Problem 36 - 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. $$\sin ^{2} x \cos ^{2} x$$

Step-by-Step Solution

Answer

The expression \( \sin^{2}x \cos^{2}x \) can be rewritten in terms of the first power of the cosine as \( \cos^{2}x - \cos^{4}x \). Graphs of both expressions confirm they are equivalent.

Step 1: Rewrite the expression using trigonometric identities.

The given expression is \( \sin^{2}x \cos^{2}x \). Start by expressing \( \sin^{2}x \) in terms of cosine. Using the Pythagorean identity \( \sin^{2}x + \cos^{2}x = 1 \), we can rearrange it to see that \( \sin^{2}x = 1 - \cos^{2}x \). Now, substitute this expression into the original formula to get \( (1 - \cos^{2}x) \cos^{2}x \).

Step 2: Simplify the new expression.

Next, multiply through the previous expression to simplify it: \( \cos^{2}x - \cos^{4}x \). This is the given expression rewritten in terms of the first power of the cosine.

Step 3: Graph both the expressions.

Using a graphing tool of choice, graph both the original expression \( \sin^{2}x \cos^{2}x \) and the rewritten one \( \cos^{2}x - \cos^{4}x \). If both forms are equivalent, the graphs should overlap. You should observe that both graphs are identical, which verifies the transformation.

Key Concepts

Cosine

The cosine function is a fundamental element of trigonometry, particularly when dealing with right triangles and periodic phenomena like waves. In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. This relationship is summarized by the equation:

• \( \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \)

Cosine values range between -1 and 1 for any angle measured in radians or degrees. When analyzing periodic patterns, especially waves, cosine helps describe the shape and oscillation.
Cosine is often used with other trigonometric functions to simplify equations or expressions. Understanding its properties and behavior is foundational for tackling problems involving trigonometric identities, like rewriting expressions in terms of the first power of another function.

Sine

Sine, alongside cosine, is one of the primary trigonometric functions. It represents the ratio of the opposite side to the hypotenuse in a right triangle. Mathematically, it is expressed as:

• \( \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} \)

The sine function oscillates between -1 and 1, similar to cosine, but it starts at zero when the angle is zero. This offset makes sine particularly useful for modeling wave forms and circular motion.
In trigonometric identities, sine is often paired or contrasted with cosine. By understanding these relationships, you can use identities like the Pythagorean identity to convert between sine and cosine or simplify expressions involving them.

Pythagorean identity

The Pythagorean identity is a cornerstone of trigonometry. It forms the basis for many transformations and simplifications in trigonometric expressions. The identity states:

• \( \sin^{2}x + \cos^{2}x = 1 \)

This equation reflects the inherent relationship between the sine and cosine of any angle, derived from the Pythagorean theorem. This identity is versatile; if you know either the sine or the cosine, you can easily calculate the other.
In expressions like \( \sin^2 x \cos^2 x \), using the Pythagorean identity allows you to substitute \( \sin^2 x \) with \( 1 - \cos^2 x \), simplifying the expression in terms of cosine. Such transformations are crucial for working through more complex trigonometric calculations.

Graphing utility

Graphing utilities, such as graphing calculators or software, are essential tools in verifying mathematical transformations. They offer visual confirmation of equations or identities, making them immensely helpful for students learning trigonometry. When you graph expressions like \( \sin^2 x \cos^2 x \) and its transformed equivalent, \( \cos^2 x - \cos^4 x \), you should see that both graphs overlap, indicating the expressions are equivalent.
Using a graphing utility allows you to experiment and understand how different expressions can look identical but are expressed differently. This visual aid is crucial for comprehending the deeper connections between trigonometric identities and functions. Such exploration not only solidifies theoretical knowledge but also builds intuition and confidence in handling trigonometric problems.