If `sin theta=(4)/(5)`, then the value of `tan theta` will be :

Options

A. `(4)/(3)`
B. `(5)/(4)`
C. `(5)/(3)`
D. 1

Answer

To find the value of \( \tan \theta \) given that \( \sin \theta = \frac{4}{5} \), we can use the definitions of trigonometric ratios and the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Understand the relationship of sine, cosine, and tangent
We know that:
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{P}{H} \]
From the problem, we have:
\[ \sin \theta = \frac{4}{5} \]
This means the opposite side (P) is 4 and the hypotenuse (H) is 5.

### Step 2: Use the Pythagorean theorem to find the adjacent side
According to the Pythagorean theorem:
\[ H^2 = P^2 + A^2 \]
where \( A \) is the length of the adjacent side. Plugging in the values we have:
\[ 5^2 = 4^2 + A^2 \]
This simplifies to:
\[ 25 = 16 + A^2 \]
Now, subtract 16 from both sides:
\[ A^2 = 25 - 16 = 9 \]
Taking the square root gives:
\[ A = \sqrt{9} = 3 \]

### Step 3: Calculate \( \tan \theta \)
Now that we have both the opposite side (4) and the adjacent side (3), we can find \( \tan \theta \):
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{P}{A} = \frac{4}{3} \]

### Final Answer
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = \frac{4}{3} \]

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