Problem 6 - Algebra and Trigonometry

Question

\(\tan ^{-1}(-1)=\) _______.

Step-by-Step Solution

Answer

\tan^{-1}(-1) = -\frac{\text{\textbackslash }pi}{4}

Step 1: Understanding the Inverse Tangent Function

The inverse tangent function, written as \(\tan^{-1} x\), gives the angle whose tangent is \(\text{x}\). The output of \(\tan^{-1} x\) is always in the range [\(-\frac{\pi}{2} \, \frac{\pi}{2}\)].

Step 2: Understanding the Input -1

We need to find an angle \(\theta\) such that \(\tan(\theta) = -1\).

Step 3: Identifying the Angle in the Correct Range

Recall that \(\tan \theta = -1\) in the second quadrant, but within the inverse function's range [\(-\frac{\pi}{2} \, \frac{\pi}{2}\)], the angle we're looking for is \(-\frac{\pi}{4} \) since \(\tan(-\frac{\pi}{4}) = -1 \).

Step 4: Conclusion

Thus, \(\tan^{-1}(-1) = -\frac{\pi}{4} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find the angle if we know the value of a trigonometric function. They are the reverse processes of the basic trigonometric functions like sine, cosine, and tangent. For example, while the tangent function gives us the ratio of the opposite side to the adjacent side of a right triangle, the inverse tangent function \(\tan^{-1} x\) tells us the angle that corresponds to this ratio.
The notation \tan^{-1} x \ is used to denote the inverse tangent function. It's important not to confuse this with the reciprocal of the tangent function, which is written as cotangent \cot x \.
When we write \tan^{-1} x \ to find an angle, we are essentially asking, 'What is the angle whose tangent is x?'

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Mathematically, it is written as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \ \]
The tangent function is periodic, with a period of \pi \. This means the function repeats its values every \pi \ radians. It is also undefined at odd multiples of \pi/2\ because the cosine of these angles is zero, which leads to division by zero.

Angle Ranges

When dealing with inverse trigonometric functions, it's crucial to know the range of output values for these functions. For instance, the range of the inverse tangent function, \tan^{-1} x\, is \[-\frac{\pi}{2}, \frac{\pi}{2}\]. This means the angles we get as a result of the inverse tangent function will always fall within this interval.
By keeping the output within this range, we ensure that each output value is uniquely determined. This avoids ambiguity, as trigonometric functions are periodic and can repeat values at multiple angles. For our exercise, we were looking for \theta \ such that \tan(\theta) = -1\ \ in the range \[-\frac{\pi}{2}, \frac{\pi}{2}\]. Hence, the angle is \-frac{\pi}{4}\.