Problem 52 - Algebra and Trigonometry

Question

(a) Use a graphing device to find all solutions of the equation, correct to two decimal places, and (b) find the exact solution. \(\sin ^{-1} x-\cos ^{-1} x=0\)

Step-by-Step Solution

Answer

The exact solution is \( x = \frac{\sqrt{2}}{2} \); approximately \( x = 0.71 \).

Step 1: Understand the Equation

The equation given is \( \sin^{-1} x - \cos^{-1} x = 0 \). This means the arcsine of \( x \) minus the arccosine of \( x \) equals zero.

Step 2: Use the Identity

Recall the identity: \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \). We can use this identity to solve the given equation.

Step 3: Set Up the Equation with Identity

Substitute the identity into the equation: \( \sin^{-1} x = \cos^{-1} x \) becomes \( 2\sin^{-1} x = \frac{\pi}{2} \).

Step 4: Solve for \( \sin^{-1} x \)

Divide both sides by 2 to solve for \( \sin^{-1} x \): \( \sin^{-1} x = \frac{\pi}{4} \).

Step 5: Determine \( x \) Using Arcsine

Since \( \sin^{-1} x = \frac{\pi}{4} \), it implies that \( x = \sin\left(\frac{\pi}{4}\right) \). Calculate that \( x = \frac{\sqrt{2}}{2} \).

Step 6: Graph the Functions

Graph the functions \( y = \sin^{-1} x \) and \( y = \cos^{-1} x \) using a graphing device. Determine where they intersect by finding where \( \sin^{-1} x = \cos^{-1} x \).

Step 7: Verify the Intersection

Using the graph, verify that the intersection occurs at \( x \approx 0.71 \), confirming the calculated value of \( x = \frac{\sqrt{2}}{2} \approx 0.7071 \).

Step 8: Record the Solutions

The exact solution is \( x = \frac{\sqrt{2}}{2} \), and numerically it is approximately \( x = 0.71 \) after rounding to two decimal places.

Key Concepts

Understanding Arcsine

The arcsine function, denoted as \( \sin^{-1} x \), is the inverse of the sine function. It gives you the angle whose sine is \( x \).
For instance, if you have \( \sin(\theta) = x \), then \( \sin^{-1}(x) = \theta \). The domain for the arcsine function is \([-1, 1]\), and the range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
This means it works on values from -1 to 1 and returns angles in radians, specifically in these intervals of the unit circle. One critical thing to remember is that the output of arcsine is restricted to these ranges to maintain it as a function.

• Arcsine outputs angles in the 1st and 4th quadrants of the circle.
• Graphically, it is the reflection of the sine function along the line \( y = x \).

This is vital when considering solutions to equations involving arcsine, as seen in our problem.

Overview of Arccosine

The arccosine function, represented by \( \cos^{-1} x \), is the inverse of the cosine function. It gives you the angle whose cosine is \( x \).
If \( \cos(\theta) = x \), then \( \cos^{-1}(x) = \theta \). This function's domain is also \([-1, 1]\), but its range is different from arcsine, being \([0, \pi]\). This means:

• Arccosine gives angles only in the 1st and 2nd quadrants of the unit circle.
• The behavior is such that if \( x = 0 \), then \( \theta = \frac{\pi}{2} \). Similarly, if \( x = 1 \), \( \theta = 0 \).

Graphing arccosine shows it as the reflection of the cosine function along the line \( y = x \). Understanding these ranges and outputs is key to solving complex trigonometric equations involving inverse functions.

Trigonometric Identities and Their Application

Trigonometric identities are equations that hold true for all angles and are tools used to simplify and solve trig equations. One essential identity is \( \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \). This identity suggests that the sum of the arcsine and arccosine of the same value \( x \) always results in a right angle (\( \frac{\pi}{2} \)).
In this exercise, using this identity helped us reframe and solve the equation significantly by reducing it to a simpler form. Here’s why it was crucial:

• Applied initially, it allowed the substitution in the equation.
• By recognizing that \(\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}\), it confirmed \(\sin^{-1}(x) = \cos^{-1}(x)\).
• This elegant substitution led directly to an easy solution for \( \sin^{-1} x = \frac{\pi}{4} \).

Utilizing such trigonometric identities can often transform complex problems into manageable ones.

Graphing Techniques for Inverse Trigonometric Functions

Graphing inverse trigonometric functions can visually confirm solutions to equations. It’s a way to interpret and understand how two functions relate.
In our problem, graphing \( y = \sin^{-1}(x) \) and \( y = \cos^{-1}(x) \) allowed us to visually verify where these functions intersect. Points of intersection highlight equal values of these functions for the same \( x \).

• With a graphing device or graphing software, drawing both functions on the same coordinate system showed that intersecting points have equal values of \( \sin^{-1} x \) and \( \cos^{-1} x \).
• This visual tool was key to confirming \( x \approx 0.71 \), supporting our calculated solution of \( x = \frac{\sqrt{2}}{2} \) or approximately 0.7071.

Graphing not only helps in confirming algebraic solutions but also provides a holistic view of how functions behave in relation to each other across their domains and ranges.