Q: Which of the following is the correct verification for the identity $$(1 + \sin \theta) \sec \theta = \tan \theta + \sec \theta$$?

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

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

What is the vertical shift for the function $$y = 4 - 5 \cos\left(\frac{x}{2} + \frac{\pi}{2}\right)$$?

Accepted Answer

$$4$$

Question 4

Which of the following describes the phase shift of $$y = 3 \sin\left(2x - \frac{\pi}{2}\right) - 1$$?

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

Express $$\sin\frac{5\pi}{6} in $$terms of its reference angle and find its exact value.

Accepted Answer

$$\sin\frac{5\pi}{6} = \frac{1}{2}$$

Question 7

If the terminal side of an angle lies on $$2x - y = 0$$ with $$x \leq 0$$, which quadrant does the angle lie in?

Accepted Answer

Second quadrant

Question 8

Verify the identity: $$\sin^3$ x + \sin x \cos^2$ x = \sin x. $$Which of the following steps is correct?

Accepted Answer

$$\sin^3$ x + \sin x \cos^2$ x = \sin x (\sin^2$ x + \cos^2$ x)$$

Question 9

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

Accepted Answer

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

Question 10

Find the exact value of $$\cos 105^\circ$$ without a calculator.

Accepted Answer

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

Question 11

Find the exact value of $$\sin 15^\circ$$ without a calculator.

Accepted Answer

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

Question 12

Solve the right triangle with $$A = 35^\circ$$ and $$c = 1.00. $$What is the length of side $$a$$?

Accepted Answer

$$a = 0.574$$

Question 13

Solve the right triangle with $$a = 27 cm $$and $$c = 28.1 cm. $$What is the measure of angle $$A$$?

Accepted Answer

$$A = 73.7^\circ$$

Question 14

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

Accepted Answer

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

Question 15

For the function $$y = 4 - 5 \sec\left(\frac{x}{2} + \frac{\pi}{2}\right)$$, what is the period?

Accepted Answer

$$4\pi$$

Question 16

Which of the following is the correct verification for the identity $$(1 + \sin 	heta) \sec 	heta = 	an 	heta + \sec 	heta$$?

Accepted Answer

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

Question 17

A Ferris wheel rotates such that its height above ground can be modeled by $$y = 4 - 5 \cos\left(\frac{x}{2} + \frac{\pi}{2}\right). $$What is the maximum height reached?

Accepted Answer

$$-1$$

Question 18

If $$y = 3 \sin\left(2x - \frac{\pi}{2}\right) - 1$$ models the height of a tide above sea level, what is the minimum height?

Accepted Answer

$$-4$$

Question 19

Which of the following is the frequency of $$y = 3 \sin\left(2x - \frac{\pi}{2}\right) - 1$$?

Accepted Answer

$$2$$

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