Given that [tex]\(\tan(\theta) = -9.9\)[/tex], we want to find the angle [tex]\(\theta\)[/tex] in radians for which this is true. Here is the step-by-step process:
1. Understand the Tangent Function: The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this problem, we are given that [tex]\(\tan(\theta) = -9.9\)[/tex].
2. Inverse Tangent Function: To find the angle [tex]\(\theta\)[/tex] whose tangent is -9.9, we use the inverse tangent function, also known as the arctangent function. The arctangent function, denoted [tex]\(\arctan\)[/tex] or [tex]\(\tan^{-1}\)[/tex], will give us the angle whose tangent is the given number.
3. Calculate [tex]\(\theta\)[/tex]: Applying the arctangent function to -9.9, we have:
[tex]\[
\theta = \tan^{-1}(-9.9)
\][/tex]
4. Find the Result in Radians:
Through the arctangent function, we determine that:
[tex]\[
\theta \approx -1.4701276746370677 \text{ radians}
\][/tex]
This value of [tex]\(\theta \approx -1.4701276746370677\)[/tex] radians is the angle for which the tangent is -9.9.
