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.