Problem 37 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression, if it is defined. $$\tan ^{-1}\left(\tan \left(\frac{2 \pi}{3}\right)\right)$$

Step-by-Step Solution

Answer

The exact value is \(-\frac{\pi}{3}\).

Step 1: Understand the Range of Arctan Function

The function \( \tan^{-1}(x) \) or \( \arctan(x) \) has a range of \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \). This means the output of \( \tan^{-1}(x) \) will always be an angle within this range.

Step 2: Calculate \( \tan\left(\frac{2\pi}{3}\right) \)

The angle \( \frac{2\pi}{3} \) is in the second quadrant. In the second quadrant, the tangent function is negative. The reference angle for \( \frac{2\pi}{3} \) is \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \). The tangent of \( \frac{\pi}{3} \) is \( \sqrt{3} \), so \( \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} \).

Step 3: Apply \(\tan^{-1}\) to the \(\tan\) Result

Calculate \( \tan^{-1}(-\sqrt{3}) \). Since \(-\sqrt{3}\) is within the domain of \( \tan^{-1}(x) \), we find the angle whose tangent is \(-\sqrt{3}\) and within the range \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \). This gives \( \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \).

Step 4: Confirm the Angle is Within Range

Verify that \(-\frac{\pi}{3}\) is within the allowable range of \(\tan^{-1}(x)\). Since \(-\frac{\pi}{3} \) is between \(-\frac{\pi}{2} \) and \( \frac{\pi}{2} \), it is a valid result.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan(x) \), is a key trigonometric function that relates to the sine and cosine of a given angle. It is defined as the ratio of the sine to the cosine of that angle. In mathematical terms, it's expressed as:\[\tan(x) = \frac{\sin(x)}{\cos(x)}\]This function is periodic, with a period of \( \pi \), meaning it repeats its values every \( \pi \) units. Because of its relationship with sine and cosine, the tangent function has vertical asymptotes where the cosine of the angle equals zero, resulting in undefined values. These asymptotes appear at odd multiples of \( \frac{\pi}{2} \).In the unit circle,

• when the angle is right at \( 0 \), the tangent is \( 0 \),
• when at \( \frac{\pi}{4} \), it becomes \( 1 \),
• and when at \( \frac{\pi}{2} \), it's undefined.

The sign of the tangent depends on the quadrant of the angle. In the second quadrant, such as in our problem with \( \frac{2\pi}{3} \), the tangent is negative due to the sine being positive and cosine being negative.

Arctan Function

The arctan function, also known as the inverse tangent function and denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \), is used to determine the angle whose tangent is \( x \). It is the reverse operation of the tangent function.One crucial aspect of the arctan function is its range. It only returns values (angles) between

• \(-\frac{\pi}{2}\) and
• \(\frac{\pi}{2}\).

This means any computed angle from \( \tan^{-1} \) will always be in the first or fourth quadrants of the unit circle;
one example is

• \( \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \)

as seen in the exercise.Functionally, when you apply \( \tan^{-1} \) to a tangent value, you are essentially "un-doing" the tangent, but constrained within the set range. This translates a wider range of angles into their equivalent within the constrained output.

Angle Calculation

Calculating angles using tangent and arctan functions involves understanding angle positions and reference points on the unit circle.For an angle such as \( \frac{2\pi}{3} \), located in the second quadrant:

• The reference angle becomes \( \frac{\pi}{3} \).
• The tangent of \( \frac{\pi}{3} \) is \( \sqrt{3} \), thus \( \tan(\frac{2\pi}{3}) = -\sqrt{3} \), since tangent is negative in this quadrant.

Next,

• by utilizing the arctan function, \( \tan^{-1}(-\sqrt{3}) \), we determine the corresponding angle \(-\frac{\pi}{3}\) within the arctan's specific range.

This illustrates an essential calculation method where angles in angles' actual quadrants are converted into equivalent arctan-restricted values. By always reducing to the principal value, or primary range, the calculation remains consistent and recognizable. This results in easier comprehension and a smaller, predictable result set, crucial for solving complex problems.