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The problem in your image asks to find the exact values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for the angle \( 420^\circ \).

### Step-by-Step Solution:

1. **Standardize the angle:**
   To work with angles, we first reduce \( 420^\circ \) to an equivalent angle within the standard \( 0^\circ \) to \( 360^\circ \) range by subtracting \( 360^\circ \):
   \[
   420^\circ - 360^\circ = 60^\circ
   \]
   So, \( \sin(420^\circ) = \sin(60^\circ) \), and similarly for the other functions.

2. **Use known trigonometric values for \( 60^\circ \):**
   From the unit circle, the trigonometric values for \( 60^\circ \) are well-known:
   \[
   \sin(60^\circ) = \frac{\sqrt{3}}{2}, \quad \cos(60^\circ) = \frac{1}{2}, \quad \tan(60^\circ) = \sqrt{3}
   \]
   Now we can find the remaining functions:
   \[
   \csc(60^\circ) = \frac{1}{\sin(60^\circ)} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}, \quad \sec(60^\circ) = \frac{1}{\cos(60^\circ)} = 2
   \]
   \[
   \cot(60^\circ) = \frac{1}{\tan(60^\circ)} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}
   \]

3. **Conclusion:**
   The exact values of the six trigonometric functions for \( 420^\circ \) (or \( 60^\circ \)) are:
   - \( \sin(420^\circ) = \frac{\sqrt{3}}{2} \)
   - \( \cos(420^\circ) = \frac{1}{2} \)
   - \( \tan(420^\circ) = \sqrt{3} \)
   - \( \csc(420^\circ) = \frac{2\sqrt{3}}{3} \)
   - \( \sec(420^\circ) = 2 \)
   - \( \cot(420^\circ) = \frac{\sqrt{3}}{3} \)

Would you like further explanation on any step or the application of trigonometric identities?

### Relative Questions:
1. How do we calculate trigonometric functions for angles greater than \(360^\circ\)?
2. Why do we subtract \(360^\circ\) to reduce the angle to a standard form?
3. What is the importance of the unit circle in trigonometry?
4. How do you find the trigonometric values for angles like \(45^\circ\) or \(30^\circ\)?
5. What is the relationship between the sine and cosecant functions?

### Tip:
Always simplify angles greater than \(360^\circ\) by subtracting multiples of \(360^\circ\) to bring the angle within the first full rotation of the unit circle!

Find the exact values of the six trigonometric functions of the angle 420°.

Learn how to find the exact values of the six trigonometric functions for the angle 420°. This problem involves standardizing the angle and using known trigonometric values for 60° based on the unit circle. Suitable for high school students in Grades 10-12.

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