Problem 38 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression, if it is defined. $$\sin ^{-1}\left(\sin \left(\frac{11 \pi}{4}\right)\right)$$

Step-by-Step Solution

Answer

The expression equals \( \frac{\pi}{4} \).

Step 1: Simplify the Angle

The first step is to simplify the angle \( \frac{11\pi}{4} \) within the sine function. Since the cycle of sine is \( 2\pi \), we reduce the angle modulo \( 2\pi \).\[\frac{11\pi}{4} - 2\pi = \frac{11\pi}{4} - \frac{8\pi}{4} = \frac{3\pi}{4}\]The equivalent angle within \( 0 \) to \( 2\pi \) is \( \frac{3\pi}{4} \).

Step 2: Evaluate the Sine Function

Evaluate the sine of the simplified angle \( \frac{3\pi}{4} \). Using the unit circle, the sine of \( \frac{3\pi}{4} \) is \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

Step 3: Apply the Inverse Sine Function

Next, apply the inverse sine (\( \sin^{-1}\)) to the result from Step 2. The inverse sine function returns values in the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). Since \( \frac{\sqrt{2}}{2} \) is achievable within this range, \[ \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4} \]

Step 4: Check if Expression is Defined

Consider if the expression is defined: \( \sin^{-1}\left(\sin\left(\frac{11\pi}{4}\right)\right) \). Since \( \sin^{-1} \) outputs the original angle in the first or fourth quadrants and \( \frac{\pi}{4} \) is within this range, the expression is defined.

Key Concepts

Angle Simplification

When dealing with angles in trigonometry, it is often necessary to simplify the angle to make calculations easier. An angle (\(\frac{11\pi}{4}\)) can seem daunting at first glance, particularly because it is larger than \(2\pi\), the standard cycle length of trigonometric functions.
Reducing this angle modulo \(2\pi\) means you are finding an equivalent angle within a full cycle on the unit circle.

• You first express \(2\pi\) in terms of the same denominator: \(\frac{8\pi}{4}\).
• Then, subtract this from the original angle: \(\frac{11\pi}{4} - \frac{8\pi}{4} = \frac{3\pi}{4}\).
• This gives you \(\frac{3\pi}{4}\), a simplified angle within the full cycle.

The angle \(\frac{3\pi}{4}\) is easier to evaluate using sine or any other trigonometric function because it's less than \(2\pi\).
This simplification allows us to work within the bounds of the unit circle, a crucial concept in trigonometry.

Unit Circle

The unit circle is a vital component of trigonometry, especially when working with angle measurements and trigonometric functions.
It provides a clear and easy way to understand how trigonometric functions behave at different angles. The unit circle is a circle with a radius of one centered at the origin of a coordinate plane.

• Every point on the unit circle corresponds to a unique angle, measured from the positive x-axis.
• Angles are often given in radians, with a full circle being \(2\pi\) radians. Halfway around the circle is \(\pi\), a quarter way is \(\frac{\pi}{2}\), and so forth.
• The coordinates of each point (x, y) on the unit circle are the values of the cosine and sine of the angle \(\theta\), respectively.

For the angle \(\frac{3\pi}{4}\), we look up its position on the unit circle.
Here, the sine value (y-coordinate) is \(\frac{\sqrt{2}}{2}\). Understanding this helps in applying trigonometric functions quickly and accurately.

Sine Function

The sine function is a fundamental trigonometric function that measures the y-coordinate of a point on the unit circle.
Given an angle \(\theta\), the sine function (\(\sin(\theta)\)) gives us the y-value of the point on the circle that corresponds to the angle. This function has some distinct properties that are essential to understand:

• The sine function is periodic with a period of \(2\pi\), meaning, it repeats its values every \(2\pi\) radians.
• The range of the sine function is between -1 and 1, inclusive.
• Specifically for this exercise, \(\sin\left(\frac{3\pi}{4}\right)\) evaluated on the unit circle yields \(\frac{\sqrt{2}}{2}\).

These characteristics make the sine function predictable and consistent, crucial for solving trigonometric equations like the one presented in the original problem.

Range of Inverse Sine

The inverse sine function, denoted as (\(\sin^{-1}\)), is used to determine the angle whose sine is a given number.
This function is the reverse of the sine function, thus essential in solving trigonometric equations where you need to find an angle from its sine value. It has a particular range that limits its output to ensure it is a function:

• The range of \(\sin^{-1}\) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
• This means it returns angles situated between these values, inclusive of them, which correspond to the first and fourth quadrants of the unit circle.

In the given exercise, we applied \(\sin^{-1}\) to \(\frac{\sqrt{2}}{2}\), and the result is \(\frac{\pi}{4}\), which clearly lies within the possible output range.
This is crucial to ensure that for any given sine value, the inverse sine will provide a valid angle, making our calculations precise and reliable.