To find the exact values of the six trigonometric functions for the given angle $\theta$, we start by analyzing the information provided:

1. Point on the Terminal Side: The point $(-4, 4)$ is on the terminal side of $\theta$.
2. Calculate the Radius (r): Using the Pythagorean theorem, we find the distance from the origin to the point $(-4, 4)$, which is the radius $r$: $r = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

Now we can find the values of the six trigonometric functions.

1. $\sin(\theta)$

$\sin(\theta) = \frac{y}{r} = \frac{4}{4\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

2. $\cos(\theta)$

$\cos(\theta) = \frac{x}{r} = \frac{-4}{4\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}$

3. $\tan(\theta)$

$\tan(\theta) = \frac{y}{x} = \frac{4}{-4} = -1$

4. $\csc(\theta)$

$\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$

5. $\sec(\theta)$

$\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}$

6. $\cot(\theta)$

$\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{-1} = -1$

Final Answers

• $\sin(\theta) = \frac{\sqrt{2}}{2}$
• $\cos(\theta) = -\frac{\sqrt{2}}{2}$
• $\tan(\theta) = -1$
• $\csc(\theta) = \sqrt{2}$
• $\sec(\theta) = -\sqrt{2}$
• $\cot(\theta) = -1$

Would you like further explanations on any of these steps or additional details?

Here are five related questions to explore this further:

1. How would these values change if the point were in a different quadrant?
2. What is the significance of the sign changes in trigonometric functions in different quadrants?
3. How can we calculate trigonometric values if given just the angle in degrees or radians instead of a point?
4. Why is the radius $r$ always positive in trigonometric calculations?
5. What are the general values of trigonometric functions for angles of 45°, 135°, 225°, and 315°?

Tip: Remember that the sign of trigonometric functions depends on the quadrant of the angle.