If `sintheta=(sqrt(3))/(2)and0^(@)ltthetalt90^(@)`, then the value of `tan(theta-15^(@))` is

Options

A. a.1
B. b.`sqrt(3)`
C. c.`(1)/(sqrt(3))`
D. d.`sqrt(2)`

Answer

To solve the problem step by step, we start with the given information:

**Step 1: Identify the value of theta.**
We know that:
\[ \sin \theta = \frac{\sqrt{3}}{2} \]
From trigonometric values, we recognize that:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
Since \(0^\circ < \theta < 90^\circ\), we conclude:
\[ \theta = 60^\circ \]

**Step 2: Calculate \(\tan(\theta - 15^\circ)\).**
Now we need to find:
\[ \tan(\theta - 15^\circ) = \tan(60^\circ - 15^\circ) = \tan(45^\circ) \]

**Step 3: Find the value of \(\tan(45^\circ)\).**
Using the known value:
\[ \tan 45^\circ = 1 \]

Thus, the final answer is:
\[ \tan(\theta - 15^\circ) = 1 \]

### Summary of Steps:
1. Identify \(\theta\) using \(\sin \theta = \frac{\sqrt{3}}{2}\).
2. Calculate \(\tan(\theta - 15^\circ)\) using the value of \(\theta\).
3. Find the value of \(\tan(45^\circ)\).