Exam 6: Applications of Trigonometry

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 $z$ . $\text { List the other four roots of } z \text { in trigonometric form. }$

What is the unit vector in the direction of $\mathrm{v}=\langle 6,-8\rangle ?$

Determine the rectangular coordinates of the point with polar coordinates $\left(8,325^{\circ}\right)$

Plot the complex number $-2+3 i$ in the complex plane and find its absolute value.

Find the fifth roots of unity.

Point A has rectangular coordinates (-4,4) . What are two sets of polar coordinates for point A , where $0 \leq \theta \leq 2 \pi ?$ ?

Find the unit vector in the direction of $\langle-16,12\rangle$

Find the components of the vector V with direction angle $242^{\circ}$ and length 5 .

Which parametric equations describe a spiral beginning at the origin for $t \geq 0 ?$

An NFL punter at the 15 -yard line kicks a football with initial velocity of 90 feet per second at an angle of elevation of $70^{\circ}$ Let t be the elapsed time since the football is kicked. (a) Write parametric equations that represent the distance the football travels. Assume that x=0 represents the 15 -yard line. (b) What interval represents the possible values for t ? (c) What is the distance the ball travels in feet downfield (to the nearest whole number)?

Let $\mathbf{u}=\langle 1,1\rangle$ Find the vector v such that $\mathbf{u} * \mathbf{v}=8$ and $|\mathbf{v}|=\sqrt{32}$

The scalar k is equal to 5 . Vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are defined as $\mathbf{v}_{1}=\langle 2,-6\rangle$ and $\mathbf{v}_{2}=\langle-3,4\rangle$ Evaluate $\mathbf{v}_{1}+\mathbf{k} \mathbf{v}_{2}$

An NFL punter at the 15 -yard line kicks a football with initial velocity of 95 feet per second at an angle of elevation of $65^{\circ}$ Let t be the elapsed time since the football is kicked. (a) Write parametric equations that represent the distance the football travels. Assume that x=0 represents the 15 -yard line. (b) What interval represents the possible values for t ? (c) What is the distance the ball travels in feet downfield (to the nearest whole number)?

Let $\mathbf{u}=\langle-1,-1\rangle$ Find the vector v such that $\mathbf{u} * \mathbf{v}=-6$ and $|\mathbf{v}|=\sqrt{18}$

The parametric equations x(t)=t-2 and $y(t)=\frac{t}{t-2}, \text { where } t \neq 2$ represent what curve? What is the rectangular form of this curve?

What is the vector projection of $\mathbf{u}=\langle 7,3\rangle$ onto $\mathbf{v}=\langle 3,-3\rangle ?$

Find $(2+\sqrt{5} i)^{3}$ using De Moivre's theorem.

Find the quotient $\frac{z_{1}}{z_{2}}$ for the complex numbers $z_{1}=9\left(\cos 240^{\circ}+i \sin 240^{\circ}\right)$ and $z_{2}=4\left(\cos 105^{\circ}+i \sin 105^{\circ}\right)$

Find the quotient $\frac{z_{1}}{z_{2}}$ for the complex numbers $z_{1}=7\left(\cos 220^{\circ}+i \sin 220^{\circ}\right)$ and $z_{2}=2\left(\cos 160^{\circ}+i \sin 160^{\circ}\right)$

A plane is flying on a bearing of $35^{\circ}$ east of north at 550 mph. Express the velocity of the plane as a vector. Assume that there is no wind.