Exam 8: Applications of Trigonometry

Solve the problem. -A painter is going to apply a special coating to a triangular metal plate on a new building. Two sides measure are $10.7 \mathrm {~m}$ and $6.4 \mathrm {~m}$ . She knows that the angle between these sides is $25.3 ^ { \circ }$ . What is the area of the surface she plans to cover with the coating? Round to the nearest square meter.

Find the product. Write the product in rectangular form, using exact values. -[4 $\left. \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) \right] \left[ 5 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right) \right]$

Solve the problem. -A pilot wants to fly on a bearing of $60.1 ^ { \circ }$ . By flying due east, he finds that a $51 - \mathrm { mph }$ wind, blowing from the south, puts him on course. Find the ground speed of the plane.

Use a table of values to graph the plane curve defined by the following parametric equations. Find a rectangular equation for the curve. -

Find the indicated vector. -Let $\mathbf { a } = \langle 5 , - 1 \rangle$ . Find $8 \mathbf { a }$ .

Solve the problem. -Find the area of a triangular-shaped field with sides of $244.5 \mathrm {~m}$ and $163.9 \mathrm {~m}$ , and the included angle between them measuring $64.65 ^ { \circ }$ . Round to the nearest square meter.

Find the given power. Write answer in rectangular form. - $( - 5 \sqrt { 2 } + 5 \sqrt { 2 } i ) ^ { 3 }$

Find all solutions of the equation. Leave answers in trigonometric form. - $x ^ { 2 } - i = 0$

Solve. -A complex number $\mathrm { z }$ does not belong to the Mandelbrot set if any of the complex numbers in the sequence $z , z ^ { 2 } + z , \left( z ^ { 2 } + z \right) ^ { 2 } + z , \left[ \left( z ^ { 2 } + z \right) ^ { 2 } + z \right] ^ { 2 } + z , \ldots$ has modulus exceeding 2 . Does $z = 0.7$ belong to the Mandelbrot set?

Solve the problem. -A wind has an easterly component of $6 \mathrm {~km} / \mathrm { h }$ and a northerly component of $17 \mathrm {~km} / \mathrm { h }$ . Find the direction of the wind.

Solve the problem. -Suppose that a radio signal transmission pattern can be modeled by $r = 48 + 48 \sin \theta$ , for $0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$ , where the units are miles. Assume that the positive $x$ -axis points east, and determine the distances of transmission in the east, north, west, and south directions.

Determine the number of triangles ABC possible with the given parts. - $\mathrm { b } = 44 , \mathrm { c } = 54 , \mathrm {~B} = 100 ^ { \circ }$

Solve the problem. -A box weighing $77 \mathrm { lb }$ is hanging from the end of a rope. The box is pulled sideways by a horizontal rope with a force of $20 \mathrm { lb }$ . What angle, to the nearest degree, does the first rope make with the vertical?

Solve the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. - =15. =100. =22.4

Solve the problem. -If $\mathbf { u } = \langle - 5,3 \rangle , \mathbf { v } = \langle 3 , - 6 \rangle$ , and $\mathbf { w } = \langle - 3,12 \rangle$ , evaluate $\mathbf { u } \cdot \mathbf { w } + \mathbf { v } \cdot \mathbf { w }$ .

Use the figure to find the specified vector. -Find -a.

Solve the problem. -In a parallel electrical circuit, the impedance $Z$ can be calculated using the equation $\frac { 1 } { Z } = \left( \frac { 1 } { Z _ { 1 } } + \frac { 1 } { Z _ { 2 } } \right)$ where $\mathrm { Z } _ { 1 }$ and $\mathrm { Z } _ { 2 }$ are the impedances for the branches of the circuit. The phase angle measures the phase difference between the voltage and the current in an electrical circuit. If the impedance $\mathrm { Z }$ is expressed in the form $a + b i , \theta$ can be determined by the equation $\tan \theta = b / a$ . Determine the phase angle $\theta$ (in degrees) for a parallel circuit in which $\mathrm { Z } _ { 1 } = 10 + 20 \mathrm { i }$ and $\mathrm { Z } _ { 2 } = 30 + 10 \mathrm { i }$ .

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle - 5,3 \rangle , \mathbf { v } = \langle 2,4 \rangle$ . Find $2 \mathbf { u } + 5 \mathbf { v }$ .

Solve the problem. -Two people are carrying a box. One person exerts a force of 129 pounds at an angle of $46.2 ^ { \circ }$ with the horizontal. The other person exerts a force of 93 pounds at an angle of $52.5 ^ { \circ }$ . Find the weight of the box.

Find sum of the pair of complex numbers. -8,9