# Lesson 11

Extending the Domain of Trigonometric Functions

- Let’s think about the value of cosine and sine for all types of inputs.

### Problem 1

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

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

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

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

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

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

### Problem 2

The clock reads 3:00 p.m.

Which of the following are true? Select **all** that apply.

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.

### Problem 3

Plot each point on the unit circle.

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

### Problem 4

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.

### Problem 5

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.

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

### Problem 6

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

first

second

third

fourth

### Problem 7

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

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

\(y = e^x\)

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

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

\(y = 3\)

### Problem 8

- List three different counterclockwise angles of rotation around the center of the circle that take \(P\) to \(Q\).
- 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?