https://file.aiquickdraw.com/mathbot/file/1724725857928m015cftc.jpg

for the cos, the options are -1, 1, -1/2 and 1/2
for the sin, the options are the same as that

Let's solve the trigonometric expressions step by step.

### 1. **Cosine Calculation:**
   \[
   \cos(420^\circ)
   \]
   To simplify \( 420^\circ \), subtract \( 360^\circ \) (since one full circle is \( 360^\circ \)):
   \[
   420^\circ - 360^\circ = 60^\circ
   \]
   Therefore:
   \[
   \cos(420^\circ) = \cos(60^\circ)
   \]
   The value of \( \cos(60^\circ) \) is \( \frac{1}{2} \).

### 2. **Sine Calculation:**
   \[
   \sin(450^\circ)
   \]
   Subtract \( 360^\circ \) from \( 450^\circ \) to find the corresponding angle within the standard range:
   \[
   450^\circ - 360^\circ = 90^\circ
   \]
   Therefore:
   \[
   \sin(450^\circ) = \sin(90^\circ)
   \]
   The value of \( \sin(90^\circ) \) is \( 1 \).

### **Final Answers:**
- \(\cos(420^\circ) = \frac{1}{2}\)
- \(\sin(450^\circ) = 1\)

Would you like more details or have any questions? Here are some related questions for further understanding:

1. What is the general approach to solving trigonometric functions with angles greater than 360°?
2. How do the unit circle and reference angles help in solving trigonometric functions?
3. Can you explain the symmetry properties of trigonometric functions?
4. What is the significance of negative angles in trigonometry?
5. How do you calculate trigonometric functions for angles that are multiples of \(30^\circ\) and \(45^\circ\)?

**Tip:** Always reduce angles larger than \(360^\circ\) by subtracting \(360^\circ\) repeatedly until the angle is within the standard \(0^\circ\) to \(360^\circ\) range. This makes finding trigonometric values straightforward.

Learn how to solve trigonometric expressions involving angles greater than 360°. This detailed guide covers cosine and sine calculations for angles like 420° and 450°, using unit circle trigonometry and symmetry properties of trigonometric functions. Suitable for high school students studying trigonometry.

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