https://file.aiquickdraw.com/mathbot/file/17291009382710kqvg05x.jpg

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.

Decide if the given triangle is a right triangle. If it is, find the sine, cosine, and tangent for the angle B.

Learn how to verify if a triangle with sides 3, 4, and 5 is a right triangle using the Pythagorean Theorem. Also, find the sine, cosine, and tangent for angle B in this triangle. Suitable for students in Grades 9-10, this problem covers key trigonometric concepts and theorems.

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