https://file.aiquickdraw.com/mathbot/file/17258997770259dftxbdn.jpg

To solve for \(\sin A\) and \(\cos A\), we can use the information given in the problem.

### Step 1: Identify the sides using \(\tan A\)
The given ratio is \(\tan A = \frac{9}{12}\), where:
- \(\tan A = \frac{\text{opposite}}{\text{adjacent}}\).

Thus:
- Opposite side to \(A = 9\),
- Adjacent side to \(A = 12\).

### Step 2: Use the Pythagorean theorem to find the hypotenuse
We can now find the hypotenuse \(h\) using the Pythagorean theorem:
\[
h = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15
\]
So, the hypotenuse is 15.

### Step 3: Calculate \(\sin A\) and \(\cos A\)
- \(\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{15} = 0.6\),
- \(\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{15} = 0.8\).

Thus:
\[
\sin A = 0.6 \quad \text{and} \quad \cos A = 0.8
\]

Would you like any further details on this?

Here are 5 related questions for practice:
1. What is the value of \(\tan A\) when given \(\sin A\) and \(\cos A\)?
2. How do you calculate the remaining angles in a right triangle if \(\sin A\) is known?
3. How does the Pythagorean theorem help in trigonometry?
4. What is the formula for \(\cot A\) in terms of \(\sin A\) and \(\cos A\)?
5. How would you find the angle \(A\) if you know \(\sin A\)?

**Tip:** Remember, the trigonometric ratios are always based on the sides relative to the angle you're working with!

In the triangle below, if tan(A) = 9/12, then find sin(A) and cos(A).

Find sin(A) and cos(A) in a right triangle with tan(A) = 9/12. This solution involves using trigonometric ratios and the Pythagorean theorem. Suitable for students in Grades 8-10.

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