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We are asked to find the exact values of the six trigonometric functions for the angle $-330^\circ$.

Step 1: Determine the reference angle

The given angle is $-330^\circ$, which is a negative angle. To find the reference angle, we first need to convert the negative angle into a positive one by adding $360^\circ$:

$-330^\circ + 360^\circ = 30^\circ$

So, the reference angle is $30^\circ$.

Step 2: Determine the quadrant

Since $-330^\circ$ is negative, it means we rotate clockwise from the positive x-axis. Adding $360^\circ$ placed the terminal side of the angle in the fourth quadrant.

In the fourth quadrant, the sine function is negative, and cosine and tangent functions are positive.

Step 3: Find the exact trigonometric values

We now use the reference angle $30^\circ$ to determine the values of the six trigonometric functions. From the unit circle, the exact values for the trigonometric functions at $30^\circ$ are:

• $\sin(30^\circ) = \frac{1}{2}$
• $\cos(30^\circ) = \frac{\sqrt{3}}{2}$
• $\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
• $\csc(30^\circ) = 2$
• $\sec(30^\circ) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$
• $\cot(30^\circ) = \sqrt{3}$

Step 4: Apply the signs for the fourth quadrant

Since we are in the fourth quadrant, we need to adjust the signs of the functions based on their behavior in that quadrant:

• $\sin(\theta)$ is negative in the fourth quadrant.
• $\cos(\theta)$ is positive in the fourth quadrant.
• $\tan(\theta)$ is negative in the fourth quadrant.

So, the trigonometric values for $-330^\circ$ will be:

• $\sin(-330^\circ) = -\frac{1}{2}$
• $\cos(-330^\circ) = \frac{\sqrt{3}}{2}$
• $\tan(-330^\circ) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$
• $\csc(-330^\circ) = -2$
• $\sec(-330^\circ) = \frac{2\sqrt{3}}{3}$
• $\cot(-330^\circ) = -\sqrt{3}$

Final Answer:

• $\sin(-330^\circ) = -\frac{1}{2}$
• $\cos(-330^\circ) = \frac{\sqrt{3}}{2}$
• $\tan(-330^\circ) = -\frac{\sqrt{3}}{3}$
• $\csc(-330^\circ) = -2$
• $\sec(-330^\circ) = \frac{2\sqrt{3}}{3}$
• $\cot(-330^\circ) = -\sqrt{3}$

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Let me know if you'd like more detailed steps or further explanations.

Further Questions:

1. What is the significance of the reference angle when determining trigonometric values?
2. How do we determine the sign of trigonometric functions in different quadrants?
3. What are the differences between secant, cosecant, tangent, and cotangent functions?
4. How would you find trigonometric values for angles beyond 360°?
5. What role does the unit circle play in solving trigonometric problems?

Tip:

When solving for trigonometric values of angles, always first find the reference angle and identify the quadrant where the angle lies. This will help you determine the correct signs for the functions!