Problem 31 - Precalculus Mathematics for Calculus

Question

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\cos ^{2} 34^{\circ}-\sin ^{2} 34^{\circ}\) (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 the Formula

The expressions \(\cos^2 \theta - \sin^2 \theta\) for both parts of the problem are recognized as applications of the double-angle formula for cosine: \(\cos(2\theta) = \cos^2 \theta - \sin^2 \theta\). We will use this formula to simplify the given expressions.

Step 2: Apply the Formula to Part (a)

For part (a), \(\theta = 34^\circ\). Using the formula \(\cos(2\theta) = \cos^2 \theta - \sin^2 \theta\), we substitute \(\theta\) with \(34^\circ\) to get \(\cos(68^\circ) = \cos^2 34^\circ - \sin^2 34^\circ\). Thus, \(\cos^2 34^\circ - \sin^2 34^\circ\) simplifies to \(\cos(68^\circ)\).

Step 3: Apply the Formula to Part (b)

In part (b), we have \(\theta = 5\theta\). Substituting into the double-angle formula, \(\cos(2 \times 5 \theta) = \cos^2 5 \theta - \sin^2 5 \theta\), this simplifies to \(\cos(10 \theta)\). Therefore, \(\cos^2 5 \theta - \sin^2 5 \theta\) is equivalent to \(\cos(10 \theta)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are vital tools in mathematics. They allow us to simplify expressions and solve problems concerning angles and triangles. These identities show relationships between different trigonometric functions like sine (\( \sin \theta \)) and cosine (\( \cos \theta \)).

One of the most useful trigonometric identities is the Double-Angle Formula. This formula can be used to express a trigonometric function of double an angle in terms of the square of the function, such as:

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

This specific identity is pivotal for simplifying expressions like \( \cos^2 \theta - \sin^2 \theta \), as seen in our original problem. Knowing and applying these identities can make solving trigonometric equations much more manageable.

An understanding of trigonometric identities not only aids in academic work but has practical applications too, such as in physics, engineering, and computer science.

Radians and Degrees

Trigonometry often requires the conversion between radians and degrees to solve problems effectively. Remember, radians and degrees are both units to measure angles. However, they are used in different contexts and can be interchangeable if needed.

The circle division is the basis for these units:

• 360 degrees is equal to a full circle.
• 2\(\pi\) radians is also a complete circle measurement.

To convert between these measurements, you can use the relationship: \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \). This means for converting angles measured in degrees to radians, the formula is \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

In our exercise, angles were given in degrees, like \(34^{\circ}\). It's often necessary to switch to radians, especially when using certain mathematical functions or calculators that require radian inputs. It's useful to become comfortable with these conversions, as they are frequent in trigonometric calculations.

Cosine Function

The cosine function is a fundamental trigonometric function that represents the adjacent side over the hypotenuse in a right triangle. It is crucial in the study of trigonometry since it appears in various identities and formulas.

In the context of the Double-Angle Formula, the cosine function helps simplify expressions by transforming a problem into a more approachable form. Specifically, the reduction of a complex expression like \( \cos^2 \theta - \sin^2 \theta \) into a simple cosine of double the angle, \( \cos(2\theta) \).

• The cosine function, being periodic, repeats its values every \(360^{\circ}\) or \(2\pi\) radians.
• Its values range from -1 to 1, alternately being positive or negative depending on the angle's quadrant.

A deep understanding of the cosine function, including its graph and properties, is important in tackling problems involving trigonometric expressions and identities.