Use the unit circle to define cos θ and sin θ for any number θ between 0 and 360,

inclusive. Then explain how to use cos θ and sin θ to define tan θ.

To define \(\cos \theta\) and \(\sin \theta\) using the unit circle, we can think of the unit circle as a circle with a radius of 1, centered at the origin \((0, 0)\) of the coordinate plane. The equation of the unit circle is:

\[
x^2 + y^2 = 1
\]

### Step 1: Understanding the Unit Circle and Angles

- **Position of an Angle (\(\theta\)):** An angle \(\theta\) is measured from the positive x-axis, counterclockwise if \(\theta\) is positive and clockwise if \(\theta\) is negative.
- The angle \(\theta\) can be any value between \(0^\circ\) and \(360^\circ\) (inclusive), where \(0^\circ\) is along the positive x-axis, and \(360^\circ\) completes a full rotation back to \(0^\circ\).

### Step 2: Defining \(\cos \theta\) and \(\sin \theta\) Using the Unit Circle

For a given angle \(\theta\), the coordinates of the point on the unit circle corresponding to that angle are:

\[
(x, y) = (\cos \theta, \sin \theta)
\]

Thus, for any angle \(\theta\), the **cosine** of the angle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle, and the **sine** of the angle is the y-coordinate of that point.

- **\(\cos \theta\)** corresponds to the horizontal distance from the origin.
- **\(\sin \theta\)** corresponds to the vertical distance from the origin.

### Example Values:
- \(\cos 0^\circ = 1\), \(\sin 0^\circ = 0\) (point \((1, 0)\))
- \(\cos 90^\circ = 0\), \(\sin 90^\circ = 1\) (point \((0, 1)\))
- \(\cos 180^\circ = -1\), \(\sin 180^\circ = 0\) (point \((-1, 0)\))
- \(\cos 270^\circ = 0\), \(\sin 270^\circ = -1\) (point \((0, -1)\))
- \(\cos 360^\circ = 1\), \(\sin 360^\circ = 0\) (same as \(0^\circ\))

### Step 3: Defining \(\tan \theta\) Using \(\cos \theta\) and \(\sin \theta\)

The tangent of an angle \(\theta\) is defined as the ratio of \(\sin \theta\) to \(\cos \theta\):

\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

This ratio gives us the slope of the line formed by the terminal side of the angle \(\theta\) and the x-axis. However, it's important to note that \(\tan \theta\) is undefined when \(\cos \theta = 0\), which occurs at:

- \(\theta = 90^\circ\)
- \(\theta = 270^\circ\)

At these points, the line is vertical, and the tangent function does not exist (it goes to infinity or is undefined).

### Example Values of \(\tan \theta\):
- \(\tan 0^\circ = 0\)
- \(\tan 45^\circ = 1\)
- \(\tan 135^\circ = -1\)
- \(\tan 180^\circ = 0\)
- \(\tan 270^\circ\) is undefined.

Would you like more details on any specific part of this explanation?

Here are 5 questions you could explore further:

1. How can you use the unit circle to find values of \(\sin \theta\) and \(\cos \theta\) for angles greater than \(360^\circ\)?
2. Why is \(\tan \theta\) undefined at \(90^\circ\) and \(270^\circ\)?
3. How do \(\cos \theta\) and \(\sin \theta\) change as \(\theta\) moves through different quadrants?
4. What are the exact values of \(\cos \theta\), \(\sin \theta\), and \(\tan \theta\) for commonly used angles (like \(30^\circ\), \(60^\circ\))?
5. How can you use symmetry in the unit circle to determine values of trigonometric functions in different quadrants?

**Tip:** Memorizing key values of sine and cosine for angles like \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\) will make trigonometry problems easier to solve!

Use the unit circle to define cos θ and sin θ for any number θ between 0 and 360, inclusive. Then explain how to use cos θ and sin θ to define tan θ.

Explore how to use the unit circle to define cos θ and sin θ for any angle between 0° and 360°. Learn how to calculate tan θ using the ratio of sin θ to cos θ, including key values and why tan θ is undefined at specific angles. Suitable for students in Grades 9-12.

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