Q: Verify the identity: $$-\sin 2\theta = \frac{-2 \cot \theta}{1 + \cot^2$ \theta}. Is $$this identity true?

True, the identity holds for all $$\theta$$ where both sides are defined.

Question 1

Given the function $$y = 2 \cos \left(x - \frac{\pi}{10}\right)$$, which of the following statements about its graph is correct?

Accepted Answer

The amplitude is $$2$$, the period is $$2\pi$$, and there is a phase shift of $$\frac{\pi}{10}$$ units to the right.

Question 2

Which of the following is the correct period for the function $$y = \sec\left(x - \frac{\pi}{4}\right)$$?

Accepted Answer

$$2\pi$$

Question 3

An object attached to a spring is modeled by $$s(t) = -3 \cos\left(\frac{2\pi}{7} t\right). $$What is the frequency of oscillation?

Accepted Answer

$$\frac{1}{7}$$ oscillations/sec

Question 4

Rewrite $$4 \sin 2x \sin 4x as a $$sum or difference of trigonometric functions.

Accepted Answer

$$2 \cos 2x - 2 \cos 6x$$

Question 5

Verify the identity: $$-\sin 2	heta = \frac{-2 \cot 	heta}{1 + \cot^2$ 	heta}. Is $$this identity true?

Accepted Answer

True, the identity holds for all $$	heta$$ where both sides are defined.

Question 6

Given $$\cos s = -\frac{4}{5}$$ and $$\sin t = \frac{5}{13}$$, with $$s$$ and $$t in $$quadrant II, what is $$\sin(s + t)$$?

Accepted Answer

$$-\frac{56}{65}$$

Question 7

Given $$\cos 	heta = -\frac{20}{29}$$ and $$\sin 	heta \u003c 0$$, what is $$\sin 2	heta$$?

Accepted Answer

$$-\frac{56}{841}$$

Question 8

Find the exact value: $$y = \arccos\left(\frac{\sqrt{3}}{2}\right). $$What is $$y$$?

Accepted Answer

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

Question 9

Evaluate $$\sin\left(2 \cos$$^{-1}$$\left(\frac{1}{13}\right)\right).$$

Accepted Answer

$$\frac{24}{169}$$

Question 10

Express $$\sec\left(\arccot\left(\frac{4- u^2$}{u}\right)\right) as an $$algebraic expression in $$u$$ for $$u \u003e 0.$$

Accepted Answer

$$\frac{2}{\sqrt{4-u^2$}}$$

Question 11

A Ferris wheel with a radius of $$10$$ meters makes one revolution every $$30$$ seconds. If a rider starts at the lowest point, which equation models the rider's height $$h(t)$$ above the ground at time $$t$$ seconds, assuming the center is $$12$$ meters above the ground?

Accepted Answer

$$h(t) = 12 - 10 \cos\left(\frac{2\pi}{30} t\right)$$

Question 12

Which of the following is the correct formula for the period of $$y = a \sin(bx + c) + d$$?

Accepted Answer

$$\frac{2\pi}{|b|}$$

Question 13

If $$	an \alpha = 3$$ and $$\alpha is in $$quadrant III, what is $$\sin 2\alpha$$?

Accepted Answer

$$-\frac{24}{10}$$

Question 14

Given $$\sin 	heta = -\frac{5}{13}$$ and $$	heta is in $$quadrant III, what is $$\cos 	heta$$?

Accepted Answer

$$-\frac{12}{13}$$

Question 15

Which of the following is the correct phase shift for $$y = \sin\left(2x + \frac{\pi}{3}\right)$$?

Accepted Answer

$$-\frac{\pi}{6} ($$shifted left)

Question 16

If $$y = 3 \cos(4x) + 2$$, what is the vertical translation and amplitude?

Accepted Answer

Vertical translation: $$2$$ units up; Amplitude: $$3$$

Question 17

A sound wave is modeled by $$y = 0.5 \sin(880\pi t). $$What is the frequency in Hz?

Accepted Answer

$$440 Hz$$

Question 18

If $$	an(\alpha + \beta) = \frac{	an \alpha + 	an \beta}{1 - 	an \alpha 	an \beta}$$, what is $$	an(45^\circ + 30^\circ)$$?

Accepted Answer

$$\sqrt{3}$$

Question 19

Which of the following is the correct set of vertical asymptotes for $$y = \sec\left(x - \frac{\pi}{4}\right) in $$one period?

Accepted Answer

$$x = \frac{\pi}{2} + \frac{\pi}{4} + n\pi,\ n \in \mathbb{Z}$$

Step-by-Step Guidance for Plane Trigonometry Practice Test (Ch 4–6.1)

Study Guide - Practice Questions

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