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The question in the image involves a right triangle \( \triangle ABC \) where angle \( C \) is given as \( \alpha \), and side lengths \( AB = 6 \) and \( BC = 4 \) are specified. You are asked to determine the trigonometric ratios: \( \sin \alpha \), \( \cos \alpha \), and \( \tan \alpha \).

### Solution:

#### Step 1: Calculate the hypotenuse
Using the Pythagorean theorem to find the length of the hypotenuse \( AC \):
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}
\]

#### Step 2: Determine trigonometric ratios
Since \( \alpha \) is at angle \( C \), the sides relative to angle \( \alpha \) are:
- Opposite side: \( AB = 6 \)
- Adjacent side: \( BC = 4 \)
- Hypotenuse: \( AC = 2\sqrt{13} \)

Now, the trigonometric ratios are:
- \( \sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{2\sqrt{13}} = \frac{3}{\sqrt{13}} \)
- \( \cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{2\sqrt{13}} = \frac{2}{\sqrt{13}} \)
- \( \tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{4} = 1.5 \)

If you wish to rationalize the results:
\[
\sin \alpha = \frac{3}{\sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{3\sqrt{13}}{13}
\]
\[
\cos \alpha = \frac{2}{\sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{2\sqrt{13}}{13}
\]

Thus:
- \( \sin \alpha = \frac{3\sqrt{13}}{13} \)
- \( \cos \alpha = \frac{2\sqrt{13}}{13} \)
- \( \tan \alpha = 1.5 \)

Would you like more details or further explanation?

Here are some related questions you can explore:
1. How do you calculate the hypotenuse in any right triangle?
2. What are the definitions of the basic trigonometric functions in a right triangle?
3. How can you rationalize a denominator when working with trigonometric functions?
4. What are the identities relating \( \sin \), \( \cos \), and \( \tan \)?
5. How do you find the angle \( \alpha \) if you know \( \sin \alpha \)?

**Tip:** Always check whether your triangle's side lengths match the conditions of a right triangle using the Pythagorean theorem.

Diketahui segitiga ABC dengan siku-siku di B dan sudut C adalah α. Tentukan perbandingan trigonometri (sin, cos, tan) dari segitiga tersebut.

Learn how to calculate the trigonometric ratios sin(α), cos(α), and tan(α) for a right triangle ABC with a given side length AB = 6 and BC = 4. This problem utilizes key trigonometric concepts and the Pythagorean theorem. Suitable for Grade 10-11 students.

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