$$x \neq \frac{\pi}{2} + n\pi$$, where $$n is $$any integer

$$c = \sqrt{7^2$ + 10^2$ - 2 \times 7 \times 10 \times \cos 45$$^{\circ}$$}$$

$$r = 3\sqrt{2}, \theta = \frac{7\pi}{4}$$

Question 1

Convert $$135^{\circ} to $$radians.

Accepted Answer

$$\frac{3\pi}{4}$$

Question 2

Given that $$\sin 	heta  0$$, in which quadrant does angle $$	heta$$ lie?

Accepted Answer

Quadrant IV

Question 3

Find the reference angle for $$210^{\circ}.$$

Accepted Answer

$$30^{\circ}$$

Question 4

A point $$P$$ has coordinates $$(-3, 4). $$Find $$\sin 	heta$$, where $$	heta is $$the angle the terminal side makes with the positive $$x-$$axis.

Accepted Answer

$$\frac{4}{5}$$

Question 5

What is the $$y-$$intercept of the function $$y = \sin x$$?

Accepted Answer

$$0$$

Question 6

What is the domain of the function $$y = 	an x$$?

Accepted Answer

$$x 
eq \frac{\pi}{2} + n\pi$$, where $$n is $$any integer

Question 7

What is the range of the inverse sine function $$y = \arcsin x$$?

Accepted Answer

$$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$$

Question 8

Simplify $$\sin^2$ x + \cos^2$ x.$$

Accepted Answer

$$1$$

Question 9

Given $$\cos 	heta = \frac{1}{2}$$ and $$	heta is in $$Quadrant IV, find $$\sin 	heta.$$

Accepted Answer

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

Question 10

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

Accepted Answer

$$-\frac{3}{\sqrt{10}}$$

Question 11

Solve for $$x in$$ $$2\sin x - 1 = 0$$ for $$x in$$ $$[0, 2\pi].$$

Accepted Answer

$$\frac{\pi}{6}, \frac{5\pi}{6}$$

Question 12

A triangle has sides $$a = 7$$, $$b = 10$$, and angle $$C = 45$$^{\circ}$$. Use the Law of Cosines to find the length of side c$ opposite angle C$.

Accepted Answer

$$c = \sqrt{7^2$ + 10^2$ - 2 	imes 7 	imes 10 	imes \cos 45$$^{\circ}$$}$$

Question 13

Convert the rectangular coordinates $$(3, -3) to $$polar coordinates $$(r, 	heta)$$, where $$	heta is in $$radians.

Accepted Answer

$$r = 3\sqrt{2}, 	heta = \frac{7\pi}{4}$$

Question 14

Convert the polar equation $$r = 2\cos 	heta to $$rectangular form.

Accepted Answer

$$x^2$$ + $$y^2 = 2x$

Question 15

Find the position vector from point $$A(2, -1) to $$point $$B(5, 3).$$

Accepted Answer

$$\langle 3, 4 \rangle$$

Question 16

Find a unit vector in the same direction as $$\vec{v} = \langle 6, 8 \rangle.$$

Accepted Answer

$$\langle \frac{3}{5}, \frac{4}{5} \rangle$$

Question 17

Find the magnitude of the vector $$\vec{a} = \langle -7, 24 \rangle.$$

Accepted Answer

$$25$$

Question 18

A ladder leans against a wall, making a $$60^{\circ}$$ angle with the ground. If the ladder is $$10 ft $$long, how high up the wall does it reach?

Accepted Answer

$$10 \sin 60$$^{\circ}$$ = 5\sqrt{3} ft$$

Question 19

Simplify $$\frac{1 - \cos 2x}{\sin 2x}.$$

Accepted Answer

$$	an x$$

Question 20

Given $$\sin 	heta = \frac{5}{13}$$ and $$	heta is in $$Quadrant II, find $$\cos 	heta.$$

Accepted Answer

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

Comprehensive Guidance for MAT 1560 Trigonometry Final Exam Topics

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