Question 1

Given a triangle with sides $$a = 7$$, $$b = 9$$, and included angle $$C = 40^\circ$$, what is the area of the triangle? (Round to the nearest tenth.)

Accepted Answer

$$20.3$$ square units

Question 2

Find the absolute value of the complex number $$z = -3 + 4i.$$

Accepted Answer

$$5$$

Question 3

Express the complex number $$z = 1 - i in $$trigonometric form.

Accepted Answer

$$\sqrt{2}(\cos(\frac{7\pi}{4}) + i\sin(\frac{7\pi}{4}))$$

Question 4

Convert the polar coordinates $$(4, \frac{2\pi}{3}) to $$rectangular coordinates.

Accepted Answer

$$(-2, 2\sqrt{3})$$

Question 5

Express the complex number $$z = 5	ext{cis}\left(\frac{\pi}{6}ight) in $$standard notation.

Accepted Answer

$$\frac{5\sqrt{3}}{2} + \frac{5}{2}i$$

Question 6

Given vectors $$\vec{u} = \langle 3, 4 \rangle$$ and $$\vec{v} = \langle -1, 2 \rangle$$, find $$\vec{u} + \vec{v}.$$

Accepted Answer

$$\langle 2, 6 \rangle$$

Question 7

Two forces of $$63\ \mathrm{N}$$ and $$39\ \mathrm{N}$$ act on an object at right angles. What is the magnitude of the resultant force?

Accepted Answer

$$72\ \mathrm{N}$$

Question 8

Given $$|\vec{u}| = 8$$, $$|\vec{v}| = 6$$, and the angle between them is $$60^\circ$$, find the magnitude of $$\vec{u} + \vec{v} to $$the nearest tenth.

Accepted Answer

$$13.9$$

Question 9

Given $$|\vec{u}| = 8$$, $$|\vec{v}| = 6$$, and the angle between them is $$60^\circ$$, what is the angle (to the nearest degree) that $$\vec{u} + \vec{v}$$ makes with $$\vec{u}$$?

Accepted Answer

$$32^\circ$$

Question 10

Convert the rectangular coordinates $$(-3, 3\sqrt{3}) to $$polar coordinates (express the angle in radians, $$0 \leq 	heta \u003c 2\pi).$$

Accepted Answer

$$(6, \frac{2\pi}{3})$$

Question 11

Two forces of $$300\ \mathrm{N}$$ and $$425\ \mathrm{N}$$ act on an object with an angle of $$53^\circ$$ between them. What is the magnitude of the resultant force? (Round to the nearest newton.)

Accepted Answer

$$654\ \mathrm{N}$$

Question 12

Given $$z = 2(\cos(120^\circ) + i\sin(120^\circ))$$, express $$z in $$standard notation.

Accepted Answer

$$-1 + i\sqrt{3}$$

Question 13

A pilot flies $$100 km $$east, then $$100 km $$north. What is the straight-line distance from the starting point to the ending point?

Accepted Answer

$$100\sqrt{2} km$$

Question 14

Find the area of a triangle with sides $$a = 10$$, $$b = 12$$, and included angle $$C = 75^\circ. ($$Round to the nearest tenth.)

Accepted Answer

$$58.0$$

Question 15

Given $$z = 4	ext{cis}\left(\frac{\pi}{2}ight)$$, express $$z in $$standard notation.

Accepted Answer

$$0 + 4i$$

Question 16

Given vectors $$\vec{u} = \langle 5, -2 \rangle$$ and $$\vec{v} = \langle -3, 4 \rangle$$, compute $$2\vec{u} - \vec{v}.$$

Accepted Answer

$$\langle 13, -8 \rangle$$

Question 17

A ship sails $$120 km on a $$bearing of $$30^\circ$$ and then $$80 km on a $$bearing of $$120^\circ. $$What is the straight-line distance from the starting point to the ending point? (Round to the nearest km.)

Accepted Answer

$$170 km$$

Question 18

Find the angle (to the nearest degree) that the resultant of two forces of $$300\ \mathrm{N}$$ and $$425\ \mathrm{N}$$, acting at $$53^\circ$$, makes with the $$300\ \mathrm{N}$$ force.

Accepted Answer

$$36^\circ$$

Trigonometry Exam Review: Solving Triangles, Complex Numbers, Vectors, and Forces

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