Problem 53 - Precalculus with Limits

Question

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \dfrac{\cos^2y}{1 - \sin y} \)

Step-by-Step Solution

Answer

The simplified form of the expression \( \dfrac{\cos^2y}{1 - \sin y} \) is 1.

Step 1: Rearrange Pythagorean identity

The Pythagorean identity is \( 1 = \sin^2y + \cos^2y \). Solving for \( \sin^2y \), we get \( \sin^2y = 1 - \cos^2y \).

Step 2: Substitute \( \sin^2y \) in the expression

Substitute \( \sin^2y = 1 - \cos^2y \) to the original expression \( \dfrac{\cos^2y}{1 - \sin y} \). We get \( \dfrac{\cos^2y}{1 - (1 - \cos^2y)} \).

Step 3: Simplify the expression

Simplify the above expression to \( \dfrac{\cos^2y}{\cos^2y} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the fundamental concepts in trigonometry that is essential for simplifying expressions and solving equations. This identity is expressed as:

• \( 1 = \sin^2y + \cos^2y \)

This equation provides a relationship between sine and cosine for any angle \( y \). By understanding and using this identity, we can manipulate expressions by substituting values to simplify them.
In some cases, like the exercise given, you can rearrange the Pythagorean Identity to solve for one function in terms of the other. For example, you can rearrange it to solve for \( \sin^2y \) as \( \sin^2y = 1 - \cos^2y \). Doing this allows you to replace \( \sin^2y \) in any expression, making it simpler. This technique often makes complex problems much more manageable.

Simplifying Trigonometric Expressions

When faced with complex trigonometric problems, simplifying the expressions can make them more approachable. Simplification involves manipulating the expression to make it easier to work with or solve. Here is a brief breakdown of how simplification is achieved:

• Identify identities: Use trigonometric identities like the Pythagorean Identity to substitute and reduce expressions.
• Substitute equivalent forms: Sometimes you replace one trigonometric function with another using fundamental identities for ease.
• Cancel common factors: Once substitution and basic algebraic manipulation are done, look for common factors that can be simplified or cancelled out.

In the original expression \( \dfrac{\cos^2y}{1 - \sin y} \), applying the appropriately rearranged Pythagorean Identity allows for substitution that simplifies the expression to \( \dfrac{\cos^2y}{\cos^2y} \), ultimately simplifying to \( 1 \). Recognizing these simplifications can vastly facilitate working through trigonometric equations.

Trigonometric Functions

Trigonometric functions are the core of trigonometric calculations and identities. These functions, such as sine, cosine, and tangent, describe the relationships between the angles and sides of triangles. In trigonometry, the most commonly used primary functions are:

• \( \sin y \): Sine of an angle \( y \)
• \( \cos y \): Cosine of an angle \( y \)
• \( \tan y \): Tangent of an angle \( y \)

Each function has its own range and properties that are often utilized in solving equations or proving identities. Understanding how these functions relate to one another through identities is crucial in simplifying trigonometric expressions. Aside from the basic ratio definitions, recognizing these trigonometric functions' complementary identities such as the Pythagorean Identity, enhances our ability to work with trigonometric problems efficiently.