Problem 34 - Algebra 2

Question

Simplify each expression. $$ \frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin ^{2} \theta} $$

Step-by-Step Solution

Answer

The simplified form is \(\csc^2 \theta\).

Step 1: Recall the Pythagorean identity

The Pythagorean identity states that for any angle \(\theta\), \(\sin^2 \theta + \cos^2 \theta = 1\) holds true. This identity is fundamental in trigonometry.

Step 2: Apply the Pythagorean identity

Substitute the identity into the expression: \(\frac{1}{\sin^2 \theta}\). This simplification is possible because the numerator \(\sin^2 \theta + \cos^2 \theta\) equals 1.

Step 3: Rewrite the expression

After substitution, the expression becomes \(\frac{1}{\sin^2 \theta}\). This is a more straightforward form of the expression.

Step 4: Recognize the reciprocal identity

Recall that \(\frac{1}{\sin^2 \theta}\) is equivalent to \(\csc^2 \theta\), based on the trigonometric identity for the cosecant function.

Step 5: Final simplified form

Thus, the simplified form of the original expression \(\frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta}\) is \(\csc^2 \theta\).

Key Concepts

Pythagorean Identity

In trigonometry, the Pythagorean Identity is a fundamental equation. It ties together the squares of the sine and cosine of the same angle \( \theta \). This identity is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is a direct result of the Pythagorean theorem in geometry and can be used to simplify many trigonometric expressions. Whenever you come across \( \sin^2 \theta + \cos^2 \theta \) in any trigonometric problem, you can replace it with 1. This is immensely helpful for simplifying expressions and solving trigonometric equations. Understanding and applying this identity helps in making complex expressions much simpler by reducing terms.

Reciprocal Trigonometric Functions

Trigonometric functions have their reciprocal counterparts that help simplify expressions further. In this exercise, after using the Pythagorean identity, the expression become \( \frac{1}{\sin^2 \theta} \). This can be translated into one of the reciprocal trigonometric functions. Here's what you should know:

• The reciprocal of \( \sin \theta \) is \( \csc \theta \) or cosecant, so \( \frac{1}{\sin \theta} = \csc \theta \).
• Similarly, \( \frac{1}{\sin^2 \theta} \) becomes \( \csc^2 \theta \).

Using these reciprocal relationships allows for additional simplification and interpretation of trigonometric expressions. They are particularly helpful in various branches of mathematics, including calculus, where operations could become complex without them.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing them to their simplest forms, making computations easier. The goal is to use known identities and relationships to transform an expression into one that is neat and easy to manage. Here's how:

• Use identities like the Pythagorean Identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to substitute portions of the expression.
• Convert expressions to involve similar trigonometric functions or their reciprocals when possible.
• Look for opportunities to factor or combine terms to achieve simpler results.

In our example, the original expression \( \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} \) simplifies to \( \csc^2 \theta \). This process reduces the complexity and allows for easier manipulation in further mathematical calculations. Becoming comfortable with these simplifications enhances problem-solving skills across many areas of mathematics.