Q: Which of the following is the correct definition of the sine function for any angle $$\theta in $$the unit circle?

$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$

Q: Which of the following is a correct trigonometric identity?

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

Q: Given the complex number $$z = 3 + 4i$$, what is its modulus and argument?

Modulus: $$|z| = 5$$, Argument: $$\theta = \tan$$^{-1}$$\left(\frac{4}{3}\right)$$

Q: Which of the following is the correct polar form of the complex number $$z = -2 + 2i$$?

$$z = 2\sqrt{2} \text{cis} \left(135^ ext{o}\right)$$

Question 1

Convert $$135^	ext{o} to $$radians.

Accepted Answer

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

Question 2

A pilot flying from Chicago to New York travels 800 miles northeast. If the direction is $$45^	ext{o}$$ from due east, what are the east and north components of the displacement vector?

Accepted Answer

East: $$800 \cos 45^	ext{o} = 565.7 mi$$, North: $$800 \sin 45^	ext{o} = 565.7 mi$$

Question 3

Which of the following is the correct definition of the sine function for any angle $$	heta in $$the unit circle?

Accepted Answer

$$\sin 	heta = \frac{	ext{opposite}}{	ext{hypotenuse}}$$

Question 4

Solve the right triangle where $$a = 7$$, $$b = 24$$, and $$c is $$unknown. What is $$c$$?

Accepted Answer

$$c = \sqrt{7^2$ + 24^2$} = 25$$

Question 5

A surveyor measures two sides of a plot of land and the included angle: $$a = 50 m$$, $$b = 80 m$$, $$C = 60^	ext{o}. $$What is the length of side $$c$$ using the Law of Cosines?

Accepted Answer

$$c = \sqrt{50^2$ + 80^2$ - 2 \cdot 50 \cdot 80 \cdot \cos 60^	ext{o}} = 70$$

Question 6

Which of the following is a correct trigonometric identity?

Accepted Answer

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

Question 7

A vector $$\vec{v}$$ has magnitude $$10$$ and direction $$120^	ext{o}. $$What are its rectangular components?

Accepted Answer

$$x = 10 \cos 120^	ext{o} = -5$$, $$y = 10 \sin 120^	ext{o} = 8.66$$

Question 8

Which equation represents a circle in standard form?

Accepted Answer

$$(x - h)^2$ + (y - k)^2$ = r^2$

Question 9

Given the complex number $$z = 3 + 4i$$, what is its modulus and argument?

Accepted Answer

Modulus: $$|z| = 5$$, Argument: $$	heta = 	an$$^{-1}$$\left(\frac{4}{3}ight)$$

Question 10

Which of the following is the correct polar form of the complex number $$z = -2 + 2i$$?

Accepted Answer

$$z = 2\sqrt{2} 	ext{cis} \left(135^	ext{o}ight)$$

Question 11

A parabola has equation $$y = 2(x - 3)^2$ + 4. $$What are its vertex and direction?

Accepted Answer

Vertex: $$(3, 4)$$, opens upward

Question 12

Which conic section is represented by the equation $$\frac{(x - 2)^2$}{9} + \frac{(y + 1)^2$}{16} = 1$$?

Accepted Answer

Ellipse

Question 13

Solve for $$x in$$ $$\sin x = \frac{1}{2}$$ for $$0 \leq x \u003c 2\pi.$$

Accepted Answer

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

Question 14

A medical imaging device rotates $$270^	ext{o}$$ counterclockwise. What is this angle in radians?

Accepted Answer

$$\frac{3\pi}{2}$$

Question 15

Which of the following is the correct graph for $$y = \sin x$$?

Accepted Answer

A wave oscillating between $$-1$$ and $$1$$ with period $$2\pi$$

Question 16

If $$	an 	heta = 3$$ and $$	heta is in $$the second quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = \frac{3}{\sqrt{10}}$$

Question 17

Convert the point $$(4, 4)$$ from rectangular to polar coordinates.

Accepted Answer

$$(r, 	heta) = (4\sqrt{2}, 45^	ext{o})$$

Question 18

Which formula is used to solve an oblique triangle when given two sides and a non-included angle?

Accepted Answer

Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Question 19

Which of the following is a valid trigonometric equation to solve for $$x$$?

Accepted Answer

$$2\sin x - 1 = 0$$

Trigonometry with Analytic Geometry: Course Structure and Key Topics (Syllabus Study Guide)

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