Answer:
The principal value of \(\tan^{-1}(-44)\) refers to the angle whose tangent is \(-44\). Since tangent is a periodic function, there are multiple angles that have the same tangent value. However, the principal value is the angle within a specific range.
To find the principal value of \(\tan^{-1}(-44)\), we can use a scientific calculator or trigonometric identity.
Using a scientific calculator:
1. Press the "tan^-1" or "arctan" button.
2. Enter "-44" as the input.
3. Press the "equals" button.
The principal value of \(\tan^{-1}(-44)\) is approximately \(-88.1\) degrees (rounded to the nearest tenth).
Using a trigonometric identity:
1. Identify the reference angle, which is the positive angle that has the same tangent value. In this case, the reference angle with a tangent of \(44\) is \(88.1\) degrees.
2. Determine the quadrant where the angle lies by considering the sign of the tangent value. Since \(\tan^{-1}(-44)\) is negative, the angle lies in either the third or the fourth quadrant.
3. Determine the angle in the third or fourth quadrant by subtracting the reference angle from \(180\) degrees.
- In the third quadrant: \(180 - 88.1 = 91.9\) degrees.
- In the fourth quadrant: \(360 - 88.1 = 271.9\) degrees.
Both \(91.9\) degrees and \(271.9\) degrees are valid answers for \(\tan^{-1}(-44)\).
Step-by-step explanation:
