Problem 25 - Algebra 2

Question

Simplify each expression. \(\frac{\sin \theta \csc \theta}{\cot \theta}\)

Step-by-Step Solution

Answer

The simplified expression is \(\tan \theta\).

Step 1: Rewrite Trigonometric Functions in Terms of Sine and Cosine

Start by rewriting the trigonometric functions in the expression \(\frac{\sin \theta \csc \theta}{\cot \theta}\). We have:- \(\csc \theta = \frac{1}{\sin \theta}\)- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)Substituting these into the expression, we get:\[\frac{\sin \theta \left(\frac{1}{\sin \theta}\right)}{\frac{\cos \theta}{\sin \theta}}.\]

Step 2: Simplify the Expression

Next, we simplify the expression:- \(\sin \theta \cdot \frac{1}{\sin \theta} = 1\), so the numerator becomes 1.Thus, the expression simplifies to:\[\frac{1}{\frac{\cos \theta}{\sin \theta}}.\]

Step 3: Convert Division to Multiplication

To further simplify, convert the division of fractions to multiplication by the reciprocal:- \(\frac{1}{\frac{\cos \theta}{\sin \theta}} = 1 \cdot \frac{\sin \theta}{\cos \theta}\)So, the expression becomes:\[\frac{\sin \theta}{\cos \theta}.\]

Step 4: Recognize as a Standard Trigonometric Ratio

Recognize that \(\frac{\sin \theta}{\cos \theta}\) is the definition of the tangent function:Therefore, \(\frac{\sin \theta}{\cos \theta} = \tan \theta\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the involved variables, usually angles. These identities form the backbone of trigonometry, allowing us to simplify and solve trigonometric equations. For example, the Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \), help us understand the fundamental relationships between different trigonometric functions.
They provide tools for transforming and simplifying expressions, which is essential for solving geometry and calculus problems involving angles.
During the process of simplifying \( \frac{\sin \theta \csc \theta}{\cot \theta} \), recognizing the identity \( \csc \theta = \frac{1}{\sin \theta} \) is crucial. This enables us to substitute equivalent forms, leading to further simplifications.

Sine and Cosine

Sine and cosine are the primary trigonometric functions and are deeply interconnected. They are defined as ratios of the sides of a right triangle with respect to a given angle. Specifically:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• \( \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \)

The relationship between sine and cosine is foundational in trigonometry. The identity \( \sin^2 \theta + \cos^2 \theta = 1 \) comes directly from the Pythagorean theorem.
In our expression, \( \frac{\sin \theta}{\cos \theta} \) emerges from simplification steps, upon realizing that the complex fraction reduces to the tangent function.

Tangent Function

The tangent function connects sine and cosine through the simple relationship \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). It is another basic trigonometric function defined as the ratio of the sine and cosine of an angle. In the context of a right triangle:

• \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \)

Understanding this function is critical because it allows us to relate angles to sides in geometric settings.
Recognizing \( \frac{\sin \theta}{\cos \theta} \) as \( \tan \theta \) can often simplify problems by reducing them to a well-known form. This concept is evident in the final step of our expression simplification.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are inverse relationships derived from sine, cosine, and tangent. They add to the "toolbox" used for transformations and solving equations, described as:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

These functions are useful for converting expressions and revealing hidden identities, often simplifying complex trigonometric expressions.
In our particular problem, the use of \( \csc \theta \) and \( \cot \theta \) is pivotal as substituting their known equivalents helps in breaking down and simplifying the given complex expression. This illustrates how reciprocal identities play a significant role in trigonometric simplification.