Problem 117 - Precalculus with Limits

Question

In Exercises 111 - 124, verify the identity. \( \cos 4\alpha = \cos^2 2\alpha - \sin^2 2\alpha \)

Step-by-Step Solution

Answer

Yes, the identity \( \cos 4\alpha = \cos^2 2\alpha - \sin^2 2\alpha \) is verified using trigonometric identities.

Step 1: Use the double angle identity for cosine

First, recall the double angle identity for cosine: \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\). Apply this identity to our given problem, replacing \(2\theta\) with \(2\alpha\), we get \(\cos 2(2\alpha) = \cos^2(2\alpha) - \sin^2(2\alpha)\). But \(\cos 2(2\alpha)\) simplifies to \(\cos 4\alpha\). So we find that \(\cos 4\alpha = \cos^2(2\alpha) - \sin^2(2\alpha)\).

Step 2: Simplifying the right side (optional)

Another way to confirm this identity is by simplifying the right side. Recall the identity \(\cos^2 \theta = 1 - \sin^2 \theta\). Substitute this into the right side, to get \(\cos^2(2\alpha) - \sin^2(2\alpha) = (1 - \sin^2(2\alpha)) - \sin^2(2\alpha) = 1 - 2\sin^2(2\alpha)\). Now you can use the sine double angle identity \(\sin^2 \theta = 1 - \cos^2 \theta\) and the previous substitution to get the equation into the original form of \(\cos 4\alpha = \cos^2 2\alpha - \sin^2 2\alpha\).

Key Concepts

Understanding Double Angle Identity

The double angle identity is a fundamental formula in trigonometry that helps us simplify expressions involving trigonometric functions of double angles. For cosine, the double angle identity is expressed as \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \). This formula is particularly useful when trying to simplify expressions involving angles that are multiples of another angle.

These identities allow us to transform and manipulate trigonometric functions, making them easier to work with. For example, in the given exercise, we see how the double angle identity helps verify the expression \( \cos 4\alpha = \cos^2 2\alpha - \sin^2 2\alpha \). By recognizing that \( 4\alpha \) is double \( 2\alpha \), we can apply the identity to simplify and verify the expression.

• These identities are crucial because they connect single-angle trigonometric functions with those of double angles.
• They help in solving complex trigonometric equations and can be applied in various fields such as physics and engineering.

Key Aspects of Cosine

Cosine is one of the basic trigonometric functions, often abbreviated as cos. It is fundamentally related to the x-coordinate of a point on the unit circle. For any angle \( \theta \), \( \cos \theta \) gives the horizontal distance from the origin to the point on the circle.

The double angle identity for cosine, \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \), can also be expressed in other forms such as:

• \( \cos 2\theta = 2\cos^2 \theta - 1 \)
• \( \cos 2\theta = 1 - 2\sin^2 \theta \)

These variations allow for greater flexibility when solving trigonometric problems.

Cosine values range from -1 to 1, and its periodicity means the function repeats every \( 2\pi \) radians. The cosine function is symmetric around the vertical axis, which makes it an even function. This property, plus its identities, make cosine a versatile and widely used function in mathematics and science.

Insights Into Sine

Sine is another primary trigonometric function, abbreviated as sin. It represents the y-coordinate of a point on the unit circle for any given angle \( \theta \). This translates to the vertical distance from the origin.

When exploring double angle identities, it's important to understand how sine and cosine interact. For sine, the main double angle identity is \( \sin 2\theta = 2\sin \theta \cos \theta \). This illustrates how sine of a double angle can be broken down into its sine and cosine components.

• Sine values also range from -1 to 1.
• Unlike cosine, sine is an odd function, meaning \( \sin(-\theta) = -\sin \theta \).

The concept actually further contributes to establishing various trigonometric identities. Understanding the relationship between sine and cosine is key for solving more complex trigonometric identities and equations.