Q: If $$\cos(\theta) = -\frac{1}{2}$$ and $$0 \leq \theta \u003c 2\pi$$, what are all possible solutions for $$\theta$$?

$$\theta = \frac{2\pi}{3}, \frac{4\pi}{3}$$

Question 1

Given the function $$y = 3 \sin(2x) - 1$$, what is the amplitude, period, and midline of the graph?

Accepted Answer

Amplitude: $$3$$, Period: $$\pi$$, Midline: $$y = -1$$

Question 2

Which transformation occurs when the equation $$y = 2 \cos(x - \frac{\pi}{2}) + 1 is $$compared to $$y = \cos(x)$$?

Accepted Answer

Vertical stretch by $$2$$, horizontal shift right by $$\frac{\pi}{2}$$, vertical shift up by $$1$$

Question 3

The population of rabbits in a park is modeled by $$R(t) = 500 \sin\left(\frac{\pi}{12} t\right) + 800. $$What is the period of the population cycle?

Accepted Answer

$$24$$ months

Question 4

Solve for $$x in$$ $$\sin(x) = \frac{\sqrt{2}}{2}$$ for $$x in$$ $$[0, 2\pi].$$

Accepted Answer

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

Question 5

Which of the following best describes the effect of increasing the coefficient $$B in$$ $$y = A \sin(Bx)$$?

Accepted Answer

It decreases the period of the wave.

Question 6

Given $$y = 4 \sin(2x) - 2 = 0$$, what are the solutions for $$x in$$ $$[0, 2\pi]$$?

Accepted Answer

$$x = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{13\pi}{12}, \frac{17\pi}{12}$$

Question 7

If $$\cos(	heta) = -\frac{1}{2}$$ and $$0 \leq 	heta \u003c 2\pi$$, what are all possible solutions for $$	heta$$?

Accepted Answer

$$	heta = \frac{2\pi}{3}, \frac{4\pi}{3}$$

Question 8

What is the midline of the function $$y = 5 \cos(3x) + 4$$?

Accepted Answer

$$y = 4$$

Question 9

A fox population in a park is modeled by $$F(t) = 50 \cos\left(\frac{\pi}{6} t\right) + 100. $$What is the maximum population of foxes?

Accepted Answer

$$150$$

Question 10

Which of the following equations represents a cosine function with amplitude $$7$$, period $$\pi$$, and midline $$y = -2$$?

Accepted Answer

$$y = 7 \cos(2x) - 2$$

Question 11

Solve the trigonometric equation $$2\sin(x) + 1 = 0$$ for $$x in$$ $$[0, 2\pi].$$

Accepted Answer

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

Question 12

Which of the following best describes the reference angle for $$	heta = \frac{5\pi}{6}$$?

Accepted Answer

$$\frac{\pi}{6}$$

Question 13

If $$y = 2 \sin(x - \frac{\pi}{3}) + 1$$, what is the horizontal shift of the graph?

Accepted Answer

Right $$\frac{\pi}{3}$$ units

Question 14

Which of the following is the correct period for the function $$y = \sin(4x)$$?

Accepted Answer

$$\frac{\pi}{2}$$

Question 15

Given $$y = 2 \sin(x) \cos(x)$$, which trigonometric identity can be used to rewrite this function?

Accepted Answer

$$y = \sin(2x)$$

Question 16

If $$\sin(	heta) is $$negative and $$\cos(	heta) is $$positive, in which quadrant does $$	heta$$ lie?

Accepted Answer

Quadrant IV

Question 17

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

Accepted Answer

$$x = 0, \frac{\pi}{2}, \frac{5\pi}{4}$$

Question 18

Which of the following is the correct equation for a sine function with amplitude $$4$$, period $$\frac{\pi}{2}$$, and midline $$y = 3$$?

Accepted Answer

$$y = 4 \sin(8x) + 3$$

Question 19

Which of the following best explains why trigonometric functions are appropriate for modeling populations that oscillate over time, such as rabbits and foxes?

Accepted Answer

Because trigonometric functions are periodic and can model regular cycles.

Study Guide: Graphs and Equations of Trigonometric Functions (Period, Amplitude, Midline, Transformations, and Solving Trig Equations)

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