Problem 48 - 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. $$\cos ^{4} \frac{x}{2}$$

Step-by-Step Solution

Answer

The original expression \( \cos ^{4} \frac{x}{2} \) can be rewritten as \( \frac{1+ 2\cos x + \cos^{2} x}{4} \). Verification through graphing shows the two expressions to be equivalent.

Step 1: Applying the Double Angle Formula

First, recognize that the double-angle formula, \(\cos ^{2} \theta = \frac{1 + \cos(2\theta)}{2}\), can be applied. Substitute \( \frac{x}{2} \) for \( \theta \) in the formula to get \( \cos ^{2} \frac{x}{2} = \frac{1 + \cos x}{2} \).

Step 2: Substitute the value of \( \cos ^{2} \theta \)

Next, considering that \( \cos ^{4} \frac{x}{2} = (\cos ^{2} \frac{x}{2})^{2} \), substitute the earlier obtained value of \( \cos ^{2} \frac{x}{2} = \frac{1 + \cos x}{2} \) into \( \cos ^{4} \frac{x}{2} \). This gives \( \cos ^{4} \frac{x}{2} = (\frac{1 + \cos x}{2})^{2} \).

Step 3: Simplify the Expression

Expand the square on the right side, leading to the simplification: \( \cos ^{4} \frac{x}{2} = \frac{1+ 2\cos x + \cos^{2} x }{4} \). Now, our expression is in terms of the first power of cosine and we'll proceed with graphing.

Step 4: Verify through Graphing

Using any graphing tool, plot the two expressions \( y = \cos ^{4} \frac{x}{2} \) and \( y = \frac{1+ 2\cos x + \cos^{2} x}{4} \). If both expressions are the same as per our assumption, their graphs should coincide.

Key Concepts

Cosine Function

The cosine function is one of the fundamental functions in trigonometry. It helps measure the horizontal length in a circle when dealing with angles.
It's particularly crucial in understanding periodic phenomena like sound waves and tides.

• The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse.
• In the unit circle, cosine defines the x-coordinate of a point on the circle.

The formula for cosine involving double angles is exceptionally useful in transformations and simplifying higher power expressions. When applying this to the double angle, we use: \[\cos 2\theta = 2\cos^2\theta - 1\]This formula can help reduce expressions with higher powers of cosine (like \(\cos^4\frac{x}{2}\)) to forms involving single degree powers, making them easier to analyze and graph.

Power of Cosine Expression

The power of a cosine expression often needs simplification in trigonometry. It's important to rewrite these powers in terms of lower powers for easier manipulation and understanding.
In our exercise, the expression \(\cos^4 \frac{x}{2}\) is simplified using known identities.
First, remember the double-angle identity for cosine: \(\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}\).

• We can replace \(\theta\) with \(\frac{x}{2}\), finding that \(\cos^2 \frac{x}{2} = \frac{1 + \cos x}{2}\).
• This substitution simplifies the calculation of powers like \(\cos^4 \frac{x}{2}\), making them easier to expand and reduce.

Finally, by expanding this reduced form, we reach the expression \(\frac{1 + 2\cos x + \cos^2 x}{4}\), which is much simpler to work with when plotting a graph.

Verifying Equivalent Expressions with Graphs

Graphs provide a visual form of verification that two expressions are equivalent. Comparing their graphs can prove useful, especially in trigonometry, where functions can be difficult to intuit without a visual aid.
Using a graphing tool helps to confirm the mathematical work behind simplifying expressions.
For instance, to ensure that \(\cos^4\frac{x}{2}\) and \(\frac{1+ 2\cos x + \cos^{2} x}{4}\) are indeed equivalent, graph both expressions. Here's how:

• Plot \(y = \cos^4\frac{x}{2}\) on a graph using a calculator or graphing software.
• On the same axes, plot \(y = \frac{1+ 2\cos x + \cos^{2} x}{4}\).

If the graphs overlap completely, it confirms that both expressions represent the same function. This graphing approach simplifies debugging and verifying complex trigonometric transformations.