Deck 10: Applications of Trigonometry and Vectors

Question 1

Find the indicated part of each triangle.
(a) $B=50.3^{\circ}, b=5.92 \mathrm{~m}, c=4.11 \mathrm{~m}$ ; find $A$ .
(b) $A=125^{\circ}, c=98 \mathrm{ft}, b=47 \mathrm{ft}$ ; find $a$ .
(c) $a=15.1 \mathrm{ft}, b=28.2 \mathrm{ft}, c=36.7 \mathrm{ft}$ ; find $B$ .

Answer 1

(a) $131 \mathrm{ft}$
(b) $5.7^{\circ}$
(c) $97.4^{\circ}$

Question 2

Can a triangle $A B C$ exist if $a=11.5, b=6.8$ , and $c=18.3$ ? Explain why or why not. Answer this question without using trigonometry.

Answer 2

No, the sum of the lengths of the two smaller sides must be greater than the length of the third side. Here, $11.5+6.8=18.3$ , which is not greater than 18.3.

Question 3

Given $a=15$ and $B=131^{\circ}$ in triangle $A B C$ , determine the values of $b$ for which $A$ has
(a) exactly one value
(b) two values
(c) no values

Answer 3

(a) $b>15$
(b) not possible
(c) $b \leq 15$

Question 4

Solve each applied problem.
(a) Find the magnitude of the resultant of forces of $151 \mathrm{lb}$ and $212 \mathrm{lb}$ that form an angle of $38.6^{\circ}$ .
(b) A regulation softball field is a square and the distance between the bases is $60 \mathrm{ft}$ . The pitcher's mound is $46 \mathrm{ft}$ from home plate. How far is the pitcher's mound from second base?
(c) Find the horizontal and vertical components of a vector with magnitude 263 that is inclined $103^{\circ} 30^{\prime}$ from the horizontal. Give your answer in the form $|a, b|$ .
(d) Two ships leave a harbor together, traveling on courses that have an angle of $122.5^{\circ}$ between them. If they each travel 527 miles, how far apart are they?

Question 5

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 $\mathbf{v}$

Question 6

Find the following for the complex numbers 4 cis $330^{\circ}$ and $\sqrt{3}-i$ :
(a) the rectangular form of 4 cis $330^{\circ}$ .
(b) the trigonometric form of $\sqrt{3}-i$ .
(c) their resultant in the form $a+b i$ .

Question 7

Perform the indicated operation. Give the answer in rectangular form.
(a) $2\left(\cos 30^{\circ}+i \sin 30^{\circ}\right) \cdot 3\left(\cos 180^{\circ}+i \sin 180^{\circ}\right)$
(b) $\frac{3 \operatorname{cis} 90^{\circ}}{4 \operatorname{cis} 30^{\circ}}$
(c) $(-3+2 i)^{3}$
(d) Find the fourth roots of $\frac{3 \sqrt{2}}{2}-\frac{3 \sqrt{2}}{2} i$ . Leave your answers in trigonometric form.

Question 8

Graph the parametric equations $x=3.5 \cos 2 t, y=3.5 \sin 2 t$ , for $t$ in $[0, \pi]$ .

Question 9

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

Question 10

For the polar equation $r=3 \cos \theta$ , 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.

Question 11

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 $A$ .

Question 12

Can a triangle $A B C$ exist if $a=6.8, b=16.1$ , and $c=17.6$ ? Explain why or why not. Answer this question without using trigonometry.

Question 13

Given $a=27$ and $B=141^{\circ}$ in triangle $A B C$ , determine the values of $b$ for which $A$ has
(a) exactly one value
(b) two values
(c) no values

Question 14

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?

Question 15

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 $\mathbf{v}$

Question 16

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 $a+b i$ .

Question 17

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.

Question 18

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

Question 19

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

Question 20

For the polar equation $r=5 \cos \theta$ , 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.

Question 21

Find the indicated part of each triangle.
(a) $A=138^{\circ}, b=132 \mathrm{yd}, c=74.7 \mathrm{yd}$ ; find $a$ .
(b) $A=23.4^{\circ}, a=8.31 \mathrm{~km}, b=10.75 \mathrm{~km}$ ; find $C$ .
(c) $a=11.1 \mathrm{~km}, b=13.5 \mathrm{~km}, c=3.8 \mathrm{~km}$ ; find $B$ .

Question 22

Can a triangle $A B C$ exist if $a=7.4, b=8.3$ , and $c=15.9$ ? Explain why or why not. Answer this question without using trigonometry.

Question 23

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

Question 24

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 $A B$ .

Question 25

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 $\mathbf{v}$

Question 26

Find the following for the complex numbers 3 cis $240^{\circ}$ and $2+2 i \sqrt{3}$ :
(a) the rectangular form of 3 cis $240^{\circ}$ .
(b) the trigonometric form of $2+2 i \sqrt{3}$ .
(c) their resultant in the form $a+b i$ .

Question 27

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

Question 28

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

Question 29

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

Question 30

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.

Question 31

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 $B$ .

Question 32

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.

Question 33

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

Question 34

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?

Question 35

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 $\mathbf{v}$

Question 36

Find the following for the complex numbers 4 cis $135^{\circ}$ and $5 \sqrt{2}-5 i \sqrt{2}$ :
(a) the rectangular form of 4 cis $135^{\circ}$ .
(b) the trigonometric form of $5 \sqrt{2}-5 i \sqrt{2}$ .
(c) their resultant in the form $a+b i$ .

Question 37

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.

Question 38

Graph the parametric equations $x=3 \cos 2 t, y=3 \sin 2 t$ , for $t$ in $[0, \pi]$

Question 39

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

Question 40

For the polar equation $r=3 \sin \theta$ , 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.