Q: Convert the rectangular coordinates $$(x, y) = (3, -4) to $$polar coordinates $$(r, \theta)$$, where $$0 \leq \theta \u003c 2\pi.$$

$$r = 5, \theta = 5.355$$ radians$

Question 1

Given triangle ABC with sides $$a = 7$$, $$b = 10$$, and angle $$A = 30^\circ$$, how many possible triangles can be formed using the Law of Sines (SSA case)?

Accepted Answer

Two triangles

Question 2

A ship leaves port and travels on a bearing of $$165^\circ$$ for 100 miles. Which direction is the ship traveling relative to due North?

Accepted Answer

S $$15^\circ E$$

Question 3

Which of the following is the correct Law of Cosines formula for finding side $$a in $$triangle $$ABC$$?

Accepted Answer

$$a^2$$ = $$b^2$$ + $$c^2 - 2bc$$ \cos A$$

Question 4

Given a triangle with sides $$a = 8$$, $$b = 15$$, $$c = 17$$, what is the area using Heron's formula?

Accepted Answer

$$A = \sqrt{20(12)(5)(3)}$$

Question 5

If a vector $$\vec{v}$$ has components $$\langle 3, 4 \rangle$$, what is its magnitude?

Accepted Answer

$$|\vec{v}| = 5$$

Question 6

Which of the following statements about vectors is TRUE?

Accepted Answer

The magnitude of a vector is always a positive scalar quantity.

Question 7

Given two vectors $$\vec{A} = \langle 2, 3 \rangle$$ and $$\vec{B} = \langle -1, 4 \rangle$$, what is their dot product?

Accepted Answer

$$\vec{A} \cdot \vec{B} = 10$$

Question 8

A projectile is launched from a height of 20 feet with an initial velocity of 40 ft/s at an angle of $$30^\circ$$ above the horizontal. What is the parametric equation for its vertical position $$y at $$time $$t$$ seconds?

Accepted Answer

$$y = -16t^2$ + 40 \sin 30^\circ t + 20$$

Question 9

Convert the rectangular coordinates $$(x, y) = (3, -4) to $$polar coordinates $$(r, 	heta)$$, where $$0 \leq 	heta \u003c 2\pi.$$

Accepted Answer

$$r = 5, 	heta = 5.355$$ radians$

Question 10

Which of the following is the correct formula for the area of a triangle given two sides $$a$$ and $$b$$ and the included angle $$C$$?

Accepted Answer

$$A = \frac{1}{2} ab \sin C$$

Question 11

A vector has magnitude $$10$$ and direction angle $$120^\circ. $$What are its horizontal and vertical components?

Accepted Answer

Horizontal: $$10 \cos 120^\circ$$, Vertical: $$10 \sin 120^\circ$$

Question 12

Given the parametric equations $$x = 3t + 2$$, $$y = 2t - 1$$, eliminate the parameter $$t to $$find the rectangular equation.

Accepted Answer

$$y = \frac{2}{3}x - \frac{7}{3}$$

Question 13

If two vectors are perpendicular, what is true about their dot product?

Accepted Answer

The dot product is zero.

Question 14

A chord of a circle has endpoints on the circle and subtends a central angle $$\alpha. If $$the radius of the circle is $$r$$, what is the length of the chord?

Accepted Answer

$$a = r \sqrt{2 - 2 \cos \alpha}$$

Question 15

Which of the following is NOT a property of parallelograms?

Accepted Answer

Diagonals bisect the angles.

Question 16

Given the vector $$\vec{v} = \langle -6, 8 angle$$, what is the direction angle $$	heta (in $$degrees) measured counterclockwise from the positive x-axis?

Accepted Answer

$$126.87^\circ$$

Question 17

If a triangle has angles $$A = 40^\circ$$, $$B = 60^\circ$$, and side $$a = 10$$, what is the length of side $$b$$ using the Law of Sines?

Accepted Answer

$$b = \frac{10 \sin 60^\circ}{\sin 40^\circ}$$

Question 18

Which of the following is the correct conversion from polar to rectangular coordinates for $$(r, 	heta) = (4, 150^\circ)$$?

Accepted Answer

$$x = 4 \cos 150^\circ, y = 4 \sin 150^\circ$$

Question 19

A force of 50 N acts at a bearing of $$38^\circ$$ and another force of 30 N acts at a bearing of $$165^\circ. $$What is the best method to find the magnitude of the resultant force?

Accepted Answer

Resolve each force into components and add the vectors.

Oblique Triangles, Law of Sines & Cosines, Vectors, Polar & Parametric Equations: Precalculus Study Guide

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