https://file.aiquickdraw.com/mathbot/file/172568440725116v1hlvf.jpg

The problem asks to:

1. Draw a small sketch of each angle, labeling the angle and the reference angle.
2. Find the decimal approximation of the sine or cosine using a calculator.

Let's break down the solution for each part:

### Part (a): \(\sin\left(\frac{11\pi}{5}\right)\)

1. **Determine the angle in radians:**
   The given angle is \(\frac{11\pi}{5}\) radians. To understand where this angle lies, let's convert it into degrees:
   
   \[
   \text{Degrees} = \frac{11\pi}{5} \times \frac{180}{\pi} = \frac{11 \times 180}{5} = 396^\circ
   \]

   This angle is more than one full rotation (360 degrees). To find the equivalent angle between \(0^\circ\) and \(360^\circ\), subtract \(360^\circ\):

   \[
   396^\circ - 360^\circ = 36^\circ
   \]

   So, the reference angle is \(36^\circ\).

2. **Determine the sine of the angle:**
   Since \(396^\circ\) is in the first quadrant, where the sine is positive, we have:

   \[
   \sin\left(\frac{11\pi}{5}\right) = \sin(36^\circ)
   \]

3. **Calculate the decimal approximation:**

   Using a calculator:

   \[
   \sin(36^\circ) \approx 0.5878
   \]

### Part (b): \(\cos(100^\circ)\)

1. **Identify the angle:**
   The angle is \(100^\circ\), which is in the second quadrant (between \(90^\circ\) and \(180^\circ\)).

2. **Find the reference angle:**
   The reference angle for \(100^\circ\) is:

   \[
   180^\circ - 100^\circ = 80^\circ
   \]

3. **Determine the cosine of the angle:**
   In the second quadrant, cosine is negative, so:

   \[
   \cos(100^\circ) = -\cos(80^\circ)
   \]

4. **Calculate the decimal approximation:**

   Using a calculator:

   \[
   \cos(100^\circ) \approx -0.1736
   \]

### Summary:
- **(a)** \(\sin\left(\frac{11\pi}{5}\right) \approx 0.5878\)
- **(b)** \(\cos(100^\circ) \approx -0.1736\)

Would you like further details, or do you have any specific questions?

### Related Questions:
1. How do you find reference angles for any given angle in radians?
2. What are the properties of trigonometric functions in different quadrants?
3. How can you determine the sign of trigonometric functions based on their quadrant?
4. How do you convert between degrees and radians?
5. What are the key trigonometric values for special angles?

**Tip:** Always remember that the reference angle is the smallest angle between the terminal side of the given angle and the x-axis.

Draw a small sketch of each angle, labeling the angle and the reference angle (in the measurement type given). Using a calculator, find the decimal approximation of the sine or cosine:
a) sin(11π/5)
b) cos(100°)

Learn how to calculate the sine and cosine of angles, such as sin(11π/5) and cos(100°), by converting radians to degrees, finding reference angles, and using a calculator for decimal approximations. Perfect for students in Grades 10-12.

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