Problem 32 - Precalculus Mathematics for Calculus

Question

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\cos ^{2} \frac{\theta}{2}-\sin ^{2} \frac{\theta}{2}\) (b) \(2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\)

Step-by-Step Solution

Answer

(a) \(\cos(\theta)\); (b) \(\sin(\theta)\).

Step 1: Identify the Formulae

First, recognize the parts of the expression that relate to known trigonometric identities. For example, knowing that the cosine and sine squared terms could relate to the double angle formula.

Step 2: Rewrite Using Double Angle Formulae

For part (a), we use the identity: \[\cos^2 x - \sin^2 x = \cos(2x)\] Applying it to \(\theta/2\) gives us: \[\cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2} = \cos(\theta)\]

Step 3: Simplify the Expression in Part (a)

Conclude that for part (a) the expression simplifies directly to \( \cos(\theta) \).

Step 4: Identify Half Angle Product Formula for Part (b)

Recognize that \(2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\) is a known identity related to the sine of the sum of angles. Use the identity: \[2 \sin x \cos x = \sin(2x)\]

Step 5: Apply the Half Angle Product Formula

For part (b), apply the identity to \(\theta/2\), giving: \[2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} = \sin(\theta)\]

Step 6: Simplify the Expression in Part (b)

Conclude that for part (b) the expression simplifies directly to \( \sin(\theta) \).

Key Concepts

Double-Angle Formulas

Double-Angle Formulas are powerful tools in trigonometry that can simplify complicated expressions. They relate trigonometric functions of double angles to functions of single angles. Common double-angle formulas include:

• Cosine double-angle: \[ \cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x \]
• Sine double-angle: \[ \sin(2x) = 2 \sin x \cos x\]
• Tangent double-angle: \[ \tan(2x) = \frac{2 \tan x}{1 - \tan^2 x} \]

In our given problem, part (a), the expression \( \cos^2 \frac{\theta}{2} - \sin^2 \frac{\theta}{2} \) matches the cosine double-angle identity, leading us to rewrite it as \( \cos(\theta) \). This process exemplifies how recognizing and applying identities can streamline trigonometric expressions.

Half-Angle Formulas

Half-Angle Formulas are derived from double-angle formulas and are useful for finding trigonometric values of half angles. These formulas are:

• For sine:\[ \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{2}}\]
• For cosine:\[ \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}}\]
• For tangent:\[ \tan\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} = \frac{\sin x}{1 + \cos x} = \frac{1 - \cos x}{\sin x} \]

In part (b) of our exercise, we saw how the expression \( 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \) is transformed using the sine double-angle identity into \( \sin(\theta) \). This clearly demonstrates why understanding half-angle identities is critical when dealing with expressions involving half angles.

Trigonometric Simplification

Trigonometric Simplification involves transforming complex trigonometric expressions into simpler forms using various identities and formulas. This process allows for easier computation and understanding of trigonometric functions, especially in calculus and higher mathematics.
Simplification makes use of:

• Basic identities such as Pythagorean identities: \( \sin^2 x + \cos^2 x = 1 \)
• Reciprocal identities: \( \csc x = \frac{1}{\sin x} \), \( \sec x = \frac{1}{\cos x} \), \( \cot x = \frac{1}{\tan x} \)
• Double-Angle and Half-Angle Formulas as discussed earlier

In our exercise, both parts make use of simplification techniques by recognizing special trigonometric patterns and applying the appropriate formulae. Engagement in these exercises will increase agility in modifying and simplifying trigonometric expressions, an essential skill in solving many real-world and theoretical problems.