If `sin theta=(a)/(b)`,then find the value of (i) `costheta`, (ii) `tantheta`

Answer

To find the values of \( \cos \theta \) and \( \tan \theta \) given that \( \sin \theta = \frac{a}{b} \), we can follow these steps:

### Step 1: Understand the relationship of sine, cosine, and tangent
We know that:
- \( \sin \theta = \frac{\text{perpendicular}}{\text{hypotenuse}} \)
- \( \cos \theta = \frac{\text{base}}{\text{hypotenuse}} \)
- \( \tan \theta = \frac{\text{perpendicular}}{\text{base}} \)

### Step 2: Identify the sides of the triangle
From the given \( \sin \theta = \frac{a}{b} \):
- The hypotenuse is \( b \).
- The perpendicular (opposite side) is \( a \).

### Step 3: Calculate the base using Pythagorean theorem
According to the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{base}^2 + \text{perpendicular}^2 \]
Substituting the known values:
\[ b^2 = \text{base}^2 + a^2 \]
Rearranging gives:
\[ \text{base}^2 = b^2 - a^2 \]
Taking the square root, we find the base:
\[ \text{base} = \sqrt{b^2 - a^2} \]

### Step 4: Calculate \( \cos \theta \)
Using the definition of cosine:
\[ \cos \theta = \frac{\text{base}}{\text{hypotenuse}} = \frac{\sqrt{b^2 - a^2}}{b} \]

### Step 5: Calculate \( \tan \theta \)
Using the definition of tangent:
\[ \tan \theta = \frac{\text{perpendicular}}{\text{base}} = \frac{a}{\sqrt{b^2 - a^2}} \]

### Final Results
Thus, the values are:
1. \( \cos \theta = \frac{\sqrt{b^2 - a^2}}{b} \)
2. \( \tan \theta = \frac{a}{\sqrt{b^2 - a^2}} \)

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