Problem 29 - Precalculus with Calculus Previews

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \tan ^{-1}(\tan \pi) $$

Step-by-Step Solution

Answer

The exact value is 0.

Step 1: Understanding the Inverse Tangent Function

The function \( \tan^{-1}(x) \) is the inverse of the tangent function. It returns an angle whose tangent is \( x \). By definition, if \( y = \tan^{-1}(x) \), then \( \tan(y) = x \), and \( y \) is typically in the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

Step 2: Evaluating \( \tan(\pi) \)

The tangent function can be represented as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). At \( \theta = \pi \), the sine function is \( \sin(\pi) = 0 \) and the cosine function is \( \cos(\pi) = -1 \). Thus, \( \tan(\pi) = \frac{0}{-1} = 0 \).

Step 3: Finding \( \tan^{-1}(\tan(\pi)) \)

Given that \( \tan(\pi) = 0 \), we are essentially asked to find \( \tan^{-1}(0) \). The angle whose tangent is 0 within the range \(-\frac{\pi}{2}, \frac{\pi}{2}\) is \( 0 \).

Step 4: Conclusion

Since \( \tan^{-1}(\tan(\pi)) = \tan^{-1}(0) \), and the angle whose tangent is 0 in the appropriate range is 0, the answer is \( 0 \).

Key Concepts

Tangent

The tangent function is one of the primary trigonometric functions, symbolized by \( \tan(\theta) \). It represents the ratio of the opposite side to the adjacent side in a right triangle:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
However, when dealing with the unit circle, tangent can also be defined through the sine and cosine functions:

\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
This formula means that for any angle \( \theta \), the tangent of \( \theta \) is the quotient of its sine and cosine values.

In the exercise, \( \tan(\pi) \) was evaluated. Since \( \sin(\pi) = 0 \) and \( \cos(\pi) = -1 \), we find that \( \tan(\pi) = \frac{0}{-1} = 0 \).

**Key Points About Tangent:**

• Tangent is periodic, meaning it repeats its values in cycles.
• The principal period of the tangent function is \( \pi \).
• Unlike sine and cosine, tangent has undefined values at specific points where cosine is zero, causing divisions by zero.

Understanding the basics of tangent can help simplify complex trigonometric expressions and solve problems like \( \tan^{-1}(\tan(\pi)) \).

Trigonometric Identities

Trigonometric identities are equations that hold true for all angle measures and are extremely useful in simplifying trigonometric expressions. They are tools that allow you to transform and solve trigonometric equations. A fundamental identity that was employed in the solution is:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
This identity illustrates the relationship between the tangent, sine, and cosine functions, as seen in Step 2 of the solution when calculating \( \tan(\pi) \).

**Common Trigonometric Identities:**

• Pythagorean Identities:
  \[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
• Quotient Identity for Tangent:
  \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
• Even-Odd Identities:
• \( \sin(-\theta) = -\sin(\theta) \)
• \( \cos(-\theta) = \cos(\theta) \)
• \( \tan(-\theta) = -\tan(\theta) \)

Using these identities, you can rewrite, simplify, and evaluate trigonometric expressions, such as the one in the exercise.

Mastering trigonometric identities is essential for success in any math course involving trigonometry.

Angle Measurement

Angle measurement in mathematics is a way of describing the size of an angle. We commonly measure angles in either degrees or radians.

**Degrees and Radians:**

• Degrees are most familiar, with one full revolution equaling 360 degrees.
• Radians relate to the radius of a circle, where one full revolution is \( 2\pi \) radians.

In trigonometry, especially when using functions like sine, cosine, and tangent, radians are often preferred. This is because they simplify many mathematical expressions and are more natural in calculus.

In the given exercise, \( \pi \) radians were used. This means an angle of 180 degrees, which is half of a full circle. Knowing this helps understand why \( \tan(\pi) = 0 \), as at this angle, you reach a point on the unit circle where the sine is 0.

**Key Points About Angle Measurement:**

• Converting between degrees and radians: \( 180^\circ = \pi \) radians.
• Radians are dimensionless, and they provide a direct measure of the arc length in the unit circle.
• Understanding the conversion and use of radians is critical for solving trigonometry problems accurately.

These insights help clarify why angular measurements are crucial in trigonometric problems and how they can be used to find exact values without a calculator.