 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

We are given a right-angled triangle \( PQR \) where \( \angle Q = 90^\circ \), \( PR + QR = 25 \, \text{cm} \), and \( PQ = 5 \, \text{cm} \). Our task is to find \( \sin P \), \( \cos P \), and \( \tan P \).

### Step 1: Use the Pythagorean theorem to find the sides

Let:
- \( PQ = 5 \, \text{cm} \)
- \( QR = b \, \text{cm} \)
- \( PR = c \, \text{cm} \) (hypotenuse)
- \( PR + QR = 25 \, \text{cm} \)

From the given condition \( PR + QR = 25 \), we can express:
\[
c + b = 25
\]
So, \( c = 25 - b \).

Now, apply the Pythagorean theorem:
\[
PQ^2 + QR^2 = PR^2
\]
Substitute the known values:
\[
5^2 + b^2 = (25 - b)^2
\]
This simplifies to:
\[
25 + b^2 = 625 - 50b + b^2
\]
Cancel out \( b^2 \) from both sides:
\[
25 = 625 - 50b
\]
Solve for \( b \):
\[
50b = 625 - 25
\]
\[
50b = 600
\]
\[
b = 12
\]

Now substitute \( b = 12 \) into \( PR = 25 - b \):
\[
PR = 25 - 12 = 13
\]

Thus, we have:
- \( PQ = 5 \, \text{cm} \)
- \( QR = 12 \, \text{cm} \)
- \( PR = 13 \, \text{cm} \) (hypotenuse)

### Step 2: Find \( \sin P \), \( \cos P \), and \( \tan P \)

- \( \sin P = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PR} = \frac{12}{13} \)
- \( \cos P = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{PQ}{PR} = \frac{5}{13} \)
- \( \tan P = \frac{\text{opposite}}{\text{adjacent}} = \frac{QR}{PQ} = \frac{12}{5} \)

### Final Answer:
\[
\sin P = \frac{12}{13}, \quad \cos P = \frac{5}{13}, \quad \tan P = \frac{12}{5}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the values of \( \sin P \), \( \cos P \), and \( \tan P \) change if \( PQ \) was different?
2. What is the value of \( \sin R \) in the same triangle?
3. Can you use trigonometric identities to verify the results?
4. How do we apply the Pythagorean theorem in non-right-angled triangles?
5. What are the relationships between trigonometric ratios in similar triangles?

**Tip**: Remember that in any right-angled triangle, the Pythagorean theorem is crucial to find the unknown side lengths!

Explore how to calculate sin P, cos P, and tan P in triangle PQR, a right-angled triangle with specific sides given. Understand the application of the Pythagorean theorem and trigonometric ratios. Detailed step-by-step solution provided.

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