Problem 48 - Algebra and Trigonometry

Question

\(43-52\) a 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

\(\theta = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\)

Step 1: Use Double-Angle Formula for Cosine

The given equation is \(2 \sin^2 \theta = 2 + \cos 2\theta\). We can relate \( \cos 2\theta\) and \(\sin^2 \theta\) using the double-angle formula for cosine:\[\cos 2\theta = 1 - 2\sin^2 \theta\]

Step 2: Substitute the Double-Angle Identity

Substitute \(\cos 2\theta = 1 - 2\sin^2 \theta\) into the equation:\[2\sin^2 \theta = 2 + (1 - 2\sin^2 \theta)\]Simplify the equation:\[2\sin^2 \theta = 3 - 2\sin^2 \theta\]

Step 3: Rearrange Terms and Simplify

Add \(2\sin^2 \theta\) to both sides to combine like terms:\[4\sin^2 \theta = 3\]Divide both sides by 4 to solve for \(\sin^2 \theta\):\[\sin^2 \theta = \frac{3}{4}\]

Step 4: Solve for \(\theta\)

Take the square root of both sides, keeping in mind to consider both the positive and negative roots:\[\sin \theta = \pm \frac{\sqrt{3}}{2}\]Determine the angles that satisfy \(\sin \theta = \frac{\sqrt{3}}{2}\) and \(\sin \theta = -\frac{\sqrt{3}}{2}\) within the interval \([0, 2\pi)\).

Step 5: Determine Specific Solutions

For \(\sin \theta = \frac{\sqrt{3}}{2}\), \(\theta = \frac{\pi}{3}, \frac{2\pi}{3}\).For \(\sin \theta = -\frac{\sqrt{3}}{2}\), \(\theta = \frac{4\pi}{3}, \frac{5\pi}{3}\).Thus, the solutions for \(\theta\) in the interval \([0, 2\pi)\) are \(\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\).

Key Concepts

Double-Angle Formula

The double-angle formula is a handy tool in trigonometry when you need to express trigonometric functions of double angles in terms of single angles. This formula simplifies complex expressions and is commonly used in solving trigonometric equations. Here, we focus on the double-angle formula for cosine, which states:

• \( \cos 2\theta = 1 - 2\sin^2 \theta \)
• Alternative forms include \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) and \( \cos 2\theta = 2\cos^2 \theta - 1 \)

In the given exercise, this formula allowed us to rewrite \( \cos 2\theta \) in terms of \( \sin^2 \theta \), which transformed the equation into a simpler form. Understanding this formula is crucial since it connects different trigonometric functions, providing flexibility when solving complex trigonometric problems.

Solving Trigonometric Equations

Solving trigonometric equations can be an intricate process involving several steps to break down and simplify. In the given problem, our primary goal was to find values of \( \theta \) that satisfy the equation within a specified interval, \([0, 2\pi)\). Here is a step-by-step approach to solving such equations:

• Use known identities or formulas to simplify the equation
• Rearrange the equation to isolate trigonometric terms
• Solve for the basic trigonometric function
• Find specific angles that satisfy the equation over the given interval

This process requires critical thinking and a good grasp of trigonometric functions' behavior over their usual cycle. The exercise illustrates that when \( \sin^2 \theta = \frac{3}{4} \), by taking the square root, we find \( \sin \theta = \pm \frac{\sqrt{3}}{2} \). Keep in mind both the positive and negative roots are important in determining all possible solutions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are essential for simplifying expressions and solving equations in trigonometry. Key identities include:

• Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Reciprocal identities, like \( \csc \theta = \frac{1}{\sin \theta} \)
• Double-angle identities, which were used in this exercise

Using these identities, you can transform complex equations into simpler forms. They help in recognizing patterns and establishing connections between different trigonometric functions. In our solved problem, the identity \( \cos 2\theta = 1 - 2\sin^2 \theta \) was pivotal because it allowed us to convert everything into sine terms, making it easier to isolate and solve for \( \theta \). Mastery of these identities empowers students to tackle diverse trigonometric problems with confidence.