# Lesson 12

Tangent

- Let’s learn more about tangent.

### Problem 1

Here is a graph of \(f\) given by \(f(\theta) = \tan(\theta)\).

- Are \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) in the domain of \(f\)? Explain how you know.
- What are the \(\theta\)-intercepts of the graph of \(f\)? Explain how you know.

### Problem 2

The function \(f\) is given by \(f(\theta) = \tan(\theta)\). Which of the statements are true? Select **all** that apply.

A:

\(f\) is a periodic function

B:

The domain of \(f\) is all real numbers.

C:

The range of \(f\) is all real numbers.

D:

The period of \(f\) is \(2\pi\).

E:

The period of \(f\) is \(\pi\).

### Problem 3

Here is the unit circle.

If \(\tan(a) > 1\) where could angle \(a\) be on the unit circle?

### Problem 4

Here is a point on the unit circle.

- Explain why the line going through \((0,0)\) and \(P\) has slope \(\frac{1}{2}\).
- What is the tangent of the angle represented by \(P\)? Explain how you know.

### Problem 5

For which angles \(\theta\) between 0 and \(2\pi\) is \(\cos(\theta) < 0\)? Explain how you know.

### Problem 6

It is 3:00 a.m.

- What angle will the hour hand rotate through in the next hour? Explain how you know.
- What angle will the hour hand rotate through in the next 12 hours? Explain how you know.
- What angle will the hour hand rotate through in the next 24 hours? Explain how you know.

### Problem 7

The function \(f\) is given by \(f(x) = x^2\).

- Write an equation for the function \(g\) whose graph is the graph of \(f\) translated 3 units left and then reflected over the \(y\)-axis.
- Write an equation for the function \(h\) whose graph is the graph of \(f\) reflected over the \(y\)-axis and then translated 3 units to the left.
- Do \(g\) and \(h\) have the same graph? Explain your reasoning.