Lesson 11

For which of these angles is the sine negative? Select all that apply.

\(\text-\frac{\pi}{4}\)

\(\text-\frac{\pi}{3}\)

\(\text-\frac{2\pi}{3}\)

\(\text-\frac{4\pi}{3}\)

\(\text-\frac{11\pi}{6}\)

In the next hour, the minute hand moves through an angle of \(2\pi\) radians.

In the next 5 minutes, the minute hand will move through an angle of \(\text-\frac{\pi}{6}\) radians.

After the minute hand moves through an angle of \(\text-\pi\) radians, it is 3:30 p.m.

When the hour hand moves through an angle of \(\text-\frac{\pi}{6}\) radians, it is 4:00 p.m.

The angle the minute hand moves through is 12 times the angle the hour hand moves through.

Plot each point on the unit circle.

1. \(A=(\cos(\text-\frac{\pi}{4}), \sin(\text-\frac{\pi}{4}))\)
2. \(B=(\cos(2\pi),\sin(2\pi))\)
3. \(C=\left(\cos(\frac{16\pi}{3}), \sin(\frac{16\pi}{3})\right)\)
4. \(D=\left(\cos(\text-\frac{16\pi}{3}), \sin(\text-\frac{16\pi}{3})\right)\)

Which of these statements are true about the function \(f\) given by \(f(\theta) = \sin(\theta)\)? Select all that apply.

The graph of \(f\) meets the \(\theta\)-axis at \(0, \pm \pi, \pm 2\pi, \pm 3\pi, \ldots\)

The value of \(f\) always stays the same when \(\pi\) radians is added to the input.

The value of \(f\) always stays the same when \(2\pi\) radians is added to the input.

The value of \(f\) always stays the same when \(\text-2\pi\) radians is added to the input.

The graph of \(f\) has a maximum when \(\theta = \frac{5\pi}{2}\) radians.

Here is a unit circle with a point \(P\) at \((1,0)\).

For each positive angle of rotation of the unit circle around its center listed, indicate on the unit circle where \(P\) is taken, and give a negative angle of rotation which takes \(P\) to the same location.

1. \(A\), \(\frac{\pi}{4}\) radians
2. \(B\), \(\frac{\pi}{2}\) radians
3. \(C\), \(\pi\) radians
4. \(D\), \(\frac{3\pi}{2}\) radians

In which quadrant are both the sine and the tangent negative?

first 

second

third

fourth

Technology required. Each equation defines a function. Graph each of them to identify which are periodic. Select all that are. 

\(y = \sin(\theta)\)

\(y = e^x\)

\(y = x^2 - 2x + 5\)

\(y = \cos(\theta)\)

\(y = 3\)

1. List three different counterclockwise angles of rotation around the center of the circle that take \(P\) to \(Q\).
2. Which quadrant(s) are the angles \(\frac{13\pi}{4}\) and \(\frac{10\pi}{3}\) radians in? Is the sine of these angles positive or negative?