Exam 6: Applications of Trigonometry

The parametric equations $x(t)=\frac{1}{\sqrt{t+1}}$ and $y(t)=\frac{t}{t+1}$ where $t \neq-1$ represent what curve? What is the rectangular form of this curve?

Identify the graph of the polar equation $r=a+b \cos \theta \text {, where } 1 \leq \frac{a}{b} \leq 2 \text {. }$

Find the unit vector in the direction of (7,-24) .

State the smallest $\theta \text {-interval }(0 \leq \theta \leq k)$ that gives a complete graph of the polar equation $r=3 \sin 2 \theta \text {. }$ Sketch the graph and label the graph to show which petals are generated by positive values of r and which by negative values of r.

In Problems $15-16$ , assume that $w=6\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)$ is one of the five roots of a complex number $z$ . $\text { Determine } z \text {. Express your answer in standard form. }$

The locations, given in polar coordinates, of two planes approaching the Toronto airport are $\left(820 \mathrm{ft}, 85^{\circ}\right)$ and $\left(460 \mathrm{ft}, 38^{\circ}\right)$ Find the distance between the planes.

The scalar k is equal to 4. Vectors $\mathbf{v}_{1}$ and $\mathbf{V}_{2}$ are defined as $\mathbf{v}_{1}=\langle-3,5\rangle \text { and } \mathbf{v}_{2}=\langle 2,-1\rangle$ Evaluate $\mathbf{k} \mathbf{v}_{1}+\mathbf{v}_{2}$

In Problems $15-16$ , assume that $w=5\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{}\right)$ is one of the six roots of a complex number $z$ . $\text { List the other five roots of } z \text { in trigonometric form. }$

An airplane is flying on a bearing of $140^{\circ}$ at 395 mph. A wind is blowing with a bearing of $190^{\circ}$ at 35 mph. Find the actual speed and direction of the plane and explain how you arrived at your answer.

What is the vector projection of $\mathbf{u}=\langle 8,2\rangle$ onto $\mathbf{v}=\langle 4,-4\rangle$ ?

The locations, given in polar coordinates, of two planes approaching the Ontario airport are $\left(7 \mathrm{mi}, 48^{\circ}\right)$ and $\left(5 \mathrm{mi}, 125^{\circ}\right)$ Find the distance between the planes.

State the smallest $\theta$ -interval $(0 \leq \theta \leq k)$ that gives a complete graph of the polar equation $r=2 \sin 3 \theta$ . Sketch the graph and label the graph to show which petals are generated by positive values of $r$ and which by negative values of $r$ .

What is the work done in lifting a 45 -pound child 5 feet off the ground?

Write the polar equation $r=3 \sin \theta$ in rectangular form.

Evaluate $2\left(\cos \frac{\pi}{12}+i \sin \frac{5 \pi}{12}\right) \cdot 3\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$ Express your answer in trigonometric form.

Determine the rectangular coordinates of the point with polar coordinates $\left(7,162^{\circ}\right)$

Which parametric equations describe a circle with its center at the origin and a radius of 7 for $0 \leq t \leq 2 \pi ?$

Evaluate $3\left(\cos \frac{\pi}{3}+i \sin \frac{2 \pi}{3}\right) \cdot 4\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{}\right)$ Express your answer in trigonometric form.

Which of the following vectors are orthogonal?

Find the angle between $\mathbf{u}=\langle-2,5\rangle \text { and } \mathbf{v}=\langle 1,-3\rangle$