The diagram shows an angle $\theta$ in standard position, where the terminal side intersects the unit circle at the point $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$. This point provides the coordinates $(x, y)$ of the terminal point on the unit circle, which can be used to evaluate the six circular (trigonometric) function values of $\theta$.

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 ($\csc \theta$) (reciprocal of sine): $\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}$

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

6. Cotangent ($\cot \theta$) (reciprocal of tangent): $\cot \theta = \frac{1}{\tan \theta} = \frac{1}{1} = 1$

Summary of the six circular function 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 further clarification on any of these values or their derivation? Here are some related questions you might explore:

1. What is the significance of the unit circle in trigonometry?
2. How do you find trigonometric values for angles that are not standard positions?
3. How do the signs of trigonometric functions change in different quadrants?
4. Why is the tangent of an angle the ratio of sine to cosine?
5. What are the applications of the reciprocal trigonometric functions?

Tip: Remember, on the unit circle, the coordinates $(x, y)$ are always $(\cos \theta, \sin \theta)$, which directly helps in finding the sine and cosine values for any angle $\theta$.