Question

Without using a calculator, evaluate or simplify the following expressions. $$\tan ^{-1}(\tan \pi / 4)$$

Short answer

Question: Evaluate the expression $\tan ^{-1}(\tan \pi / 4)$. Answer: $\pi/4$

Step-by-step solution

Recognize the inverse function

We have an expression involving an inverse trigonometric function and its respective direct trigonometric function: \(\tan ^{-1}(\tan \pi / 4)\). The goal here is to simplify this expression.

Compute the tangent of the given angle

We know that \(\tan(\pi/4) = 1\). So our expression becomes \(\tan^{-1}(1)\).

Calculate the angle with tangent 1

Now we need to find the angle whose tangent is 1. Remember that the inverse function is used to find the angle: \(\tan^{-1}(1)\). We know by definition that the angle is \(\pi/4\) or 45 degrees.

Final Answer

The simplified expression is: $$\tan ^{-1}(\tan \pi / 4) = \pi/4$$

Key concepts

Tangent Function

The tangent function, abbreviated as "tan," is one of the primary trigonometric functions. It plays a critical role in studying the properties of angles and triangles. The tangent of an angle \( \theta \) in a right triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side.

Here's a useful way to think about it:

• For any angle \( \theta \), \( \tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}} \).
• The tangent function has a repeating cycle, and it is undefined at odd multiples of \( \frac{\pi}{2} \) because the adjacent side becomes zero.
• The graph of the tangent function shows an s-shaped curve repeating every period \( \pi \), with vertical asymptotes where the tangent is undefined.

Understanding the tangent function is essential in evaluating expressions involving angles, like the one in our problem.

Angle Measurement

Angle measurement can be expressed in different units, with two of the most common being degrees and radians. In trigonometry, especially when dealing with calculus and higher-level mathematics, radians are typically preferred.

To convert between these units, remember:

• There are \( 360 \) degrees in a full circle, which is equivalent to \( 2\pi \) radians.
• A right angle, often used in trigonometric problems, is \( 90 \) degrees or \( \frac{\pi}{2} \) radians.

In the exercise example, \( \frac{\pi}{4} \) radians corresponds to \( 45 \) degrees, and this notation is crucial when dealing with inverse trigonometric functions. Inverse functions are used when you know the tangent value and need to find the angle measure.

Trigonometric Simplification

Simplifying trigonometric expressions is a key skill that involves understanding both direct and inverse trigonometric functions. The process requires recognizing known angle ratios and using inverse functions for the desired simplicity.

Steps in simplification can include:

• Applying known values: \( \tan(\pi/4) = 1 \) is a known value because \( \frac{\pi}{4} \), or 45 degrees, forms a specific right triangle where opposite and adjacent sides are equal.
• Using inverse functions: The inverse tangent function, \( \tan^{-1} \), helps find the angle when the tangent value is known. If \( \tan^{-1}(1) = \frac{\pi}{4} \), the original angle is simply recovered.
• Recognizing cycles: Knowing the periodic nature of the tangent function, you can identify angles within a principal value (usually \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \) for the tangent).

Trigonometric simplification streamlines the evaluation of more complex functions and enhances problem-solving efficiency.