Homework 6

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. When you listen to music involving electronic elements, what type of waves are you listening to?
  2. Convert the following to radians:
    1. $150^\circ$
    2. $-60^\circ$
    3. $-390^\circ$
  3. Convert $\frac{7\pi}{6}$ rad into degrees.
  4. Are $-\frac{\pi}{6}$ rad and $330^\circ$ coterminal?
  5. What is the mathematical relationship between the central angle and the arc length?
  6. Suppose a circle has radius five centimeters. A central angle of measure $40^\circ$ subtends an arc. Find the length of the arc.
  7. Suppose a circle has radius five centimeters. A central angle of measure $40^\circ$ subtends an arc. Find the area of the sector covered by the arc.
  8. Suppose the time is 1:00 PM. How many radians does the minute hand move the clock if it is 1:45 PM? (rotation is clockwise here!)
  9. A central angle $\theta$ in a circle of radius 9 cm is subtended by an arc of length 14 cm. Find the measure of $\theta$ in radians.
  10. A truck with 24 inch radius wheels are rotating at 400 revolutions per minute.
    1. What is the angular speed (in radians)?
    2. What is the linear speed (in inches)?
    3. If instead the truck had 40 inch radius wheels, does the angular speed change? How about linear speed?
  11. Suppose a right triangle $ABC$ has an acute angle $\theta$. The opposite side has length 3, adjacent length 4. Find $\sin(\theta)$.
  12. Sketch a triangle and it's acute angle $\theta$ that has $\tan(\theta) = \frac{10}{8}$. Label all three sides.
  13. A right triangle ABC has one acute angle $45^\circ$. The hypotenuse has length $16\sqrt{2}$. Solve the triangle.
  14. Suppose a truck has 40 inch radius wheels and a line is drawn from the center of the wheel to the right edge of the wheel. Let the current position of this line be the initial side. Assuming no acceleration, the wheel rotates at 0.2 rotations per second. The truck starts moving and stops after 4 seconds. Consider the final position of this drawn line the terminal side. What is the area of the sector between the terminal and initial side?
  15. Why does $t = \dfrac{\pi}{3}$ give the terminal point $P\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$?
  16. Find the side labeled $x$:
  17. A 600-ft guy wire is attached to the top of a communications tower. If the wire makes an angle of $65^\circ$ with the ground, how tall is the communications tower?
  18. From the top of a 200 feet lighthouse, the angle of depression to a ship in the ocean is $23^\circ$. How far is the ship from the base of the lighthouse?
  19. A sequoia tree casts a shadow 100 feet long. Find the height of the tree if the angle of elevation of the sun is $45^\circ$.
  20. Find the area of the shaded region.
  21. Find the area of this triangle.