Problem 129 - Precalculus

Question

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

Step-by-Step Solution

Answer

\( \sin^{-1}(\cos(\pi)) = -\frac{\pi}{2} \)

Step 1: Understanding the Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1}(x) \), is the angle whose sine is \( x \). It returns an angle in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

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

The cosine function \( \cos(x) \) calculates the horizontal coordinate of the point on the unit circle at an angle \( x \) from the positive x-axis. Thus, \( \cos(\pi) = -1 \).

Step 3: Applying the Inverse Sine Function

Now we apply the inverse sine function: \( \sin^{-1}(-1) \). We need to find an angle \( \theta \) such that \( \sin(\theta) = -1 \) within the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

Step 4: Finding the Exact Value

In the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), the only angle where \( \sin(\theta) = -1 \) is \( \theta = -\frac{\pi}{2} \). Therefore, \( \sin^{-1}(-1) = -\frac{\pi}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essentially the opposite operations of the standard trigonometric functions. They allow us to find the angle when we know the trigonometric value. The inverse sine function, also known as arcsin, is one such function. It takes a value between -1 and 1 and returns an angle in the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This is a significant aspect of the function as it helps us understand the angle corresponding to any given sine value within this domain.

• Inverse Sine Function: Denoted as \(\sin^{-1}(x)\).
• Range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• Use: Find the angle for a given sine value.

It is crucial to remember that inverse trigonometric functions can have limited domains because trigonometric functions are periodic. This limitation helps in providing unique solutions to problems like finding the exact angle of an arc for a given trigonometric value.

Cosine

The cosine function, commonly denoted by \(\cos(x)\), measures the horizontal distance from the origin to a point on the unit circle. For any angle \(x\), cosine returns the x-coordinate of the point on the unit circle positioned at that angle from the positive x-axis.

• Even Function: Cosine is symmetric across the y-axis, meaning \(\cos(-x) = \cos(x)\).
• Unit Circle Position: At \(x = \pi\), the point on the unit circle is (-1,0), hence \(\cos(\pi) = -1\).
• Value Range: \([-1, 1]\).

Remembering the positions corresponding to common angles on the unit circle helps in fast calculations and understanding of trigonometric functions without a calculator. In this case, since \(\cos(\pi)\) evaluates to -1, it serves as essential information for solving trigonometric problems involving inverse functions.

Sine

The sine function, denoted by \(\sin(x)\), is fundamental in trigonometry and deals with the vertical coordinate of a point on the unit circle. It measures how far up or down a point is on the circle compared to the origin.

• Odd Function: Meaning \(\sin(-x) = -\sin(x)\), showing symmetry along the origin.
• Sine of Special Angles: For known angles like \(\frac{\pi}{2}\) and \(-\frac{\pi}{2}\), sine gives critical values, e.g., \(\sin(-\frac{\pi}{2}) = -1\).
• Domain: Generally all real numbers; however, specific context like inverse requires restriction to \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

When applying the inverse sine function, the task is to identify which angle within \([-\frac{\pi}{2}, \frac{\pi}{2}]\) corresponds to a given sine value. Understanding the foundational values of sine at specific angles provides clarity in trigonometric calculations as seen with \(\sin^{-1}(-1)\).

Unit Circle

The unit circle is a vital tool in trigonometry that helps visualize the behavior of trigonometric functions. It is a circle with a radius of one, centered at the origin of a coordinate system.

• Basic Idea: Every point on the circle corresponds to an angle and has coordinates \((\cos(\theta), \sin(\theta))\).
• Cosine and Sine Relationship: Cosine values are x-coordinates, while sine values are y-coordinates on the unit circle.
• Applications: Provides a graphical representation and helps in understanding the trigonometric functions and in deriving values without a calculator.

For example, to understand the problem involving \(\cos(\pi)\), we use the unit circle to see that at an angle \(\pi\), the coordinates are (-1, 0), making cosine of \(\pi\) equal to -1. Similarly, any value of sine can be understood in terms of positioning on this circle, providing a complete and intuitive method to analyze trigonometric problems.