Problem 40 - Algebra 2

Question

REASONING Describe the conditions under which you would use each of the three identities for \(\cos 2 \theta .\)

Step-by-Step Solution

Answer

Use each identity based on known values: 1. \(\cos^2 \theta - \sin^2 \theta\), when both \(\cos\theta\) and \(\sin\theta\) are known. 2. \(2\cos^2 \theta - 1\), when \(\cos\theta\) is known. 3. \(1 - 2\sin^2 \theta\), when \(\sin\theta\) is known.

Step 1: Identify the Three Identities

The formula for \ \ \(\cos 2\theta\) can be expressed in three different identities: 1. \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)2. \(\cos 2\theta = 2\cos^2 \theta - 1\)3. \(\cos 2\theta = 1 - 2\sin^2 \theta\)

Step 2: First Identity – Cos and Sin Known

The identity \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\) is useful when both \(\cos\theta\) and \(\sin\theta\) values are known. This simplifies calculations when solving problems involving both cosine and sine values.

Step 3: Second Identity – Cos Known

The identity \(\cos 2\theta = 2\cos^2 \theta - 1\) is used when only \(\cos\theta\) is known or is easier to find. This situation arises often in problems focusing on cosine or where sine is more complex to compute or simply not available.

Step 4: Third Identity – Sin Known

Similarly, \(\cos 2\theta = 1 - 2\sin^2 \theta\) is beneficial in cases where \(\sin\theta\) is known. This identity eliminates the need to compute the cosine value directly, thus, simplifying the problem significantly when sine is readily available.

Key Concepts

Double Angle Formulas

The double angle formulas are important tools in trigonometry. They help us find the sine, cosine, or tangent of double angles—that is, angles that are twice another angle. Understanding these formulas makes it easier to solve a variety of trigonometric problems.

Here are the three identities for the cosine of double an angle, denoted as \( \cos 2\theta \):

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

Each identity has its specific conditions for use, depending on the values or expressions available. The first identity is ideal when both cosine and sine values are known, allowing for straightforward calculations. The second identity proves advantageous when only the cosine value is known, minimizing the need to determine sine. Lastly, the third identity is helpful when only sine is known, as it avoids needing cosine calculations entirely.

Understanding when and how to use these formulas can greatly simplify complex trigonometric problems.

Cosine Function

The cosine function, often written as \( \cos \theta \), is a fundamental trigonometric function. It helps relate the angle and the adjacent side in a right-angled triangle to the hypotenuse. The function can take values from -1 to 1, repeating its pattern every 360 degrees or \( 2\pi \) radians.

Here are some key points about the cosine function:

• It's even, meaning \( \cos(-\theta) = \cos \theta \).
• It's periodic with a period of \( 2\pi \).
• The maximum value is 1, and the minimum is -1.
• The amplitude of a cosine wave is the distance from the centerline to the peak or trough of the wave.

In the context of double angle formulas, the cosine function is especially useful in equations where knowing the angle's cosine value simplifies mathematical computations. When only the cosine value is given, using the identity \( \cos 2\theta = 2\cos^2 \theta - 1 \) makes the process more direct, bypassing the need to determine the sine value.

The simplicity of the cosine's algebraic properties supports solving trigonometric problems efficiently.

Sine Function

The sine function, denoted \( \sin \theta \), is another critical trigonometric function. Like the cosine, it is essential for modeling periodic phenomena and relating the side opposite an angle in a right-angle triangle to the hypotenuse.

Here are several significant attributes of the sine function:

• It's odd, so \( \sin(-\theta) = -\sin \theta \).
• It also has a period of \( 2\pi \), meaning it repeats every 360 degrees.
• Values range between -1 and 1.
• The phase shift in comparison to the cosine function is \( \frac{\pi}{2} \) radians.

The sine function is crucial in applying the double angle formula when the sine value is known and cosine is not easily accessed. In such scenarios, the identity \( \cos 2\theta = 1 - 2\sin^2 \theta \) is particularly beneficial.

Overall, understanding when to leverage the sine or cosine in double angle scenarios leads to clearer and more efficient problem-solving.