Q: Which of the following is an identity derived from the Pythagorean identities?

$$1 + \tan^2$(\theta) = \sec^2$(\theta)$$

Q: Which of the following is the correct formula for the area $$A of a $$sector of a circle with radius $$r$$ and central angle $$\theta (in $$radians)?

$$A = \dfrac{1}{2} r^2$ \theta$$

Q: Which of the following is the correct formula for converting degrees to radians?

$$\text{radians} = \text{degrees} \times \dfrac{\pi}{180}$$

Question 1

Which of the following is the correct definition of the sine function using the unit circle approach for an angle $$	heta in $$standard position?

Accepted Answer

The $$y-$$coordinate of the point where the terminal side of $$	heta$$ intersects the unit circle.

Question 2

A Ferris wheel with a radius of 20 meters makes one complete revolution every 2 minutes. What is the angular velocity of the Ferris wheel in radians per minute?

Accepted Answer

$$\pi$$ radians per minute

Question 3

Which of the following is the correct domain of the function $$f(x) = \arccos(x)$$?

Accepted Answer

$$[-1, 1]$$

Question 4

Given $$\sin(	heta) = \dfrac{3}{5}$$ and $$	heta is in $$Quadrant II, what is $$\cos(	heta)$$?

Accepted Answer

$$-\dfrac{4}{5}$$

Question 5

Which of the following is an identity derived from the Pythagorean identities?

Accepted Answer

$$1 + 	an^2$(	heta) = \sec^2$(	heta)$$

Question 6

A triangle has sides $$a = 7$$, $$b = 10$$, and angle $$C = 45^\circ. $$Using the Law of Cosines, what is the length of side $$c$$ opposite angle $$C$$?

Accepted Answer

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

Question 7

Which of the following is the rectangular form of the complex number $$z = 5\operatorname{cis}\left(\dfrac{\pi}{3}\right)$$?

Accepted Answer

$$2.5 + 4.33i ($$rounded to two decimals)

Question 8

Which of the following equations represents an ellipse centered at the origin with a major axis along the $$x-$$axis?

Accepted Answer

$$\dfrac{x^2$}{a^2$} + \dfrac{y^2$}{b^2$} = 1$$, where $$a \u003e b$$

Question 9

If $$\vec{u} = \langle 3, 4 \rangle$$ and $$\vec{v} = \langle -2, 5 \rangle$$, what is $$\vec{u} + \vec{v}$$?

Accepted Answer

$$\langle 1, 9 \rangle$$

Question 10

Which of the following is the sum of the infinite geometric series $$\sum_{n=0}^{\infty} 5 \left(\dfrac{1}{3}\right)^n$$?

Accepted Answer

$$\dfrac{5}{1 - \frac{1}{3}} = 7.5$$

Question 11

A satellite dish is shaped like a paraboloid. If the cross-section of the dish is described by $$y^2 = 8x$, what is the focus of the parabola?

Accepted Answer

$$(2, 0)$$

Question 12

Convert the polar equation $$r = 4 \cos(	heta) to $$rectangular form.

Accepted Answer

$$x^2$$ + $$y^2 = 4x$

Question 13

Which of the following is the correct formula for the area $$A of a $$sector of a circle with radius $$r$$ and central angle $$	heta (in $$radians)?

Accepted Answer

$$A = \dfrac{1}{2} r^2$ 	heta$$

Question 14

Given the arithmetic sequence $$2, 5, 8, 11, \ldots$$, what is the formula for the $$nth $$term?

Accepted Answer

$$a_n = 2 + 3(n-1)$$

Question 15

Which of the following is the correct sum and difference formula for $$\sin(A - B)$$?

Accepted Answer

$$\sin(A - B) = \sin A \cos B - \cos A \sin B$$

Question 16

If $$	an(\alpha) = 2$$ and $$\alpha is in $$Quadrant III, what is $$\sin(\alpha)$$?

Accepted Answer

$$-\dfrac{2}{\sqrt{5}}$$

Question 17

Which of the following is the correct binomial expansion for $$(x + y)^3$?

Accepted Answer

$$x^3$$ + $$3x^2y$$ + $$3xy^2$$ + $$y^3$$

Question 18

Which of the following is the correct formula for converting degrees to radians?

Accepted Answer

$$	ext{radians} = 	ext{degrees} 	imes \dfrac{\pi}{180}$$

Question 19

If $$f(x) = \sin(x)$$ and $$g(x) = \cos(x)$$, what is $$f(x) \cdot g(x)$$ simplified using a double-angle identity?

Accepted Answer

$$\dfrac{1}{2} \sin(2x)$$

Precalculus Course Syllabus and Study Guide: Trigonometry, Polar Coordinates, Complex Numbers, and Analytic Geometry

Guida di studio - Domande di pratica

Study guide - Flashcard