Problem 40 - Algebra and Trigonometry

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$ \tan ^{-1}\left[\tan \left(-\frac{\pi}{3}\right)\right] $$

Step-by-Step Solution

Answer

\(-\frac{\pi}{3}\)

Step 1: Understanding the problem

The problem has given the expression \(\tan ^{-1}\left[\tan \left(-\frac{\pi}{3}\right)\right]\) and we have to find its value. 'tan' and 'tan^-1' are inverses of each other, meaning they cancel each other out. This implies that the expression ultimately simplifies to the angle \(-\frac{\pi}{3}\). However, it's important to note that the result is valid only if \(-\frac{\pi}{3}\) falls within the range of the inverse tangent function, which is \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 2: Evaluate the value of the angle

The angle given here is \(-\frac{\pi}{3}\). To check whether it's in the appropriate range, we must convert it into decimal form, \(-\frac{\pi}{3}\) is approximately -1.04 radians.

Step 3: Compare the angle with the range of inverse tangent function

Our obtained angle \(-\frac{\pi}{3}\) is less than \(\frac{\pi}{2}\) but more than \(-\frac{\pi}{2}\). Hence it lies within the range of the inverse tangent function and the expression can be simplified as \(-\frac{\pi}{3}\).

Key Concepts

Tangent function

The tangent function, denoted as \( \tan(x) \), is one of the six fundamental trigonometric functions. It relates the angles of a right triangle to the ratios of two sides. Specifically, for an angle \( x \) in a right triangle, the tangent is defined as the ratio of the opposite side to the adjacent side. This can be expressed mathematically as:

• \( \tan(x) = \frac{\text{opposite}}{\text{adjacent}} \)

When we use the tangent function in expression such as \( \tan\left(-\frac{\pi}{3}\right) \), it provides the tangent of the angle \( -\frac{\pi}{3} \). Understanding the behavior of tangent helps in grasping why certain values are used or omitted, especially in inverse trigonometric functions. The tangent function is periodic with a period of \( \pi \), which means it repeats its values every \( \pi \) radians.
The tangent function has asymptotes at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer, because it approaches infinity there.

Angle evaluation

Evaluating angles requires understanding both radians and degrees. In trigonometry, angles are often measured in radians.
One complete revolution around a circle is \( 2\pi \) radians, equivalent to 360 degrees. In the expression \( \tan^{-1}\left[\tan\left(-\frac{\pi}{3}\right)\right] \), the angle \( -\frac{\pi}{3} \) is in radians, approximately \(-1.04 \) radians, or about \(-60\) degrees.

• Converting between degrees and radians is done using the relationship: \( 180^\circ = \pi \text{ radians} \).

Angle evaluation ensures that we check whether the angle falls within appropriate boundaries, especially important for inverse functions.
For inverse tangent, the resulting angle must lie within \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \). As \( -\frac{\pi}{3} \) lies within this interval, the simplified expression remains \( -\frac{\pi}{3} \). This clarity is essential in correcting potential mistakes.

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angle values. They provide methods to simplify and solve trigonometric equations. A key identity involving tangent is that \( \tan(\theta) \) and its inverse \( \tan^{-1}(\theta) \) effectively "undo" one another when the angle \( \theta \) is within the correct range.
This is because:

• \( \tan^{-1}(\tan(x)) = x \), for \( x \) within \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

However, the expression \( \tan^{-1}\left[\tan(x)\right] \) does not simply revert to \( x \) if \( x \) is outside this range. Understanding trigonometric identities and their limitations helps in manipulating and solving equations effectively.
When we apply this to \( \tan^{-1}\left[\tan\left(-\frac{\pi}{3}\right)\right] \), the identity confirms the angle is \( -\frac{\pi}{3} \) since it fits within the specified range for inverse tangent.