The image contains solutions to trigonometry problems involving right triangles. The problems require calculating unknown sides of the triangles using trigonometric ratios. Let's analyze each problem individually.

Problem 1:

• Given: One angle (31°) and the adjacent side (10 units).
• Required: The length of the opposite side (let's denote it as $x$).

Using the tangent ratio: $\tan(31^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{10}$ $x = 10 \times \tan(31^\circ)$

Problem 2:

• Given: Angle (25°) and the hypotenuse (12 units).
• Required: The length of the opposite side (let's denote it as $x$).

Using the sine ratio: $\sin(25^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{12}$ $x = 12 \times \sin(25^\circ)$

Problem 3:

• Given: Angle (40°) and the adjacent side (8 units).
• Required: The length of the opposite side (let's denote it as $x$).

Using the tangent ratio: $\tan(40^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{8}$ $x = 8 \times \tan(40^\circ)$

Problem 4:

• Given: One angle (55°) and the adjacent side (9 cm).
• Required: The length of the opposite side (let's denote it as $x$).

Using the tangent ratio: $\tan(55^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{9}$ $x = 9 \times \tan(55^\circ)$

Summary:

• The problems involve using basic trigonometric ratios: sine, cosine, and tangent.
• Each solution follows the pattern of using the given angle and side to find the unknown side.
• All the calculations involve simple multiplication after determining the correct trigonometric ratio.

Would you like detailed calculations for any specific problem? Or perhaps some questions to deepen your understanding?

Here are five related questions:

1. How would you find the adjacent side if the opposite side and an angle are given?
2. What is the cosine ratio and when is it used in solving right triangles?
3. How can you use the Pythagorean theorem in conjunction with trigonometry?
4. What happens to the tangent ratio as the angle increases?
5. How can you determine whether to use sine, cosine, or tangent in a problem?

Tip: Always double-check your calculator settings to ensure it's in degree mode when solving trigonometric problems involving angles in degrees.