https://r2.erweima.ai/mathbot/image/17265901842597yp1floq.jpg

Let's break down the image to solve the problems:

### Problem 1:
We have two parts:

#### a) \( \cos 150^\circ \cdot \sin 240^\circ \)

To solve this:
- \( \cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2} \)
- \( \sin 240^\circ = -\sin 60^\circ = -\frac{\sqrt{3}}{2} \)

Thus, the product is:
\[
\cos 150^\circ \cdot \sin 240^\circ = \left( -\frac{\sqrt{3}}{2} \right) \cdot \left( -\frac{\sqrt{3}}{2} \right) = \frac{3}{4}
\]

#### b) \( (-1/2)^5 \cdot (-1/2)^3 \)

To solve this:
- Use the property of exponents: \( a^m \cdot a^n = a^{m+n} \)
\[
(-1/2)^5 \cdot (-1/2)^3 = (-1/2)^{5+3} = (-1/2)^8
\]
\[
(-1/2)^8 = \frac{1}{2^8} = \frac{1}{256}
\]

### Problem 2:
There's a geometric illustration where the angle is \( 150^\circ \) with a line \( l \) making that angle with the x-axis.

Would you like further details or help with understanding the geometric diagram?

### Follow-Up Questions:
1. How do you compute trigonometric functions for angles like \( 150^\circ \) and \( 240^\circ \)?
2. What are the properties of exponents when multiplying powers with the same base?
3. How do you interpret a geometric figure with an angle in the coordinate plane?
4. Can you apply these principles to other angles or exponentiation problems?
5. How does the quadrant affect the sign of trigonometric functions?

#### Tip:
Remember that for angles in different quadrants, the signs of sine and cosine change according to the quadrant the angle is in.

Solve the following:
1) a) cos(150°) * sin(240°)
b) (-1/2)^5 * (-1/2)^3
2) Identify the angle on the geometric diagram (150°).

This page explains how to solve a trigonometric product involving cos(150°) and sin(240°), as well as simplifying an exponentiation expression (-1/2)^5 * (-1/2)^3. Additionally, it discusses geometric interpretation of a 150° angle. Suitable for students in Grades 10-12.

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