 In triangle PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P

Given the triangle PQR, which is right-angled at Q, with \( PR + QR = 25 \) cm and \( PQ = 5 \) cm, we need to determine the values of \(\sin P\), \(\cos P\), and \(\tan P\).

Firstly, we will use the Pythagorean theorem to find the lengths of the sides PR and QR.

### Step-by-Step Solution:

1. **Using the Pythagorean theorem:**

   Since the triangle is right-angled at Q, we can write:
   \[
   PR^2 = PQ^2 + QR^2
   \]
   Let's denote:
   \[
   PR = x \quad \text{and} \quad QR = y
   \]
   We know:
   \[
   x + y = 25 \quad \text{and} \quad PQ = 5
   \]

2. **Express \(y\) in terms of \(x\):**
   \[
   y = 25 - x
   \]

3. **Substitute \(y\) into the Pythagorean theorem:**
   \[
   x^2 = 5^2 + (25 - x)^2
   \]
   \[
   x^2 = 25 + (625 - 50x + x^2)
   \]
   \[
   x^2 = 25 + 625 - 50x + x^2
   \]
   \[
   0 = 650 - 50x
   \]
   \[
   50x = 650
   \]
   \[
   x = 13
   \]
   Therefore, \( PR = 13 \).

4. **Find \( QR \):**
   \[
   QR = 25 - PR = 25 - 13 = 12
   \]

Now, we have:
\[
PR = 13, \quad QR = 12, \quad \text{and} \quad PQ = 5
\]

### Finding the trigonometric ratios:

\[
\sin P = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PR} = \frac{12}{13}
\]

\[
\cos P = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{PQ}{PR} = \frac{5}{13}
\]

\[
\tan P = \frac{\text{opposite}}{\text{adjacent}} = \frac{QR}{PQ} = \frac{12}{5}
\]

### Summary:

- \(\sin P = \frac{12}{13}\)
- \(\cos P = \frac{5}{13}\)
- \(\tan P = \frac{12}{5}\)

Would you like more details or have any questions? 

Here are 5 related questions for further practice:
1. If \( \sin A = \frac{3}{5} \), find \( \cos A \) and \( \tan A \).
2. Given a right triangle with legs of 8 cm and 15 cm, find the hypotenuse.
3. In triangle ABC, right-angled at B, \( AB = 6 \) cm and \( BC = 8 \) cm. Find \( \sin C \), \( \cos C \), and \( \tan C \).
4. Prove the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) using the Pythagorean theorem.
5. For a right triangle with angles \( 30^\circ \), \( 60^\circ \), and \( 90^\circ \), what are the ratios of the sides?

**Tip:** Always double-check your calculations, especially when dealing with trigonometric ratios and right-angled triangles, to avoid simple errors.

Explore how to calculate the sine, cosine, and tangent of angle P in triangle PQR, which is right-angled at Q with sides PR + QR = 25 cm and PQ = 5 cm. Detailed step-by-step solution included.

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