Problem 130 - Precalculus

Question

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why. $$ \tan ^{-1}(\sin (\pi)) $$

Step-by-Step Solution

Answer

The exact value is 0.

Step 1: Understanding the Function Inside

First, we need to understand the innermost function: \(\sin(\pi)\). The sine of \(\pi\), which is the angle 180 degrees on the unit circle, is 0. Thus, \(\sin(\pi) = 0\).

Step 2: Applying the Arctangent Function

Next, we apply the arctangent (inverse tangent) function to the result from the previous step. We need to find \(\tan^{-1}(0)\).

Step 3: Evaluating \\(\tan^{-1}(0)\\)

The \(\tan^{-1}(x)\) function gives us the angle whose tangent is \(x\). We need to find an angle \(\theta\) for which \(\tan(\theta) = 0\). This occurs when \(\theta = 0\) because \(\tan(0) = 0\). The range of \(\tan^{-1}\) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), and 0 lies within this range.

Key Concepts

Inverse tangent function

When we talk about the inverse tangent function, often denoted as \( \tan^{-1}(x) \) or \( \text{arctan}(x) \), we're exploring a function that provides the angle whose tangent is a given number, \( x \). This angle is usually expressed in radians. The key feature of the inverse tangent function is its range. It outputs angles that lie between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).
Think of it like searching for the angle that, when you calculate the tangent, returns the original input number.
For instance, if you are asked to find \( \tan^{-1}(0) \), we are looking for an angle \( \theta \) such that \( \tan(\theta) = 0 \).
This angle turns out to be \( 0 \) because \( \tan(0) = 0 \).

• The inverse function is a way to "reverse" the tangent function by providing the angle instead of the ratio.
• Its output is always an angle within a specific range, namely \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Understanding this function is crucial when you're given different trigonometric expressions and need to find corresponding angles.

Sine function

The sine function is one of the fundamental trigonometric functions. Often noted as \( \sin(\theta) \), it is particularly significant when working with angles on the unit circle. The unit circle is a helpful geometric tool where the circle has a radius of 1 centered at the origin of a coordinate plane.
The sine of an angle represents the "y" coordinate (vertical position) of a point on this circle corresponding to that angle measured from the positive x-axis.
For example, when considering \( \sin(\pi) \), we need to locate the angle \( \pi \) radiants, which corresponds to 180 degrees on the unit circle. At this point, the ordinate or y-coordinate is zero, hence \( \sin(\pi) = 0 \).

• Sine function is all about measuring vertical distance on the unit circle.
• It oscillates between -1 and 1 as the angle moves around the circle.

The sine function gives crucial insights into periodic phenomena and forms a basis for further trigonometric calculations.

Unit circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin \((0,0)\) of the coordinate plane. This circle is vital because it offers a straightforward way to derive and visualize the basic trigonometric functions.
Each point on the unit circle corresponds to a specific angle from the positive x-axis, measured in radians, making it closely associated with angles and hence, trigonometric functions.
For example, on the unit circle, the point at angle \( \pi \) or 180 degrees is represented as \((-1, 0)\). This tells us directly that the cosine of \( \pi \) is -1, and the sine is 0.

• The circle allows for easy identification of fundamental trigonometric values.
• Helps in understanding how angles relate to coordinates (sine and cosine).

The unit circle simplifies the understanding of trigonometric identities and functions, making it easier to handle complex trigonometric problems.