Q: Verify the identity $$(1 + \sin\theta) \sec\theta = \tan\theta + \sec\theta. $$Which step is correct?

$$(1 + \sin\theta) \sec\theta = \sec\theta + \sin\theta \sec\theta = \sec\theta + \tan\theta$$

Question 1

Given the function $$y = 4 - 5 \cos\left(\frac{x}{2} + \frac{\pi}{2}\right)$$, what is the amplitude of the cosine component?

Accepted Answer

$$5$$

Question 2

For the function $$y = 3 \sin\left(2x - \frac{\pi}{2}\right) - 1$$, what is the period?

Accepted Answer

$$\pi$$

Question 3

Which of the following describes the vertical shift for $$y = 4 - 5 \cos\left(\frac{x}{2} + \frac{\pi}{2}\right)$$?

Accepted Answer

Up 4 units

Question 4

What is the exact value of $$\cot 330^\circ$$?

Accepted Answer

$$-\frac{1}{\sqrt{3}}$$

Question 5

Express $$\sin\left(\frac{5\pi}{6}\right) as a $$function of its reference angle and find its exact value.

Accepted Answer

$$\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$$

Question 6

If $$\sin	heta = -\frac{2}{3}$$ and $$270^\circ \u003c 	heta \u003c 360^\circ$$, what is $$\cos	heta$$?

Accepted Answer

$$-\frac{\sqrt{5}}{3}$$

Question 7

What is the exact value of $$\cos 105^\circ$$?

Accepted Answer

$$-\frac{\sqrt{6} + \sqrt{2}}{4}$$

Question 8

What is the exact value of $$\sin 15^\circ$$?

Accepted Answer

$$\frac{\sqrt{6} - \sqrt{2}}{4}$$

Question 9

Given the equation $$2x - y = 0$$ with $$x \leq 0$$, what is the value of $$	an	heta$$ for the angle whose terminal side lies on this line?

Accepted Answer

$$-2$$

Question 10

Which of the following verifies the identity $$\sin^3$ x + \sin x \cos^2$ x = \sin x$$?

Accepted Answer

$$\sin^3$ x + \sin x (1 - \sin^2$ x) = \sin x$$

Question 11

Verify the identity $$(1 + \sin	heta) \sec	heta = 	an	heta + \sec	heta. $$Which step is correct?

Accepted Answer

$$(1 + \sin	heta) \sec	heta = \sec	heta + \sin	heta \sec	heta = \sec	heta + 	an	heta$$

Question 12

For the function $$y = 3 \sin\left(2x - \frac{\pi}{2}\right) - 1$$, what is the phase shift?

Accepted Answer

$$\frac{\pi}{4}$$ units to the right

Question 13

If you are asked to graph $$y = 4 - 5 \sec\left(\frac{x}{2} + \frac{\pi}{2}\right)$$, what is the range of the function?

Accepted Answer

$$(-\infty, -1] \cup [9, \infty)$$

Question 14

A right triangle has angle $$A = 35^\circ$$ and hypotenuse $$c = 1.00. $$What is the length of side $$a$$ opposite angle $$A$$?

Accepted Answer

$$a = 0.574$$

Question 15

A right triangle has $$a = 27 cm $$and $$c = 28.1 cm. $$What is the measure of angle $$A ($$opposite side $$a$$)?

Accepted Answer

$$A = 73.7^\circ$$

Question 16

Which transformation is applied to $$y = 3 \csc\left(2x - \frac{\pi}{2}\right) - 1$$ compared to $$y = \csc x$$?

Accepted Answer

Vertical stretch by 3, period halved, phase shift right $$\frac{\pi}{4}$$, down 1 unit

Question 17

Which of the following is the frequency of $$y = 4 - 5 \cos\left(\frac{x}{2} + \frac{\pi}{2}\right)$$?

Accepted Answer

$$\frac{1}{2}$$

Question 18

If $$y = 3 \sin\left(2x - \frac{\pi}{2}\right) - 1$$ models the height of a Ferris wheel in meters as a function of time $$x in $$minutes, what is the maximum height?

Accepted Answer

$$2$$ meters

Step-by-Step Precalculus Trigonometry Guidance: Amplitude, Period, Graphs, Identities, and Exact Values

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