Problem 43 - Algebra 2

Question

Solve each equation for all values of \(\theta\) \(\sin \frac{\theta}{2}+\cos \frac{\theta}{2}=\sqrt{2}\)

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}\).

Step 1: Squaring Both Sides

First, square both sides of the equation to eliminate the square root and trigonometric functions:\[(\sin \frac{\theta}{2} + \cos \frac{\theta}{2})^2 = (\sqrt{2})^2\]Simplifying the right side, we get:\[2\]

Step 2: Apply the Identity

Use the identity \((a + b)^2 = a^2 + 2ab + b^2\) to expand the left-hand side:\[\sin^2 \frac{\theta}{2} + 2\sin \frac{\theta}{2}\cos \frac{\theta}{2} + \cos^2 \frac{\theta}{2} = 2\]We know from the Pythagorean identity that \(\sin^2 \alpha + \cos^2 \alpha = 1\). Therefore,\[1 + 2\sin \frac{\theta}{2}\cos \frac{\theta}{2} = 2\]

Step 3: Simplify the Equation

Subtract 1 from both sides of the equation:\[2\sin \frac{\theta}{2}\cos \frac{\theta}{2} = 1\]Use the double angle identity for sine, \(\sin \alpha \cos \alpha = \frac{1}{2}\sin 2\alpha\), to rewrite the left side:\[\sin \theta = 1\]

Step 4: Solve for \(\theta\)

Find \(\theta\) such that \(\sin \theta = 1\). We know \[\theta = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}\]This gives all possible solutions for \(\theta\).

Key Concepts

Angle Identities

Understanding angle identities is essential when working with trigonometric equations. These identities are mathematical expressions involving angles and trigonometric functions that hold true for all values of the involved angles.

• Common angle identities include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)).
• They can be used to simplify and transform equations, allowing for easier manipulation and solution.

In our problem, we dealt with half-angle identities where \( \theta/2 \) was the primary focus. This involves concepts that can sometimes make the problem appear more complex but are solvable using these identities. We combined \( \sin \frac{\theta}{2} \) and \( \cos \frac{\theta}{2} \) by squaring both sides to get rid of the square root for easier calculations.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship between the trigonometric functions sine and cosine. It states that \( \sin^2 \alpha + \cos^2 \alpha = 1 \) for any angle \( \alpha \). This identity is incredibly useful in solving trigonometric equations, as it can simplify expressions by replacing a sine term squared with a corresponding cosine term squared and vice versa.

• In our exercise, the equation \( \sin^2 \frac{\theta}{2} + \cos^2 \frac{\theta}{2} = 1 \) was utilized.
• This simplification led us to convert a complex expression into a much simpler one that could easily be handled: \( 1 + 2\sin \frac{\theta}{2}\cos \frac{\theta}{2} = 2 \).

By substituting these known identities, we reduced the complexity of the equation in question.

Double Angle Formulas

The double angle formulas are trigonometric identities that express functions of double angles in terms of functions of single angles. For sine, the double angle formula is \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \). These formulas are practical for transforming and simplifying certain trigonometric equations.

• In the given exercise, after applying the Pythagorean identity, we derived \( 2\sin \frac{\theta}{2}\cos \frac{\theta}{2} = 1 \).
• This is a direct application of the double angle formula where \( \sin \theta = \sin 2(\frac{\theta}{2}) \).

This allowed us to rewrite the equation in the format \( \sin \theta = 1 \), which was much simpler to solve. Understanding these formulas assists in more efficiently tackling such trigonometric problems.

Solving Equations

Solving trigonometric equations involves finding the values of the variable angles that satisfy the equation. This requires a comprehensive understanding of trigonometric identities and formulas. When solving for \( \theta \), consider:

• Identify and isolate trigonometric functions. In our problem, we ended up with the equation \( \sin \theta = 1 \).
• Recognize special angles where these expressions have known values.

For instance, \( \sin \theta = 1 \) is a direct indication that \( \theta = \frac{\pi}{2} + 2k\pi \), where \( k \) is any integer, providing all solutions due to the periodic nature of the sine function. Understanding how to manipulate and address these equations effectively is key to gaining proficiency in mathematics.