Composition of a trigonometric function with the inverse of another trigonometric function: Problem type 2

To solve the expression ( \sin(\tan^{-1}(-6)) ), follow these steps:

1. Use Right Triangle Definition: Start by letting ( \theta = \tan^{-1}(-6) ). This means that ( \tan(\theta) = -6 ). In a right triangle, the tangent is defined as the ratio of the opposite side to the adjacent side. Therefore:

   • Opposite side = -6 (since we're dealing with negative tangent)

   • Adjacent side = 1 (we can choose this value for simplicity)

2. Calculate the Hypotenuse: To find the hypotenuse ( r ), use the Pythagorean theorem: [ r = \sqrt{(-6)^2 + (1)^2} = \sqrt{36 + 1} = \sqrt{37} ]

3. Determine Sine: Now find ( \sin(\theta) ) using the definition of sine, which is: [ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{-6}{\sqrt{37}} ]

4. Final Value: So, the exact value is: [ \sin(\tan^{-1}(-6)) = \frac{-6}{\sqrt{37}} ]