answer

$$x = 0,\ \pi,\ \dfrac{\pi}{3},\ \dfrac{5\pi}{3}$$

Question 6

Solve for $$x in$$ $$[0, 2\pi)$$: $$\cos x = 1.$$

Accepted Answer

$$x = 0$$

Question 7

Solve for $$x in$$ $$[0, 2\pi)$$: $$	an x = -\sqrt{3}.$$

Accepted Answer

$$x = \dfrac{2\pi}{3},\ \dfrac{5\pi}{3}$$

Question 8

Solve for $$x in$$ $$[0, 2\pi)$$: $$2\sin^2$ x - 1 = 0.$$

Accepted Answer

$$x = \dfrac{\pi}{4},\ \dfrac{3\pi}{4},\ \dfrac{5\pi}{4},\ \dfrac{7\pi}{4}$$

Question 9

Solve for $$x in$$ $$[0, 2\pi)$$: $$\cos x + 1 = 0.$$

Accepted Answer

$$x = \pi$$

Question 10

Solve for $$x in$$ $$[0, 2\pi)$$: $$\sin x = 1.$$

Accepted Answer

$$x = \dfrac{\pi}{2}$$

Question 11

Solve for $$x in$$ $$[0, 2\pi)$$: $$\sin x = 0.$$

Accepted Answer

$$x = 0,\ \pi$$

Question 12

Solve for $$x in$$ $$[0, 2\pi)$$: $$\cos 2x = 1.$$

Accepted Answer

$$x = 0,\ \pi$$

Question 13

Solve for $$x in$$ $$[0, 2\pi)$$: $$\sin x = \dfrac{\sqrt{3}}{2}.$$

Accepted Answer

$$x = \dfrac{\pi}{3},\ \dfrac{2\pi}{3}$$

Question 14

Solve for $$x in$$ $$[0, 2\pi)$$: $$\cos x = -\dfrac{1}{2}.$$

Accepted Answer

$$x = \dfrac{2\pi}{3},\ \dfrac{4\pi}{3}$$

Question 15

Solve for $$x in$$ $$[0, 2\pi)$$: $$	an^2$ x = 3.$$

Accepted Answer

$$x = \dfrac{\pi}{3},\ \dfrac{4\pi}{3}$$

Question 16

Solve for $$x in$$ $$[0, 2\pi)$$: $$\sin x = -1.$$

Accepted Answer

$$x = \dfrac{3\pi}{2}$$

Question 17

Solve for $$x in$$ $$[0, 2\pi)$$: $$\cos x = 0.$$

Accepted Answer

$$x = \dfrac{\pi}{2},\ \dfrac{3\pi}{2}$$

Question 18

Solve for $$x in$$ $$[0, 2\pi)$$: $$\sin 2x = 0.$$

Accepted Answer

$$x = 0,\ \dfrac{\pi}{2},\ \pi,\ \dfrac{3\pi}{2}$$

Question 19

A Ferris wheel rotates such that the height $$h (in $$meters) of a passenger above the ground at time $$t (in $$seconds) is modeled by $$h(t) = 10 + 8\sin\left(\dfrac{\pi}{15} t\right). At $$what times in $$[0, 30) is $$the passenger at a height of $$10$$ meters?

Accepted Answer

$$t = 0,\ 15$$

Solving Trigonometric Equations 5.5

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