To evaluate the six trigonometric function values for angle $\theta$ based on the given diagram, we need to use the coordinates provided for the point on the unit circle where the terminal side of $\theta$ intersects.

The coordinates of the point are: $\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$ These coordinates represent the $(x, y)$ values on the unit circle for angle $\theta$, where:

• $x = -\frac{\sqrt{2}}{2}$
• $y = \frac{\sqrt{2}}{2}$

Given that the radius $r$ of the unit circle is 1, we can now calculate each of the six trigonometric functions:

1. Sine ($\sin \theta$): $\sin \theta = y = \frac{\sqrt{2}}{2}$

2. Cosine ($\cos \theta$): $\cos \theta = x = -\frac{\sqrt{2}}{2}$

3. Tangent ($\tan \theta$): $\tan \theta = \frac{y}{x} = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1$

4. Cosecant ($\csc \theta$): $\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$

5. Secant ($\sec \theta$): $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}$

6. Cotangent ($\cot \theta$): $\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1$

Summary of the Values

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

Would you like more details on how any of these values were derived or additional explanations?

Here are five related questions that might help expand your understanding:

1. How do we determine the signs of trigonometric functions in different quadrants?
2. What is the significance of a point on the unit circle for determining trigonometric values?
3. How do we calculate the trigonometric functions for special angles like $45^\circ$ or $\frac{\pi}{4}$?
4. Why does the unit circle have a radius of 1, and how does that simplify calculations?
5. How would the values change if $\theta$ were in a different quadrant?

Tip: Remember that each trigonometric function has a reciprocal function, making it easy to find values like $\csc \theta$ once you know $\sin \theta$.