https://file.aiquickdraw.com/mathbot/file/1725378949293213ycil1.jpg

The image contains a right triangle and two trigonometric questions that need to be solved.

### 1. Analyzing the Triangle:
The right triangle has the following measurements:
- Hypotenuse (\(PR\)) = 13 cm
- One leg (\(PQ\)) = 5 cm
- Other leg (\(QR\)) = \(x\) (to be determined)

We can find \(QR\) using the Pythagorean theorem:
\[
PR^2 = PQ^2 + QR^2
\]
Substituting the given values:
\[
13^2 = 5^2 + QR^2
\]
\[
169 = 25 + QR^2
\]
\[
QR^2 = 144
\]
\[
QR = \sqrt{144} = 12 \text{ cm}
\]
So, \(QR = 12\) cm.

### 2. Trigonometric Questions:

The questions on the right side ask to find the sine and cosine of angle \(\theta\).

#### (a) Find \(\sin \theta\):
\[
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PR} = \frac{12}{13}
\]

#### (b) Find \(\cos \theta\):
\[
\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{PQ}{PR} = \frac{5}{13}
\]

### Final Answers:
- \(QR = 12\) cm
- \(\sin \theta = \frac{12}{13}\)
- \(\cos \theta = \frac{5}{13}\)

Would you like more details or have any questions?

### Relative Questions:
1. How do you calculate the tangent of \(\theta\) for this triangle?
2. Can you find the length of the other leg if only one leg and the hypotenuse are known?
3. What are the values of \(\sin \theta\) and \(\cos \theta\) for \(\theta = 45^\circ\)?
4. How does the Pythagorean theorem apply to non-right triangles?
5. What is the relationship between sine, cosine, and tangent for any angle \(\theta\)?

### Tip:
Always remember the basic trigonometric identities: \(\sin^2 \theta + \cos^2 \theta = 1\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). These can help simplify many problems involving right triangles.

A right triangle is given with a hypotenuse of 13 cm, one leg of 5 cm, and an unknown leg. The task is to find the length of the unknown leg and to calculate the sine and cosine of a given angle θ in the triangle.

This problem involves calculating the sine and cosine of angle θ in a right triangle with a hypotenuse of 13 cm and a leg of 5 cm. Using the Pythagorean theorem, we find the missing leg and apply trigonometric formulas. Ideal for Grades 9-10 students.

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