Problem 31 - Algebra and Trigonometry

Question

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \text { (a) } \cos ^{2} 34^{\circ}-\sin ^{2} 34^{\circ} \quad \text { (b) } \cos ^{2} 5 \theta-\sin ^{2} 5 \theta $$

Step-by-Step Solution

Answer

(a) \(\cos(68^\circ)\); (b) \(\cos(10\theta)\).

Step 1: Identify Formula for Part (a)

The expression for part (a) is \( \cos^2 34^\circ - \sin^2 34^\circ \). Recognize that this expression can be simplified using the double-angle identity for cosine: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \). Here, \( 2\theta = 68^\circ \).

Step 2: Apply Double-Angle Formula to Part (a)

Apply the formula, \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \), to simplify the expression. Thus, \( \cos^2 34^\circ - \sin^2 34^\circ = \cos(68^\circ) \).

Step 3: Simplify Part (b) Using Appropriate Formula

The expression for part (b) is \( \cos^2 5\theta - \sin^2 5\theta \). As in part (a), apply the double-angle formula: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \). Substituting the angle gives \( 2(5\theta) = 10\theta \).

Step 4: Complete Simplification of Part (b)

Using the double-angle identity applied to the expression, conclude that \( \cos^2 5\theta - \sin^2 5\theta = \cos(10\theta) \).

Key Concepts

Double-Angle Formula

The double-angle formula is a powerful tool in trigonometry that simplifies expressions involving angles. It essentially allows you to express trigonometric functions of double angles in terms of single angles. The most common double-angle formulas relate to sine, cosine, and tangent:

• The double-angle formula for cosine is expressed as: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \).
• For sine, the formula is \( \sin(2\theta) = 2\sin \theta \cos \theta \).
• For tangent, it is represented as: \( \tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta} \).

In our given problem, the expression \( \cos^2 34^\circ - \sin^2 34^\circ \) fits perfectly with the double-angle formula for cosine. This identity helps you to rewrite this expression as \( \cos(2 \times 34^\circ) \), which equals \( \cos(68^\circ) \). Similarly, in the expression \( \cos^2 5\theta - \sin^2 5\theta \), applying the same formula transforms it into \( \cos(10\theta) \). Breaking down these identities makes them much simpler to use in various trigonometric problems.

Half-Angle Formula

The half-angle formulas are special cases in trigonometry used for finding the values of trigonometric functions at half of a given angle. These are very useful when you need to solve problems where you have angles that are not readily expressible using common angles. The half-angle formulas can be derived from the double-angle formulas and are as follows:

• For sine, \( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} \).
• For cosine, \( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}} \).
• For tangent, \( \tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \).

Even though the problem at hand did not directly require using half-angle formulas, understanding them provides broader insight into managing expressions where specific half-angles occur.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions often involves reducing complex expressions into simpler or more easily recognizable forms. This can make solving problems or integrating functions more manageable. Here are some common strategies used in simplification:

• Utilize trigonometric identities such as Pythagorean, double-angle, and half-angle formulas.
• Convert between trig functions: use identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \) to switch function types.
• Apply symmetry and periodicity of trigonometric functions. For example, recognizing that \( \cos(-\theta) = \cos \theta \) or \( \sin(-\theta) = -\sin \theta \).

In the provided exercise, you were tasked with using the double-angle formula. The simplification process highlights how an equation can be transformed into a simpler equivalent form. By recognizing and applying the right identities, complex trig expressions can become straightforward calculations like \( \cos(68^\circ) \) from \( \cos^2 34^\circ - \sin^2 34^\circ \). Consistently practicing these techniques will improve your ability to quickly simplify and solve trigonometric expressions.