Problem 39 - 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. $$\sin ^{2} x \cos ^{4} x$$

Step-by-Step Solution

Answer

The expression \(\sin ^{2} x \cos ^{4} x\) simplified in terms of the first power of the cosine is \( \frac{1 + \cos (4x)}{2} - \frac{1 + \cos (6x)}{2}\)

Step 1: Apply the Pythagorean identity

Use the Pythagorean identity to express \(\sin^2x\) in terms of \(\cos^2x\). Thus, \(\sin ^{2} x = 1 - \cos ^{2} x\). Substitute this into the original expression: \((1 - \cos ^{2} x) \cos ^{4} x\).

Step 2: Expand the Expression

Expand the expression you obtained in step 1, resulting in \( \cos ^{4} x - \cos ^{6} x \).

Step 3: Apply the Power Reduction Formula

For simplifying \(\cos ^{4} x\) and \(\cos ^{6} x\), apply the power-reduction formula: \(\cos ^{2n} x = \frac{1 + \cos (2nx)}{2}\). Apply this to \(\cos ^{4} x = \frac{1 + \cos (2*2x)}{2} = \frac{1 + \cos (4x)}{2}\) and \(\cos ^{6} x = \frac{1 + \cos (2*3x)}{2} = \frac{1 + \cos (6x)}{2}\). Sub these values in our equation from Step 2.

Step 4: Result

From Step 3, the final expression in terms of the first power of the cosine is \( \frac{1 + \cos (4x)}{2} - \frac{1 + \cos (6x)}{2}\).

Step 5: Verification using Graphic Utility

For verifying if both of the original and manipulated forms are same, graph both expressions using a graphing utility. If the graphs overlap completely or are identical, then the manipulation is correct and both forms are indeed the same.

Key Concepts

Pythagorean Identity

When simplifying trigonometric expressions, the Pythagorean identity is an essential tool. It originates from the Pythagorean theorem relating the sides of a right triangle. For trigonometric functions, the identity states that for any angle \(x\), \(\sin^2x + \cos^2x = 1\). This is a reflection of the fact that the sum of the squares of the sine and cosine of an angle equals the square of the hypotenuse (which is 1 in a unit circle).

By rearranging the identity, you can express \(\sin^2x\) in terms of \(\cos^2x\) as \(\sin^2x = 1 - \cos^2x\), and vice versa. This conversion is extremely useful in simplifying higher powers of sine or cosine into expressions involving only the first power of one of the functions, which is what we did in our example problem by substituting \(\sin^2x\) with \(1 - \cos^2x\).

Power-Reduction Formulas

Power-reduction formulas are a part of trigonometric identities that enable the simplification of even powers of sine and cosine into expressions containing only the first powers of these functions. These formulas arise from the double-angle formulas and are particularly helpful in reducing the complexity of integrals and other trigonometric expressions.

The power-reduction formula for cosine is \(\cos^{2n}x = \frac{1 + \cos(2nx)}{2}\), where \(n\) is a positive integer. This allows us to rewrite \(\cos^4x\) and \(\cos^6x\) from our original problem as \(\frac{1 + \cos(4x)}{2}\) and \(\frac{1 + \cos(6x)}{2}\), respectively. These formulas are instrumental in simplifying the original expression into a more manageable format.

Graphing Trigonometric Functions

Graphing is a powerful way to visualize and confirm the identities and transformations of trigonometric functions. Using a graphing utility, students can plot the original trigonometric expression and its simplified counterpart to ensure they are equivalent.

To graph a function like \(\sin^2x\cos^4x\), you would look for characteristic patterns, such as symmetry, periodicity, and amplitude. After simplifying the expression using identities, graph the new expression to see if it matches the original. If the graphs are identical or overlap completely, this visual confirmation serves as a strong verification of the work done in simplifying the trigonometric expression.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. They serve as the foundation for simplifying expressions, solving equations, and performing calculations involving triangles. Besides the Pythagorean identity and power-reduction formulas, there are sum and difference formulas, double and half-angle formulas, and many others.

Mastering these identities is crucial for students as they allow for the transformation and simplification of complex trigonometric expressions. By understanding how to apply these identities appropriately, you can rewrite trigonometric expressions, like the one presented in the original exercise, in various equivalent forms, aiding in solving a wider range of problems.