Problem 55 - Contemporary Precalculus

Question

Simplify the given expression. $$2 \cos 2 y \sin 2 y(\text { Think } !)$$

Step-by-Step Solution

Answer

Question: Simplify the trigonometric expression $$2 \cos 2y \sin 2y$$. Answer: $$4\sin{y}\cos{y} - 8\sin^3{y}\cos{y}$$

Step 1: Apply the double angle formulas for sine and cosine

By utilizing the identities mentioned earlier, we have: $$2 \cos 2y \sin 2y = 2 (1 - 2\sin^2{y})(2\sin{y}\cos{y})$$

Step 2: Simplify the expression

Distribute the 2 and simplify the expression: $$2(1 - 2\sin^2{y})(2\sin{y}\cos{y}) = (4\sin{y}\cos{y}) - (8\sin^3{y}\cos{y})$$ Therefore, the simplified expression is: $$4\sin{y}\cos{y} - 8\sin^3{y}\cos{y}$$

Key Concepts

Double Angle Formulas

Double Angle Formulas help to simplify expressions involving trigonometric functions by using identities for angles that are twice a given angle. These identities are used in conjunction with angles present in a trigonometric expression to transform and simplify the expression effectively.

For cosine and sine, the double angle formulas are:

• For cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• You can also express it as \( \cos 2\theta = 2\cos^2 \theta - 1 \) or \( \cos 2\theta = 1 - 2\sin^2 \theta \)
• For sine: \( \sin 2\theta = 2\sin \theta \cos \theta \)

These formulas come in handy when dealing with expressions like the one in our exercise where the product of \( \cos 2y \) and \( \sin 2y \) can be expressed and worked upon using these identities.

In the given solution, the expression \( 2 \cos 2y \sin 2y \) was restructured using these formulas to examine possible simplifications. Starting with these basic double angle identities can pave the way for easy navigation through seemingly complex trigonometric expressions.

Simplifying Expressions

Simplifying expressions is about reducing the complexity of mathematical expressions to a more manageable form. This process usually involves combining like terms, reducing fractions, or utilizing identities to condense expressions. In trigonometry, the goal of simplifying is to make the expressions easier to understand and compute.

In the provided exercise, the expression \( 2(1 - 2\sin^2{y})(2\sin{y}\cos{y}) \) was simplified by applying distribution, a common mathematical process.

• First, multiply out the terms: distribute \( 2 \sin{y}\cos{y} \) to \( 1 - 2\sin^2{y} \)
• Combine and simplify: \( 2(2\sin{y}\cos{y}) - 2(2\sin^3{y}\cos{y}) \)
• Result in simplified form: \( 4\sin{y}\cos{y} - 8\sin^3{y}\cos{y} \)

Following these steps, we derived a cleaner and more workable expression. Engaging with the simplification process step-by-step is often about understanding how to efficiently employ algebraic tactics to streamline analysis or calculations in mathematical problems.

Trigonometric Functions

Trigonometric Functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The basic trigonometric functions in a right triangle context are sine, cosine, and tangent, but they extend far beyond just triangles in their applications. Each function has specific defining properties based on the unit circle, making them essential in a variety of fields like physics, engineering, and even art.

Key functions include:

• Sine \( \sin \theta \) represents the ratio of the length of the side opposite to the angle to the hypotenuse.
• Cosine \( \cos \theta \) represents the ratio of the length of the adjacent side to the angle to the hypotenuse.
• Tangent \( \tan \theta \) is the ratio of opposite to adjacent sides, basically \( \sin \theta / \cos \theta \).

In the exercise, \( \cos 2y \) and \( \sin 2y \) were specifically referred to using their trigonometric function properties. Understanding these base functions also provides the baseline for exploring and applying trigonometric identities and formulas for advanced simplifications and transformations.