Q: Which of the following is the rectangular form of the complex number $$-2 + 2i in $$polar form?

$$2\sqrt{2}\text{cis}\left(\dfrac{3\pi}{4}\right)$$

Question 1

Convert $$135^\circ to $$radians. Which of the following is correct?

Accepted Answer

$$\dfrac{5\pi}{4}$$

Question 2

A Ferris wheel has a radius of 20 meters. If a rider travels through a central angle of $$\dfrac{\pi}{3}$$ radians, what is the length of the arc the rider travels?

Accepted Answer

$$\dfrac{20\pi}{3}$$ meters

Question 3

Which of the following is the correct value for $$\sin(225^\circ)$$?

Accepted Answer

$$-\dfrac{\sqrt{2}}{2}$$

Question 4

Given a right triangle where one leg is 7 units and the hypotenuse is 25 units, what is the sine of the angle opposite the 7-unit leg?

Accepted Answer

$$\dfrac{7}{25}$$

Question 5

A surveyor measures two sides and the included angle of a triangular plot of land: $$a = 120 m$$, $$b = 150 m$$, and $$C = 60^\circ. $$What is the length of side $$c$$ using the Law of Cosines?

Accepted Answer

$$c = \sqrt{120^2$ + 150^2$ - 2 \cdot 120 \cdot 150 \cdot \cos 60^\circ}$$

Question 6

Which of the following is the rectangular form of the complex number $$5	ext{cis}\left(\dfrac{\pi}{3}ight)$$?

Accepted Answer

$$5\left(\dfrac{1}{2} + i\dfrac{\sqrt{3}}{2}\right)$$

Question 7

Which equation represents a circle with center at $$(2, -3)$$ and radius $$5$$?

Accepted Answer

$$(x - 2)^2$ + (y + 3)^2$ = 25$$

Question 8

Which of the following is an identity?

Accepted Answer

$$\sin^2$ x + \cos^2$ x = 1$$

Question 9

A vector has initial point $$(1,2)$$ and terminal point $$(4,6). $$What is its component form?

Accepted Answer

$$\langle 3, 4 \rangle$$

Question 10

Which of the following is the correct graph for $$y = 2\sin(x)$$ compared to $$y = \sin(x)$$?

Accepted Answer

It has the same period but twice the amplitude.

Question 11

Solve for $$x in$$ $$2\sin x - 1 = 0$$ for $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}$$

Question 12

Convert the polar coordinate $$(4, \dfrac{\pi}{2}) to $$rectangular form $$(x, y). $$Which is correct?

Accepted Answer

$$(0, 4)$$

Question 13

Which of the following is the equation of a parabola that opens upward with vertex at $$(0,0)$$ and focus at $$(0,2)$$?

Accepted Answer

$$x^2 = 8y$

Question 14

If $$z = 3 + 4i$$, what is $$|z|$$?

Accepted Answer

$$5$$

Question 15

Which of the following is the correct Law of Sines formula for triangle $$ABC$$?

Accepted Answer

$$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$$

Question 16

A satellite dish is shaped like a paraboloid. If the cross-section is described by $$y = \dfrac{1}{8}x^2$, what is the focal length?

Accepted Answer

$$2$$

Question 17

Which of the following is the rectangular form of the complex number $$-2 + 2i in $$polar form?

Accepted Answer

$$2\sqrt{2}	ext{cis}\left(\dfrac{3\pi}{4}ight)$$

Question 18

Which of the following is the equation of a hyperbola centered at the origin with foci on the x-axis and vertices at $$(\pm 3, 0)$$?

Accepted Answer

$$\dfrac{x^2$}{9} - \dfrac{y^2$}{b^2$} = 1$$

Question 19

A pilot flies 100 miles east, then 100 miles north. What is the magnitude of the resultant displacement vector?

Accepted Answer

$$100\sqrt{2}$$ miles

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