Triangle PQR has right angle Q. If sin `R=4/5`, what is the value of tan P ?

Explanation

To solve the problem, we need to find the value of \( \tan P \) in triangle \( PQR \) where \( \angle Q \) is the right angle and \( \sin R = \frac{4}{5} \).

### Step-by-Step Solution:

1. **Understanding the Triangle**:
In triangle \( PQR \), since \( \angle Q \) is the right angle, we can identify the sides relative to angles \( P \) and \( R \). The side opposite angle \( R \) is \( PQ \) and the hypotenuse is \( PR \).

**Hint**: Remember that in a right triangle, the sides are related to the angles in specific ways.

2. **Using the Sine Definition**:
Given \( \sin R = \frac{4}{5} \), we can express this in terms of the sides:
\[ \sin R = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{PQ}{PR} \]
Here, \( PQ \) is the side opposite angle \( R \) and \( PR \) is the hypotenuse.

**Hint**: Recall that sine is defined as the ratio of the opposite side to the hypotenuse.

3. **Assigning Values**:
From \( \sin R = \frac{4}{5} \), we can assign:
- \( PQ = 4 \) (opposite side)
- \( PR = 5 \) (hypotenuse)

**Hint**: You can assign values based on the sine ratio and scale them appropriately.

4. **Finding the Third Side**:
To find the length of side \( QR \) (the adjacent side to angle \( P \)), we can use the Pythagorean theorem:
\[ PR^2 = PQ^2 + QR^2 \]
Plugging in the values:
\[ 5^2 = 4^2 + QR^2 \]
This simplifies to:
\[ 25 = 16 + QR^2 \]
Rearranging gives:
\[ QR^2 = 25 - 16 = 9 \]
Therefore:
\[ QR = \sqrt{9} = 3 \]

**Hint**: The Pythagorean theorem relates the sides of a right triangle; ensure to isolate the unknown side correctly.

5. **Calculating \( \tan P \)**:
Now we can find \( \tan P \):
\[ \tan P = \frac{\text{Opposite to } P}{\text{Adjacent to } P} = \frac{QR}{PQ} = \frac{3}{4} \]

**Hint**: Remember that tangent is defined as the ratio of the opposite side to the adjacent side.

6. **Final Answer**:
Thus, the value of \( \tan P \) is:
\[ \tan P = \frac{3}{4} \]