Problem 34 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \cos 2 x+\sin ^{2} x=1 $$

Step-by-Step Solution

Answer

The solutions are \( x = n\pi \) in radians or \( x = n \times 180^\circ \) in degrees, where \( n \) is any integer.

Step 1: Recognize Trigonometric Identity

Use the trigonometric identity that states that \( \cos 2x = 1 - 2\sin^2 x \). This will help rewrite the equation in terms of \( \sin^2 x \).

Step 2: Substitute Identity

Substitute the identity \( \cos 2x = 1 - 2\sin^2 x \) into the equation. This gives: \( 1 - 2\sin^2 x + \sin^2 x = 1 \).

Step 3: Simplify the Equation

Simplify the equation to get: \( 1 - \sin^2 x = 1 \).

Step 4: Solve for \( \sin^2 x \)

Subtract 1 from both sides to find \( 1 - \sin^2 x = 0 \), which simplifies to \( \sin^2 x = 0 \).

Step 5: Find \( \sin x \)

Solve \( \sin^2 x = 0 \) to get \( \sin x = 0 \).

Step 6: Determine Solutions for \( x \)

The general solutions for \( \sin x = 0 \) are \( x = n\pi \), where \( n \) is any integer. In degrees, \( x = n \times 180^\circ \).

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are essential tools in solving trigonometric equations. They are standard equations involving trigonometric functions that are true for every valid value of the variable. In this case, the identity used is \( \cos 2x = 1 - 2\sin^2 x \). This identity is called a double angle identity, which helps us express one trigonometric function in terms of another.

Using these identities can simplify the equation significantly. It transforms the given equation into a form that is easier to solve. This is particularly helpful when working with complex trigonometric equations. Remember that knowing when and how to apply these identities is key to mastering trigonometry.

Importance of Angle Measurements

Angle measurements can often be expressed in degrees or radians, two common units in mathematics. In trigonometric problems, switching between these units based on the required solution form is critical. Given that the solutions sought in the exercise can be expressed in both degrees and radians, it’s important to understand how to convert between them.

Here are some quick conversion tips:

• To convert from degrees to radians, multiply by \( \frac{\pi}{180} \).
• To convert from radians to degrees, multiply by \( \frac{180}{\pi} \).

This problem uses degrees, but expressing solutions in radians, like \( x = n \pi \), is equally valid and often required in higher-level mathematics.

Determining General Solutions

When solving trigonometric equations, the goal is not just to find a specific solution, but a general solution that includes all possible solutions. This means accounting for the periodic nature of trigonometric functions.

The equation given, \( \sin x = 0 \), is periodic with a period of \( 180^\circ \) (or \( \pi \) radians). Because of this repetition, the solutions are expressed in a form that reflects this periodicity. Thus, the general solutions are \( x = n \times 180^\circ \) in degrees or \( x = n\pi \) in radians, where \( n \) is an integer.

This representation captures all possible solutions by considering the infinite cycle of the sine function crossing zero, giving a comprehensive answer to the equation.