Question

In Exercises \(27-30\) , give the measure of the angle in radians and degrees. Give exact answers whenever possible. $$\tan ^{-1}(-5)$$

Short answer

\(\tan^{-1}(-5)\) radians is exactly equal to -\(\tan^{-1}(5)\) radians or -\(\tan^{-1}(5) * \frac{180}{\pi}^{\circ}\) degrees in exact form.

Step-by-step solution

Find the Angle in Radians

To find the angle in radians that gives a tangent of -5, use the inverse tangent function (\(\tan^{-1}\)). So, calculate \(\tan^{-1}(-5)\). The inverse tangent of any number -n is equal to the negative of the inverse tangent of that number n. In this case, \(\tan^{-1}(-5) = - \tan^{-1}(5)\). Typically, this would be computed using a scientific calculator, but since the exercise asks for exact answers, this might imply that the solutions should be left in terms of \(\tan^{-1}(5)\).

Convert Radians to Degrees

To convert radians to degrees, use the conversion factor that π radians is equal to \(180^{\circ}\). However, in this case, since the exact form is asked, convert \(- \tan^{-1}(5)\) radians into degrees: \(- \tan^{-1}(5) * \frac{180}{\pi}^{\circ}\)