Q: If $$\theta is an $$angle in the second quadrant, which of the following statements is true about $$\sin \theta$$ and $$\cos \theta$$?

$$\sin \theta \u003e 0$$ and $$\cos \theta \u003c 0$$

Q: Which of the following is the correct formula for the $$nth $$roots of a complex number $$z = r(\cos \theta + i \sin \theta)$$?

$$\sqrt[n]{r} \left( \cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} \right)$$, $$k = 0, 1, ..., n-1$$

Question 1

Given the unit circle diagram, which of the following angles corresponds to the point $$\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) on $$the unit circle?

Accepted Answer

$$\frac{\pi}{3}$$

Question 2

If $$	heta is an $$angle in the second quadrant, which of the following statements is true about $$\sin 	heta$$ and $$\cos 	heta$$?

Accepted Answer

$$\sin 	heta \u003e 0$$ and $$\cos 	heta \u003c 0$$

Question 3

Using the double angle formula, what is $$\sin(2	heta) if$$ $$\sin 	heta = \frac{3}{5}$$ and $$	heta is in $$the first quadrant?

Accepted Answer

$$\frac{24}{25}$$

Question 4

Which of the following is the correct expression for $$\cos(\alpha + \beta)$$?

Accepted Answer

$$\cos \alpha \cos \beta - \sin \alpha \sin \beta$$

Question 5

If $$	an \alpha = 2$$ and $$	an \beta = 3$$, what is $$	an(\alpha + \beta)$$?

Accepted Answer

$$-1$$

Question 6

Given $$\cos 	heta = -\frac{1}{2}$$ and $$	heta is in $$the third quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$-\frac{\sqrt{3}}{2}$$

Question 7

Which formula correctly expresses $$\cos(2	heta) in $$terms of $$\sin 	heta$$?

Accepted Answer

$$1 - 2\sin^2$ 	heta$$

Question 8

A vector $$\vec{u}$$ has coordinates $$(3, 4)$$ and $$\vec{v}$$ has coordinates $$(4, -3). $$What is $$\cos 	heta$$ where $$	heta is $$the angle between $$\vec{u}$$ and $$\vec{v}$$?

Accepted Answer

$$0$$

Question 9

If $$z = 2\left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$$, what is $$z^3 in $$polar form?

Accepted Answer

$$8\left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right)$$

Question 10

Which of the following is the correct half-angle formula for $$\sin \frac{\alpha}{2}$$?

Accepted Answer

$$\pm \sqrt{\frac{1 - \cos \alpha}{2}}$$

Question 11

A triangle has sides $$a = 7$$, $$b = 5$$, $$c = 8. $$What is $$\cos A$$ using the Law of Cosines?

Accepted Answer

$$\frac{5^2$ + 8^2$ - 7^2$}{2 \cdot 5 \cdot 8}$$

Question 12

If $$\sin \alpha = \frac{4}{5}$$ and $$\alpha is in $$the first quadrant, what is $$\cos(2\alpha)$$?

Accepted Answer

$$\frac{9}{25}$$

Question 13

Which of the following is the correct formula for the $$nth $$roots of a complex number $$z = r(\cos 	heta + i \sin 	heta)$$?

Accepted Answer

$$\sqrt[n]{r} \left( \cos \frac{	heta + 2k\pi}{n} + i \sin \frac{	heta + 2k\pi}{n} ight)$$, $$k = 0, 1, ..., n-1$$

Question 14

If $$	an 	heta = 1$$, which of the following could be a possible value for $$	heta in $$the interval $$[0, 2\pi)$$?

Accepted Answer

$$\frac{\pi}{4}$$

Question 15

Given $$\vec{u} = (2, -1)$$ and $$\vec{v} = (1, 2)$$, what is the cosine of the angle between them?

Accepted Answer

Question 16

If $$\sin \alpha = \frac{5}{13}$$ and $$\alpha is in $$the second quadrant, what is $$\cos \alpha$$?

Accepted Answer

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

Key Trigonometric Formulas and the Unit Circle

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