If `costheta=(3)/(4)`,then find the value of `sinthetaandtantheta`.

Answer

To solve the problem where \( \cos \theta = \frac{3}{4} \), we need to find the values of \( \sin \theta \) and \( \tan \theta \).

### Step-by-step Solution:

1. **Understanding Cosine**:
We know that \( \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} \). In this case, we can let the base be 3 and the hypotenuse be 4.

\[ \cos \theta = \frac{3}{4} \]

2. **Using Pythagorean Theorem**:
To find the perpendicular side (opposite side), we can use the Pythagorean theorem, which states:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]
Plugging in the values we have:
\[ 4^2 = 3^2 + \text{Perpendicular}^2 \]
\[ 16 = 9 + \text{Perpendicular}^2 \]
\[ \text{Perpendicular}^2 = 16 - 9 = 7 \]
\[ \text{Perpendicular} = \sqrt{7} \]

3. **Finding Sine**:
Now that we have the perpendicular side, we can find \( \sin \theta \):
\[ \sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{\sqrt{7}}{4} \]

4. **Finding Tangent**:
Next, we can find \( \tan \theta \):
\[ \tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{\sqrt{7}}{3} \]

### Final Answers:
- \( \sin \theta = \frac{\sqrt{7}}{4} \)
- \( \tan \theta = \frac{\sqrt{7}}{3} \)