Problem 34 - Precalculus Essentials

Question

In Exercises \(23-34\), verify each identity. $$\cos 4 t=8 \cos ^{4} t-8 \cos ^{2} t+1$$

Step-by-Step Solution

Answer

Identity \(\cos 4 t=8 \cos ^{4} t-8 \cos ^{2} t+1\) has been verified successfully.

Step 1: Break down \(\cos 4t\) using double angle formulas

Let's break down \(\cos 4t\) into \(\cos 2(2t)\). We now can use the trigonometric double angle identity for cosine: \(\cos 2A = 2\cos^2A - 1\). This gives \(\cos 4t = 2\cos^2(2t) - 1\).

Step 2: Break down \(\cos^2(2t)\) using double angle formulas

We now have \(\cos^2(2t)\) in our equation, which is the cosine of double of \(t\). This can again be broken down using the double angle identity for cosine, which gives: \(\cos 4t = 2[2\cos^2(t) - 1]^2 - 1\).

Step 3: Simplify

The next step is simplifying the equation above. With algebra, the equation can be simplified into: \(\cos 4t = 8\cos^4(t) - 8\cos^2(t) + 1\) which verifies the identity.

Key Concepts

Cosine Double Angle Formula

The cosine double angle formula is an essential tool in trigonometry. It allows us to express the cosine of double an angle in terms of the square of the cosine of the original angle. This is particularly helpful when simplifying expressions involving trigonometric functions.
Let's focus on the formula:

• The standard cosine double angle formula is \( \cos 2A = 2\cos^2 A - 1 \).
• It shows that the cosine of twice an angle can be rewritten using the square of the cosine of that angle.

To apply this in problem-solving, it's crucial to recognize when you can break down an angle and use this formula. In the initial exercise, \( \cos 4t \) is expressed using \( \cos 2(2t) \). This transformation allows the equation to be simplified step-by-step, revealing its core structure through known identities.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to achieve a desired form or solution. This is often necessary when working with trigonometric identities.

• In the given exercise, we start with the expression \( \cos 4t = 2[2\cos^2(t) - 1]^2 - 1 \), which needs simplification.
• We first expand the term \([2\cos^2(t) - 1]^2\).
• Then, multiply the terms to distribute and combine like terms. This expansion and simplification are pivotal in verifying identities and making complex expressions more manageable.

Through careful manipulation, we arrive at \( 8\cos^4(t) - 8\cos^2(t) + 1 \), which demonstrates the power of manipulation in trigonometry.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, form the backbone of trigonometric identities and equations. Understanding these functions is critical when dealing with identities in trigonometry.

• The cosine function, specifically used in this exercise, relates an angle of a right triangle to the ratio of the adjacent side over the hypotenuse in the unit circle.
• These functions are periodic, which means they repeat values at regular intervals, making them very useful for modeling cyclical phenomena.
• Mastery of these basic functions, their properties, and their relationships to each other helps in both solving and understanding trigonometric identities.

Trigonometric identities, such as the one proven in the problem above, leverage these functions’ inherent properties to transform and simplify equations, making them invaluable tools for students and mathematicians alike.