Problem 38 - Algebra and Trigonometry

Question

Use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. $$ 8 \sin ^{2} x \cos ^{2} x $$

Step-by-Step Solution

Answer

The equivalent expression that does not contain powers of trigonometric functions greater than 1 is \[\frac{3}{2} - \frac{\cos(4x)}{2}\]

Step 1: Apply Power-Reducing Formulas

Firstly, apply the power-reducing formulas to each of the trigonometric terms in the expression. This transforms the expression into\[8 \times \frac{1 - \cos(2x)}{2} \times \frac{1 + \cos(2x)}{2}\]

Step 2: Perform the Multiplication

The expression can now be simplified by multiplying out the terms. The result is\[\frac{8[1 - (\cos^2(2x))]}{4}\]

Step 3: Simplify the Expression

Finally, simplify the expression by dividing 8 by 4 and replacing \[\cos^2(2x) with \frac{1 + \cos(4x)}{2}\]to ensure all trigonometric powers are not greater than 1. This results in\[2 - \frac{2 + 2\cos(4x)}{4} = 2 - \frac{1}{2} - \frac{\cos(4x)}{2} = \frac{3}{2} - \frac{\cos(4x)}{2}\]

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding and solving many mathematical problems. These functions include \(\sin\), \(\cos\), \(\tan\), and their reciprocal functions like \(\csc\), \(\sec\), and \(\cot\). In the given exercise, we focus mainly on \(\sin\) and \(\cos\). These functions relate an angle in a right triangle to the ratios of the triangle's sides.

• \(\sin(x)\): Represents the ratio of the opposite side to the hypotenuse.
• \(\cos(x)\): Represents the ratio of the adjacent side to the hypotenuse.

The challenge here was to simplify an expression with terms \(\sin^2(x)\) and \(\cos^2(x)\). Having powers greater than one can complicate solving these. That’s where the power-reducing formulas come in handy, simplifying these terms into equivalent expressions without raising powers above one.

Algebraic Expressions

Algebraic expressions involve variables and numbers combined using operations like addition, subtraction, multiplication, and division. In this exercise, the algebraic expression initially included the term \(8 \sin^2 x \cos^2 x\). It's important to manipulate these expressions using algebraic principles to reach the simplest form.
To handle the term, we:

• Applied formulas to reduce complex terms into simpler ones.
• Performed multiplication and algebraic simplification to combine terms.
• Subsequently, arrived at a more comprehensible expression: \(\frac{3}{2} - \frac{\cos(4x)}{2}\).

Understanding how to manipulate such expressions using basic algebraic techniques is critical in reducing more complex mathematical problems to their simplest forms.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. They are vital tools for simplifying expressions in trigonometry. Some common identities include Pythagorean and power-reducing identities.

Power-Reducing Formulas

These specific identities convert powers of trigonometric functions into expressions with lowered powers:

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

Applying these, we simplified the expression in question by substituting \(\sin^2(x)\) and \(\cos^2(x)\) with these identities.

Why Use Power-Reducing Identities?

• Reduces the degree of powers in trigonometric expressions, making them easier to manage.
• Simplifies complicated expressions, facilitating solving equations or performing integrations.
• Helps in finding equivalent, more straightforward forms of trigonometric expressions.

Equipped with such identities, you can tackle complex trigonometric problems more effectively, as shown in the steps applied in this exercise.