If `tan theta = 3//4` and `theta` acute, then `cosec theta` is equal to

Options

A. `5/3`
B. `5/4`
C. `4/3`
D. `4/3`

Answer

To find the value of \( \csc \theta \) given that \( \tan \theta = \frac{3}{4} \) and \( \theta \) is acute, we can follow these steps:

### Step 1: Understand the definition of tangent
The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side (perpendicular) to the length of the adjacent side (base). Therefore, we can set:
- Opposite side (perpendicular) = 3
- Adjacent side (base) = 4

### Step 2: Use the Pythagorean theorem to find the hypotenuse
To find the hypotenuse \( h \), we use the Pythagorean theorem:
\[ h^2 = \text{perpendicular}^2 + \text{base}^2 \]
Substituting the values:
\[ h^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
Taking the square root:
\[ h = \sqrt{25} = 5 \]

### Step 3: Find \( \csc \theta \)
The cosecant of an angle is defined as the ratio of the hypotenuse to the opposite side:
\[ \csc \theta = \frac{h}{\text{perpendicular}} = \frac{5}{3} \]

### Conclusion
Thus, the value of \( \csc \theta \) is \( \frac{5}{3} \).

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