Problem 14 - Precalculus Mathematics for Calculus

Question

Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4. $$\cos ^{4} x \sin ^{2} x$$

Step-by-Step Solution

Answer

Rewrite using power reduction and substitute: \( \frac{\cos^4 x (1 - \cos 2x)}{2} \).

Step 1: Apply Double Angle Formulas

To rewrite the expression involving higher powers of trigonometric functions, start by applying the double angle identity for sine: \( \sin^2 x = \frac{1 - \cos 2x}{2} \). This helps reduce the power of sine.

Step 2: Substitute into the Expression

Substitute \( \sin^2 x \) in the original expression \( \cos^4 x \sin^2 x \) with \( \frac{1 - \cos 2x}{2} \). This gives: \( \cos^4 x \left( \frac{1 - \cos 2x}{2} \right) \).

Step 3: Simplify the Expression

Multiply \( \cos^4 x \) with the substituted term: \[ \frac{\cos^4 x (1 - \cos 2x)}{2} = \frac{\cos^4 x - \cos^4 x \cos 2x}{2} \].

Step 4: Expand Using Power-Reducing Formula for Cosine

Utilize the power-reducing identity for cosine: \( \cos^2 x = \frac{1 + \cos 2x}{2} \), and hence \( \cos^4 x = \left( \frac{1 + \cos 2x}{2} \right)^2 \).

Step 5: Rewrite \( \cos^4 x \)

Expand \( \left( \frac{1 + \cos 2x}{2} \right)^2 \) to get \( \frac{1 + 2\cos 2x + \cos^2 2x}{4} \).

Step 6: Substitute Back into the Expression

Substitute the expanded form of \( \cos^4 x \) back into the expression: \[ \frac{1 + 2\cos 2x + \cos^2 2x}{4} \cdot \frac{1 - \cos 2x}{2} \].

Step 7: Further Simplify

Distribute \( \frac{1 - \cos 2x}{2} \) across each term in \( \frac{1 + 2\cos 2x + \cos^2 2x}{4} \) to fully simplify the expression.

Key Concepts

Double Angle Formulas

Double angle formulas are important tools in trigonometry that help to simplify expressions involving trigonometric functions raised to powers. The double angle formula for sine is particularly useful and is given by:

• \( \sin^2 x = \frac{1 - \cos 2x}{2} \)

This formula allows us to express \( \sin^2 x \) in terms of \( \cos 2x \), effectively reducing the power of sine from two to one. In the given exercise, using the double angle formula for \( \sin^2 x \) simplifies the process of rewriting the original expression, \( \cos^4 x \sin^2 x \), into a more manageable form.
By substituting \( \sin^2 x \) with \( \frac{1 - \cos 2x}{2} \), we reduce the power of sine and set the stage to manipulate the expression further using cosine functions. This substitution transforms a complex expression into a simpler form, making it easier to handle and understand.

Power-Reducing Formulas

Power-reducing formulas in trigonometry are used to express higher powers of trigonometric functions in terms of the first power of cosine or sine. These identities help in simplifying expressions and integrating functions that have higher powers. For cosine, the power-reducing formula is:

• \( \cos^2 x = \frac{1 + \cos 2x}{2} \)

Applying this formula twice, as shown in the exercise, we find \( \cos^4 x \) by squaring \( \cos^2 x \) to get:

• \( \cos^4 x = \left( \frac{1 + \cos 2x}{2} \right)^2 \)

This helps in expressing \( \cos^4 x \) in terms of the first power of \( \cos 2x \). Breaking down complex expressions using power-reducing identities allows us to write them in a simpler, unified form. This makes further simplification much more straightforward, leading ultimately to an expression solely in terms of the first power of cosine.

Cosine and Sine Functions

The cosine and sine functions are the building blocks of trigonometry and are involved in numerous identities, including double angle and power-reducing formulas. Comprehending these functions deeply is key to understanding and applying trigonometric identities effectively in problems.
Cosine, represented as \( \cos(x) \), is the x-coordinate of a point on the unit circle corresponding to an angle \( x \). Sine, represented as \( \sin(x) \), is the y-coordinate of that same point. These functions help to describe the relationship between angles and the lengths of sides in right triangles, but their applications extend beyond geometry to calculus, physics, and engineering.

• Understanding how these functions relate to each other through identities helps in simplifying and solving complex trigonometric expressions.
• For instance, using the identities to move between powers or to convert products into sums prepares the ground for deeper analytical work.

As fundamental concepts, mastering cosine and sine functions is essential for tackling not just exercises and homework assignments, but also for grasping more advanced mathematical concepts.