Given that the point $P = \left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$ corresponds to the real number $t$ on the unit circle, we need to find the values of the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent for the angle $t$.

1. Cosine ( $\cos t$ ):

On the unit circle, the $x$-coordinate corresponds to the value of cosine. $\cos t = -\frac{\sqrt{3}}{2}$

2. Sine ( $\sin t$ ):

On the unit circle, the $y$-coordinate corresponds to the value of sine. $\sin t = \frac{1}{2}$

3. Tangent ( $\tan t$ ):

Tangent is the ratio of sine to cosine: $\tan t = \frac{\sin t}{\cos t} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}$ We can rationalize the denominator: $\tan t = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

4. Cosecant ( $\csc t$ ):

Cosecant is the reciprocal of sine: $\csc t = \frac{1}{\sin t} = \frac{1}{\frac{1}{2}} = 2$

5. Secant ( $\sec t$ ):

Secant is the reciprocal of cosine: $\sec t = \frac{1}{\cos t} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}$ We can rationalize the denominator: $\sec t = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}$

6. Cotangent ( $\cot t$ ):

Cotangent is the reciprocal of tangent: $\cot t = \frac{1}{\tan t} = \frac{1}{-\frac{1}{\sqrt{3}}} = -\sqrt{3}$

Final Answer:

\cos t &= -\frac{\sqrt{3}}{2} \\ \sin t &= \frac{1}{2} \\ \tan t &= -\frac{\sqrt{3}}{3} \\ \csc t &= 2 \\ \sec t &= -\frac{2\sqrt{3}}{3} \\ \cot t &= -\sqrt{3} \end{aligned}$$ Would you like more detailed explanations for any of these steps or have any further questions? --- Here are 5 related questions for further exploration: 1. How do the values of sine, cosine, and tangent change in different quadrants of the unit circle? 2. What is the significance of the unit circle in trigonometry? 3. How do you derive the reciprocal identities for cosecant, secant, and cotangent? 4. How does rationalizing the denominator help simplify trigonometric expressions? 5. What are some common angles on the unit circle and their corresponding trigonometric values? **Tip**: Always remember that on the unit circle, the point $$P = (x, y)$$ corresponds to $$(\cos t, \sin t)$$, and this can help you directly find the values for trigonometric functions.