Problem 26 - Algebra 2

Question

Simplify each expression. \(\frac{1-\cos ^{2} \theta}{\sin ^{2} \theta}\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Identify Trigonometric Identity

The given expression is \( \frac{1-\cos ^{2} \theta}{\sin ^{2} \theta} \). Recognize that \(1 - \cos^2 \theta\) resembles the identity for \( \sin^2 \theta \) since \( \sin^2 \theta = 1 - \cos^2 \theta \).

Step 2: Substitute the Trigonometric Identity

Replace \(1 - \cos^2 \theta\) in the numerator with \(\sin^2 \theta\), resulting in the expression \( \frac{\sin^2 \theta}{\sin^2 \theta} \).

Step 3: Simplify the Expression

Divide \(\sin^2 \theta\) by \(\sin^2 \theta\), which simplifies to 1 since any non-zero number divided by itself equals 1. Thus, the expression simplifies to 1.

Key Concepts

Algebra Simplification

Algebraic simplification is all about making expressions easier to work with. It involves reducing the complexity of a mathematical expression without changing its value or its truth. In the context of trigonometric identities, simplification helps unravel equations to reveal simpler forms, often leading to an easier understanding and solution.

• Look for common factors that can be factored out.
• Use basic mathematical operations like addition, subtraction, division, and multiplication.
• Identify known identities or formulas that can replace complex parts.

Simplification can include recognizing patterns, such as knowing that adding zero or multiplying by one doesn't change the value of an expression. It helps tackle complex trigonometric expressions by breaking them down into simpler parts, setting the stage for substitution with identities like those found in trigonometry.

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the square of the sine function to the square of the cosine function. This identity is derived from the Pythagorean Theorem in geometry and is expressed as:\[sin^2 \theta + \cos^2 \theta = 1\]This equation is a powerful tool in simplifying trigonometric expressions. For example, if you have an expression that contains \(1 - \cos^2 \theta\), you can use this identity to replace it with \(\sin^2 \theta\).

• This identity is used when simplifying expressions or solving trigonometric equations.
• It helps in converting cosine expressions into sine expressions and vice versa.
• Memorizing the identity allows for quick recognition and substitution, resulting in cleaner and more easily manipulated expressions.

By employing the Pythagorean identity, tricky trigonometric expressions can often be simplified significantly, as shown in the original exercise.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between angles and sides of triangles, particularly right-angled triangles. It introduces functions like sine, cosine, and tangent, which help describe the angles' ratios to the sides of the triangle.Each trigonometric function can be associated with angles and the unit circle. For example:

• The sine of an angle \(\theta\) is the ratio of the length of the side opposite the angle to the hypotenuse in a right-angled triangle.
• The cosine of an angle \(\theta\) is the ratio of the length of the adjacent side to the hypotenuse.
• Trigonometry also extends beyond angles and triangles to model periodic phenomena, waves, and more.

The beauty of trigonometry lies in its wide range of applications, from calculating heights and distances to providing insights in physics and engineering. By using trigonometric identities such as the Pythagorean Identity, trigonometry simplifies complex equations, as illustrated in the exercise where expressions were reduced to simpler forms.