Question

use a calculator to find the value of the following expression rounded to two decimal places. tan^{-1}(-22) tan^{-1}(-22) ≈ (type your answer in radians. round to two decimal places as needed.)

Explanation:

Step1: Recall the arctangent function

The function \( \tan^{-1}(x) \) (also written as \( \arctan(x) \)) gives the angle in radians whose tangent is \( x \). For negative values of \( x \), \( \arctan(-x)=-\arctan(x) \) because the tangent function is odd. So we can first find \( \arctan(22) \) and then take the negative of it.

Step2: Use a calculator to find \( \arctan(22) \)

Using a calculator (making sure it is in radians mode), we calculate \( \arctan(22) \). A calculator will give us the value of \( \arctan(22) \approx 1.53 \) (this is an approximation, more accurately, using a calculator: \( \arctan(22)\approx1.5308 \)).

Step3: Apply the odd property

Since \( \tan^{-1}(-22)=-\tan^{-1}(22) \), we have \( \tan^{-1}(-22)\approx - 1.5308 \).

Step4: Round to two decimal places

Rounding \( - 1.5308 \) to two decimal places, we look at the third decimal place which is 0. Since \( 0<5 \), we round down. So \( -1.5308\approx - 1.53 \).

Answer:

\( -1.53 \)