Exam 10: Applications of Trigonometry and Vectors

Can a triangle $A B C$ exist if $a=12.1, b=6.8$ , and $c=19$ ? Explain why or why not. Answer this question without using trigonometry.

Solve each applied problem. (a) Find the magnitude of the resultant of forces of $432 \mathrm{lb}$ and $325 \mathrm{lb}$ that form an angle of $57.2^{\circ}$ . (b) Two boats leave a dock together, traveling on straight courses that have an angle of $132.1^{\circ}$ between them. One boat travels $37.6 \mathrm{~km} / \mathrm{hr}$ and the other travels $29.1 \mathrm{~km} / \mathrm{hr}$ . How far apart are they after 3 hours? (c) Find the horizontal and vertical components of a vector with magnitude 105 that is inclined $130^{\circ} 10^{\prime}$ from the horizontal. Give your answer in the form $|a, b|$ . (d) Points $A$ and $B$ are on opposite sides of Bear Lake. From a third point, $C$ , the angle between lines of sight to $A$ and $B$ is $46.4^{\circ}$ . If $A C$ is $16 \mathrm{~km}$ long and $B C$ is $3 \mathrm{~km}$ long, find $A B$ .

Find an equivalent equation in polar coordinates for $-5 x-8 y=10$ in the form $r=f(\theta)$ .

Solve each applied problem. (a) Find the magnitude of the resultant of forces of $600 \mathrm{lb}$ and $550 \mathrm{lb}$ that form an angle of $48.3^{\circ}$ . (b) A baseball diamond is a square, $90 \mathrm{ft}$ on a side, with home plate and three bases at the vertices. The pitcher's rubber is located $60.5 \mathrm{ft}$ from home plate. Find the distance from the pitcher's rubber to first base. (c) Find the horizontal and vertical components of a vector with magnitude 378 that is inclined $141^{\circ} 50^{\prime}$ from the horizontal. Give your answer in the form $|a, b|$ (d) Two speakers are placed in a room so that the angle formed by the cables connecting them to the stereo is $78.3^{\circ}$ . One speaker is 9 feet from the stereo and the other is 4.7 feet from the stereo. How far apart are the speakers?

Given $a=31$ and $B=128^{\circ}$ in triangle $A B C$ , determine the values of $b$ for which $A$ has (a) exactly one value (b) two values (c) no values

Given $a=13$ and $B=132^{\circ}$ in triangle $A B C$ , determine the values of $b$ for which $A$ has (a) exactly one value (b) two values (c) no values

Find the indicated part of each triangle. (a) $C=122^{\circ}, b=83 \mathrm{~km}, a=145 \mathrm{~km}$ ; find $c$ . (b) $A=37.6^{\circ}, a=8.91 \mathrm{ft}, b=6.13 \mathrm{ft}$ ; find $C$ . (c) $a=14.6 \mathrm{~m}, b=21.7 \mathrm{~m}, c=24.3 \mathrm{~m}$ ; find $B$ .

Perform the indicated operation. Give the answer in rectangular form. (a) $5\left(\cos 270^{\circ}+i \sin 270^{\circ}\right) \cdot 4\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)$ (b) $\frac{2 \operatorname{cis} 45^{\circ}}{\operatorname{cis} 90^{\circ}}$ (c) $(5-2 i)^{3}$ (d) Find the fourth roots of $-\sqrt{2}+i \sqrt{2}$ . Leave your answers in trigonometric form.

Graph the parametric equations $x=2.5 \cos 2 t, y=2.5 \sin 2 t$ , for $t$ in $[0, \pi]$ .

For the polar equation $r=4 \sin \theta$ , do the following: (a) Graph the equation. (b) Find the equivalent equation in rectangular coordinates. (c) Is the graph from part (a) what you would expect for the graph of the equation from part (b)? Explain.

Perform the indicated operation. Give the answer in rectangular form. (a) $3\left(\cos 45^{\circ}+i \sin 45^{\circ}\right) \cdot 5\left(\cos 180^{\circ}+i \sin 180^{\circ}\right)$ (b) $\frac{5 \operatorname{cis} 60^{\circ}}{4 \operatorname{cis} 30^{\circ}}$ (c) $(4-i)^{4}$ (d) Find the fourth roots of $\sqrt{3}-i$ . Leave your answers in trigonometric form.

For the vectors $\mathbf{u}=\langle 1,4\rangle$ and $\mathbf{v}=\langle 2,1\rangle$ , find the following: (a) $4 \mathbf{u}-\mathbf{v}$ (b) $\frac{1}{3} \mathbf{u}$ (c) $\mathbf{u} \cdot \mathbf{v}$ (d) the angle between $\mathbf{u}$ and $\mathbf{v}$

Solve each applied problem. (a) Find the magnitude of the resultant of forces of $336 \mathrm{lb}$ and $125 \mathrm{lb}$ that form an angle of $27.3^{\circ}$ . (b) A $712 \mathrm{ft}$ sidewalk connects the library and the physics lab. A $603 \mathrm{ft}$ sidewalk connects the library and the dining hall. If the angle between these sidewalks is $51.7^{\circ}$ , how long is the sidewalk connecting the physics lab and the dining hall? (c) Find the horizontal and vertical components of a vector with magnitude 817 that is inclined $126^{\circ} 45^{\prime}$ from the horizontal. Give your answer in the form $|a, b|$ . (d) A mother is standing 20 feet away from one of her children and 6 feet away from her other child. The angle between her lines of sight to her two children is $42.2^{\circ}$ . How far apart are the two children?

For the vectors $\mathbf{u}=\langle 2,-4\rangle$ and $\mathbf{v}=\langle 4,1\rangle$ , find the following: (a) $3 \mathbf{u}+\mathbf{v}$ (b) $\frac{1}{4} \mathbf{u}$ (c) $\mathbf{u} \cdot \mathbf{v}$ (d) the angle between $\mathbf{u}$ and $\mathbf{v}$

Find the following for the complex numbers 6 cis $225^{\circ}$ and $3 \sqrt{2}-3 i \sqrt{2}$ : (a) the rectangular form of 6 cis $225^{\circ}$ . (b) the trigonometric form of $3 \sqrt{2}-3 i \sqrt{2}$ . (c) their resultant in the form $a+b i$ .

Find an equivalent equation in polar coordinates for $-4 x+7 y=9$ in the form $r=f(\theta)$ .

Find an equivalent equation in polar coordinates for $3 x+5 y=-2$ in the form $r=f(\theta)$ .

For the vectors $\mathbf{u}=\langle 4,7\rangle$ and $\mathbf{v}=\langle-2,-1\rangle$ , find the following: (a) $4 \mathbf{u}-\mathbf{v}$ (b) $-\frac{1}{4} \mathbf{u}$ (c) $\mathbf{u} \cdot \mathbf{v}$ (d) the angle between $\mathbf{u}$ and $\mathbf{v}$

Graph the parametric equations $x=4 \cos 2 t, y=4 \sin 2 t$ , for $t$ in $[0, \pi]$ .

Find the indicated part of each triangle. (a) $A=137^{\circ}, b=122.5 \mathrm{ft}, c=91.2 \mathrm{ft}$ ; find $a$ . (b) $C=19.5^{\circ}, c=9.65 \mathrm{~cm}, a=8.17 \mathrm{~cm}$ ; find $B$ . (c) $a=16.9 \mathrm{~cm}, b=22.1 \mathrm{~cm}, c=33.5 \mathrm{~cm}$ ; find $A$ .