Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite the expression in terms of the first power of the cosine. Use a graphing utility to graph both expressions to verify that both forms are the same. $$\cos ^{8} x$$

Step-by-Step Solution

Answer

The expression \(\cos ^{8} x\) can be rewritten in terms of the first power of the cosine as \( \frac{35}{128}(1+\cos2x)^4 \)

Step 1: Apply the power-reduction identity

Firstly, the power-reduction identity needs to be applied. For cosine function, the power-reduction identity is given by \[ \cos^{2n}x = \frac{1}{2^{2n}}\binom{2n}{n}(1+\cos2x)^n \]. Hence, for the given expression, n=4 and x is x. Substituting these values in the power-reduction identity, we get \[ \cos ^{8} x = \frac{1}{2^{8}}\binom{8}{4}(1+\cos2x)^4 \]

Step 2: Simplify the expression

The next step is to simplify the obtained expression. So, \[ \cos ^{8} x = \frac{1}{256}\times70\times(1+\cos2x)^4 \]. Hence the expression is rewritten in terms of the first power of the cosine as \[ \cos^{8}x = \frac{35}{128}(1+\cos2x)^4 \]

Step 3: Verification using graphing utility

The final step is to graph both the given expression \(\cos^{8}x\) and the obtained expression \( \frac{35}{128}(1+\cos2x)^4 \) using a graphing utility. Both the graphs should coincide with each other which verifies that both the forms are equivalent. This cannot be shown here and needs to manually drawn.

Key Concepts

Power-Reduction Identity

Power-reduction identities are useful trigonometric tools that help in rewriting higher powers of trigonometric functions in terms of the first power. They are particularly beneficial when dealing with integrals or simplifying expressions. For the cosine function, the identity is expressed as \[ \cos^{2n}x = \frac{1}{2^{2n}}\binom{2n}{n}(1+\cos2x)^n. \]This equation might seem complex initially, but let's break it down:

• \(2n\) refers to twice the power of the cosine you are reducing.
• \(\binom{2n}{n}\) is the binomial coefficient, representing combinations.
• The expression \((1+\cos2x)^n\) indicates repeated use of the angle doubling formula.

Essentially, power-reduction identities make it possible to express, for example, \(\cos^8x\) as a simpler function involving \(\cos2x\). This reduction makes further calculations and verifications considerably easier.

Cosine Function

The cosine function, a fundamental part of trigonometry, represents the x-coordinate of a point on the unit circle as the angle increases. This periodic function, denoted \(\cos\), oscillates between -1 and 1.Key properties of cosine include:

• Cosine is an even function: \(\cos(-x) = \cos(x)\).
• Its period is \(2\pi\), meaning \(\cos(x + 2\pi) = \cos(x)\).
• Its graph is wave-like, starting at (1, 0) and repeating every \(2\pi\).

Understanding the cosine function's behavior is crucial for graphing and simplifying expressions involving trigonometric identities, such as in power-reduction.

Graphing Utility

Graphing utilities are powerful tools used to visually compare and analyze mathematical functions. In the context of trigonometric identities, they can confirm that different expressions represent the same function. For instance, using a graphing tool, one can plot both \(\cos^{8}x\) and \(\frac{35}{128}(1+\cos2x)^4\).The steps to use a graphing utility effectively:

• Input both functions into the graphing tool.
• Set an appropriate window that includes several periods of the cosine function to see clear repetitions.
• Observe if the graphs overlap. If the graphs coincide perfectly, it verifies the equality of both expressions.

Such verification ensures that theoretical simplifications made using identities hold true visually, reinforcing understanding.

Expression Simplification

Simplifying expressions is a fundamental aspect of problem-solving in mathematics. It involves breaking down complex expressions into more manageable forms without altering their value. In the example of \(\cos^{8}x\), using the power-reduction identity helped in expressing it as\[\cos^{8}x = \frac{35}{128}(1+\cos2x)^4.\]The steps for simplification:

• Apply the appropriate identity — here, the power-reduction identity was key.
• Simplify numerical coefficients, such as \(\frac{1}{256} \times 70\).
• Ensure that all terms are correctly reduced in terms of the lowest power available.

By simplifying, mathematical expressions become easier to handle and understand, whether for graphing, integration, or further algebraic manipulation.