Let's analyze the image you provided.

Step 1: Checking if the Triangle is a Right Triangle

The triangle has sides labeled 3, 4, and 5. To determine if this is a right triangle, we can use the Pythagorean Theorem:

$a^2 + b^2 = c^2$

Here:

• $a = 3$ (one leg),
• $b = 4$ (the other leg), and
• $c = 5$ (the hypotenuse).

Now, check if the equation holds:

$3^2 + 4^2 = 5^2$ $9 + 16 = 25$ $25 = 25$

Since the Pythagorean Theorem holds, this is indeed a right triangle.

Step 2: Finding the Trigonometric Ratios

Since it's a right triangle, we can now calculate the sine, cosine, and tangent for angle $B$, where $B$ is opposite the side of length 4.

Sine of $B$

$\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5}$

Cosine of $B$

$\cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}$

Tangent of $B$

$\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}$

Final Answers:

• $\sin(B) = \frac{4}{5}$
• $\cos(B) = \frac{3}{5}$
• $\tan(B) = \frac{4}{3}$

Would you like further details or clarification? Here are some related questions to deepen your understanding:

1. How does the Pythagorean Theorem help confirm if a triangle is right-angled?
2. What would the sine, cosine, and tangent be for angle $C$?
3. How can you derive the Pythagorean identity from sine and cosine?
4. What are the general conditions for a triangle to be classified as a right triangle?
5. How can you use trigonometric ratios to find unknown angles or sides in right triangles?

Tip: Always verify the hypotenuse is the longest side when using the Pythagorean Theorem.