Problem 33 - Algebra 2

Question

Simplify each expression. $$ \frac{1-\sin ^{2} \theta}{\sin ^{2} \theta} $$

Step-by-Step Solution

Answer

The simplified expression is \( \cot^2 \theta \).

Step 1: Recognize Pythagorean Identity

Notice that the expression in the numerator, \(1 - \sin^2 \theta\), is a recognizable Pythagorean identity. Recall that \( \sin^2 \theta + \cos^2 \theta = 1 \), so \(1 - \sin^2 \theta = \cos^2 \theta\).

Step 2: Substitute Identity into Expression

Replace \(1 - \sin^2 \theta\) with \(\cos^2 \theta\) in the expression. This simplifies the original expression to \( \frac{\cos^2 \theta}{\sin^2 \theta}\).

Step 3: Simplify Fraction

Observe that \( \frac{\cos^2 \theta}{\sin^2 \theta} \) can be rewritten as \( \left(\frac{\cos \theta}{\sin \theta}\right)^2 \). This is because \( \frac{\cos^2 \theta}{\sin^2 \theta} = (\frac{\cos \theta}{\sin \theta})(\frac{\cos \theta}{\sin \theta}) \).

Step 4: Recognize Cotangent

Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Thus, \( \left( \frac{\cos \theta}{\sin \theta} \right)^2 = (\cot \theta)^2 \).

Step 5: Final Simplified Expression

The expression \( \frac{1-\sin^2 \theta}{\sin^2 \theta} \) simplifies to \( \cot^2 \theta \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that helps link different trigonometric functions together. It comes from the Pythagorean theorem and appears as: \[ \sin^2 \theta + \cos^2 \theta = 1 \]This equation expresses the intrinsic relationship between the sine and cosine of the same angle \(\theta\).
By rearranging the terms, we can derive other useful forms, such as:

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

In the exercise, we used this identity to transform the expression \(1 - \sin^2 \theta\) into \(\cos^2 \theta\), thereby simplifying the original problem.
Understanding and recognizing these variations of the Pythagorean identity helps in efficiently simplifying complex trigonometric expressions.

Simplifying Expressions

Simplifying expressions in algebra involves finding an equivalent version of the original expression which is often more manageable or easier to work with. In the case of trigonometric expressions, this could mean reducing complexity using known identities.
In this exercise, once we substituted \(1 - \sin^2 \theta\) with \(\cos^2 \theta\), the expression turned into \(\frac{\cos^2 \theta}{\sin^2 \theta}\).
To simplify further, we see that each term in the fraction can be rewritten in simpler trigonometric terms. For instance:

• The trigonometric fraction \(\frac{\cos^2 \theta}{\sin^2 \theta}\) can be rewritten as \(\left(\frac{\cos \theta}{\sin \theta}\right)^2\)

This simplification is crucial as it makes it evident that the expression corresponds to the square of another well-known trigonometric function, the cotangent.

Cotangent

The cotangent is one of the basic trigonometric functions, denoted by \(\cot \theta\).
It is defined as the reciprocal of the tangent:\[ \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \]In this exercise, recognizing \( \frac{\cos \theta}{\sin \theta} \) as \( \cot \theta \) helped in simplifying the expression \( \frac{\cos^2 \theta}{\sin^2 \theta} = \left( \frac{\cos \theta}{\sin \theta} \right)^2 = \cot^2 \theta \).
Awareness of these straightforward relationships between trigonometric functions provides a powerful toolkit for solving and simplifying a wide range of trigonometric problems. Cotangent is less commonly used than sine and cosine, but knowing it can greatly aid in problems involving right triangles and periodic functions.

Algebra 2

In Algebra 2, students deepen their understanding of functions and equations, including more complex trigonometric relationships.
This includes applying identities and simplifying expressions as seen in this problem. Trigonometric expressions often appear in Algebra 2 curriculum, making it necessary to combine algebraic and trigonometric reasoning. Working through exercises like the one provided allows students to:

• Identify and apply trigonometric identities.
• Understand the simplification process involving fractions and powers.
• Build proficiency in transforming expressions using multiple mathematical concepts.

These skills are crucial not only for solving equations but also for preparing students for more advanced mathematics in calculus and other higher-level subjects.