Exam 10: Applications of Trigonometry and Vectors

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

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 $a+b i$ .

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 $151 \mathrm{lb}$ 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?

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

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.

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

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.

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 $a+b i$ .

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 $a+b i$ .

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

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

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.

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.

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 $B$ .

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.

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.

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 $B$ .

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.

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.