Determine the trigonometric functions for angles M and B based on the provided image.

The Problem

We are given a right-angled triangle with vertices labeled M, B, and C. The right angle is at vertex C. We need to express the sine, cosine, and tangent of angles M and B.

Solution

Let's denote the lengths of the sides opposite to angles M, B, and C as $$m$$, $$b$$, and $$c$$ respectively. In a right-angled triangle:

• The side opposite the right angle is the hypotenuse.
• The side opposite an angle is the opposite side.
• The side adjacent to an angle (that is not the hypotenuse) is the adjacent side.

In our triangle:

• The hypotenuse is the side opposite to the right angle C, which is side $$MB$$.
• For angle M:
• Opposite side is $$BC$$ (length $$m$$).
• Adjacent side is $$MC$$ (length $$b$$).
• Hypotenuse is $$MB$$ (length $$c$$).
• For angle B:
• Opposite side is $$MC$$ (length $$b$$).
• Adjacent side is $$BC$$ (length $$m$$).
• Hypotenuse is $$MB$$ (length $$c$$).

The trigonometric ratios are defined as:

• Sine (sin) = Opposite / Hypotenuse
• Cosine (cos) = Adjacent / Hypotenuse
• Tangent (tan) = Opposite / Adjacent

Trigonometric Functions for Angle M:

• sin M = Opposite / Hypotenuse = $$\frac{BC}{MB}$$ = $$\frac{m}{c}$$
• cos M = Adjacent / Hypotenuse = $$\frac{MC}{MB}$$ = $$\frac{b}{c}$$
• tg M = Opposite / Adjacent = $$\frac{BC}{MC}$$ = $$\frac{m}{b}$$

Trigonometric Functions for Angle B:

• sin B = Opposite / Hypotenuse = $$\frac{MC}{MB}$$ = $$\frac{b}{c}$$
• cos B = Adjacent / Hypotenuse = $$\frac{BC}{MB}$$ = $$\frac{m}{c}$$
• tg B = Opposite / Adjacent = $$\frac{MC}{BC}$$ = $$\frac{b}{m}$$

Summary from the image:

The image provides the setup for these calculations, indicating the labels for angles and sides, and prompts for the trigonometric functions.

Final Answer:

sin M = $$\frac{m}{c}$$, cos M = $$\frac{b}{c}$$, tg M = $$\frac{m}{b}$$

sin B = $$\frac{b}{c}$$, cos B = $$\frac{m}{c}$$, tg B = $$\frac{b}{m}$$

(Where $$m$$ is the length of side BC, $$b$$ is the length of side MC, and $$c$$ is the length of the hypotenuse MB)