Questions: Evaluate sin(tan^(-1)(1/12)), giving your answer as an exact value (no decimals).

Transcript text: Evaluate $\sin \left(\tan ^{-1}\left(\frac{1}{12}\right)\right)$, giving your answer as an exact value (no decimals).

Solution

Solution Steps

Step 1: Understand the Problem

We are tasked with evaluating $ \sin \left(\tan^{-1}\left(\frac{1}{12}\right)\right) $. This involves finding the sine of an angle whose tangent is $ \frac{1}{12} $.

Step 2: Let $ \theta = \tan^{-1}\left(\frac{1}{12}\right) $

Let $ \theta = \tan^{-1}\left(\frac{1}{12}\right) $. This means that $ \tan(\theta) = \frac{1}{12} $.

Step 3: Construct a Right Triangle

Consider a right triangle where the opposite side to angle $ \theta $ is 1 and the adjacent side is 12. By the Pythagorean theorem, the hypotenuse $ h $ is: \[ h = \sqrt{1^2 + 12^2} = \sqrt{1 + 144} = \sqrt{145}. \]

Step 4: Find $ \sin(\theta) $

The sine of angle $ \theta $ is the ratio of the opposite side to the hypotenuse: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{145}}. \]

Step 5: Rationalize the Denominator

To rationalize the denominator, multiply the numerator and the denominator by $ \sqrt{145} $: \[ \sin(\theta) = \frac{1}{\sqrt{145}} \cdot \frac{\sqrt{145}}{\sqrt{145}} = \frac{\sqrt{145}}{145}. \]

Thus, the exact value of $ \sin \left(\tan^{-1}\left(\frac{1}{12}\right)\right) $ is $ \frac{\sqrt{145}}{145} $.