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.

A:

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

B:

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

C:

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

D:

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

E:

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

Problem 2

The clock reads 3:00 p.m.

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

A clock. The hour hand is pointing to the 3. The minute hand is pointing to the 12.
A:

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

B:

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

C:

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

D:

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

E:

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.

  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. \(D=\left(\cos(\text-\frac{16\pi}{3}), \sin(\text-\frac{16\pi}{3})\right)\)
A circle with center at the origin of an x y plane.

Problem 4

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

A:

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

B:

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

C:

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

D:

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

E:

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 circle with center at the origin of an x y plane. Point P lies on the outside of the circle, on the x axis, to the right of the origin.
  1. \(A\), \(\frac{\pi}{4}\) radians
  2. \(B\), \(\frac{\pi}{2}\) radians
  3. \(C\), \(\pi\) radians
  4. \(D\), \(\frac{3\pi}{2}\) radians

Problem 6

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

A:

first 

B:

second

C:

third

D:

fourth

(From Unit 6, Lesson 6.)

Problem 7

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

A:

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

B:

\(y = e^x\)

C:

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

D:

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

E:

\(y = 3\)

(From Unit 6, Lesson 8.)

Problem 8

A circle with center at the origin of an x y plane.
  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?
(From Unit 6, Lesson 10.)