91Ó°ÊÓ

Draw a diagram of the unit circle. a) Mark two points, \(P(\theta)\) and \(P(\theta+\pi),\) on your diagram. Use measurements to show that these points have the same coordinates except for their signs. b) Choose a different quadrant for the original point, \(\mathrm{P}(\theta) .\) Mark it and \(\mathrm{P}(\theta+\pi)\) on your diagram. Is the result from part a) still true?

Determine one positive and one negative angle coterminal with each angle. a) \(72^{\circ}\) b) \(\frac{3 \pi}{4}\) c) \(-120^{\circ}\) d) \(\frac{11 \pi}{2}\) e) \(-205^{\circ}\) f) 7.8

Determine whether the angles in each pair are coterminal. For one pair of angles, explain how you know. a) \(\frac{5 \pi}{6}, \frac{17 \pi}{6}\) b) \(\frac{5 \pi}{2},-\frac{9 \pi}{2}\) c) \(410^{\circ},-410^{\circ}\) d) \(227^{\circ},-493^{\circ}\)

Draw and label an angle in standard position with negative measure. Then, determine an angle with positive measure that is coterminal with your original angle. Show how to use a general expression for coterminal angles to find the second angle.

Determine the exact measure of all angles that satisfy the following. Draw a diagram for each. a) \(\sin \theta=-\frac{1}{2}\) in the domain \(\mathbf{0} \leq \boldsymbol{\theta}<2 \pi\) b) \(\cot \theta=1\) in the domain \(-\pi \leq \theta<2 \pi\) c) \(\sec \theta=2\) in the domain \(-180^{\circ} \leq \theta<90^{\circ}\) d) \((\cos \theta)^{2}=1\) in the domain \(-360^{\circ} \leq \theta<360^{\circ}\)

Determine the exact values of the other five trigonometric ratios under the given conditions. a) \(\sin \theta=\frac{3}{5}, \frac{\pi}{2}<\theta<\pi\) b) \(\cos \theta=\frac{-2 \sqrt{2}}{3},-\pi \leq \theta \leq \frac{3 \pi}{2}\) c) \(\tan \theta=\frac{2}{3},-360^{\circ}<\theta<180^{\circ}\) d) \(\sec \theta=\frac{4 \sqrt{3}}{3},-180^{\circ} \leq \theta \leq 180^{\circ}\)

a) Explain, with reference to the unit circle, what the interval \(-2 \pi \leq \theta<4 \pi\) represents. b) Use your explanation to determine all values for \(\theta\) in the interval \(-2 \pi \leq \theta<4 \pi\) such that \(\mathrm{P}(\theta)=\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right).\) c) How do your answers relate to the word "coterminal"?

A rotating water sprinkler makes one revolution every 15 s. The water reaches a distance of \(5 \mathrm{m}\) from the sprinkler. a) What is the arc length of the sector watered when the sprinkler rotates through \(\frac{5 \pi}{3} ?\) Give your answer as both an exact value and an approximate measure, to the nearest hundredth. b) Show how you could find the area of the sector watered in part a). c) What angle does the sprinkler rotate through in 2 min? Express your answer in radians and degrees.

The measure of angle \(\theta\) in standard position is \(4900^{\circ}\) a) Describe \(\theta\) in terms of revolutions. Be specific. b) In which quadrant does \(4900^{\circ}\) terminate? c) What is the measure of the reference angle? d) Give the value of each trigonometric ratio for \(4900^{\circ}\)

Angular velocity describes the rate of change in a central angle over time. For example, the change could be expressed in revolutions per minute (rpm), radians per second, degrees per hour, and so on. All that is required is an angle measurement expressed over a unit of time. a) Earth makes one revolution every \(24 \mathrm{h}\). Express the angular velocity of Earth in three other ways. b) An electric motor rotates at 1000 rpm. What is this angular velocity expressed in radians per second? c) A bicycle wheel completes 10 revolutions every 4 s. Express this angular velocity in degrees per minute.