Problem 48 - Precalculus Mathematics for Calculus

Question

Use a Double- or Half-Angle Formula to solve the equation in the interval \([0,2 \pi)\) $$2 \sin ^{2} \theta=2+\cos 2 \theta$$

Step-by-Step Solution

Answer

The solutions are \( \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \).

Step 1: Identify Double-Angle Identity

The given equation is \( 2 \sin^2 \theta = 2 + \cos 2\theta \). Notice the right-hand side contains the double-angle cosine function \( \cos 2\theta \). Recall the identity \( \cos 2\theta = 1 - 2 \sin^2 \theta \). This identity will help us express \( \cos 2\theta \) in terms of \( \sin^2 \theta \).

Step 2: Substitute Double-Angle Identity

Replace \( \cos 2\theta \) using the double-angle identity \( \cos 2\theta = 1 - 2 \sin^2 \theta \). The equation becomes \( 2 \sin^2 \theta = 2 + (1 - 2 \sin^2 \theta) \).

Step 3: Simplify the Equation

Distribute terms on the right-hand side: \( 2 \sin^2 \theta = 2 + 1 - 2 \sin^2 \theta \). Simplify to get \( 2 \sin^2 \theta = 3 - 2 \sin^2 \theta \).

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

Add \( 2 \sin^2 \theta \) to both sides to combine like terms: \( 4 \sin^2 \theta = 3 \). Divide both sides by 4 to isolate \( \sin^2 \theta \): \( \sin^2 \theta = \frac{3}{4} \).

Step 5: Solve for \( \theta \) in the Interval

Take the square root of both sides: \( \sin \theta = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \). Find the angles \( \theta \) in the interval \([0, 2\pi)\) for which \( \sin \theta = \frac{\sqrt{3}}{2} \). These are \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \). Similarly, for \( \sin \theta = -\frac{\sqrt{3}}{2} \), the angles are \( \theta = \frac{4\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).

Step 6: Conclusion

The values of \( \theta \) that satisfy the equation are \( \theta = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \text{and} \frac{5\pi}{3} \).

Key Concepts

Double-Angle Identities

Double-angle identities are fascinating tools in trigonometry designed to simplify expressions involving trigonometric functions of angles that are double a given angle. They're incredibly handy for solving equations like the one in our exercise. These identities help convert products and powers of sine and cosine functions into more manageable forms.
For cosine, the double-angle identity is given by:

• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• It can also be expressed as \( \cos 2\theta = 1 - 2\sin^2 \theta \)
• Or as \( \cos 2\theta = 2\cos^2 \theta - 1 \)

In our solution, we utilized the identity \( \cos 2\theta = 1 - 2 \sin^2 \theta \) to manipulate the equation. These forms can transform complex trigonometric equations, making them easier to solve.

Sine Function

The sine function is a fundamental trigonometric function that describes the y-coordinate or vertical position of a point on the unit circle as it moves at angle \( \theta \) from the positive x-axis. The function is periodic, with a period of \( 2\pi \). Important values to remember are:

• \( \sin(\pi/2) = 1 \)
• \( \sin(3\pi/2) = -1 \)
• \( \sin(\pi) = 0 \)

Sine values repeat every \( 2\pi \), making it a useful function for modeling repetitive phenomena like sound waves. In the given exercise, we were able to solve for \( \sin \theta = \pm \frac{\sqrt{3}}{2} \). Understanding the sine function allows us to find corresponding angles very efficiently.

Cosine Function

Another cornerstone of trigonometry is the cosine function, which describes the x-coordinate or horizontal position of a point on the unit circle as it moves with angle \( \theta \) from the positive x-axis. Much like sine, cosine is also periodic with a period of \( 2\pi \) and has pivotal values which help in numerous math problems:

• \( \cos(0) = 1 \)
• \( \cos(\pi) = -1 \)
• \( \cos(\pi/2) = 0 \)

These key points allow the cosine function to play a significant role, especially in the double-angle identities used for transforming the initial expression in our exercise.

Trigonometric Identities

Trigonometric identities are essential equations that relate the six trigonometric functions to one another. These identities allow us to transform and simplify complex trigonometric equations. Some primary identities include:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)

In addition to these, double-angle, sum, and difference identities enable us to express angles in terms of others. In the original exercise, we utilized the identity \( \cos 2\theta = 1 - 2 \sin^2 \theta \). Understanding these identities is crucial for solving trigonometric equations efficiently, as they provide pathways to expressing equations in terms of functions that are simpler to analyze and solve.