Problem 87 - Algebra 2

Question

Simplify each expression. $$ \frac{\sin \theta+\tan \theta}{1+\cos \theta} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{\sin \theta(2 + \cos \theta)}{\cos \theta}\)

Step 1: Express Tangent in terms of Sine and Cosine

Start by replacing the tangent function with its equivalent in terms of sine and cosine functions. The expression becomes: \(\frac{\sin \theta +\frac{\sin \theta}{\cos \theta}}{1+\cos \theta} \)

Step 2: Distribute the denominator across the terms in the numerator

Distribute \(1+\cos \theta\) over \(\sin \theta\) and \(\frac{\sin \theta}{\cos \theta}\) in the numerator using 'distribute the denominator' method. This step transforms the expression into: \(\frac{\sin \theta (1+\cos \theta) + \sin \theta}{\cos \theta}\). The denominator of the second fraction in the numerator and the denominator of the whole expression cancel out.

Step 3: Simplify the numerator

Simplify the numerator of the expression, which results in: \(\frac{\sin \theta+ \sin \theta \cos \theta + \sin \theta}{\cos \theta}\). Then, combine like terms, which simplifies the expression to: \(\frac{2 \sin \theta + \sin \theta \cos \theta}{\cos \theta}\).

Step 4: Factor Out the common factor

Factor out \( \sin \theta \) from the numerator, the expression simplifies to: \(\frac{\sin \theta(2 + \cos \theta)}{\cos \theta}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics. It makes complex problems easier to understand and solve. In many cases, simplifying involves combining like terms or reducing fractions.

The purpose of simplifying expressions is to make them more manageable. Here, you need to break down the expression with multiple terms into simpler components.

Consider the expression: \( \frac{\sin \theta + \tan \theta}{1 + \cos \theta} \). The first step is replacing variables or functions with their equivalent forms. This is crucial for revealing similar terms that can be combined or canceled out.

Another key part of simplification involves factoring common factors. This reduces the overall complexity of the expression and ensures that it is expressed in its simplest form. By simplifying expressions, you can more easily perform further calculations or comparisons.

Sine and Cosine

Sine and cosine are key trigonometric functions used to model periodic phenomena. They are defined using a right triangle or the unit circle, and they are intimately connected.

The sine of an angle \( \theta \) is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle. Mathematically, it's expressed as \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

Cosine, on the other hand, is the ratio of the adjacent side to the hypotenuse. It is expressed as \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).

These functions can be transformed and manipulated through various identities. For example, cosine can be employed to simplify expressions involving sine by using known identities. Being able to switch between these forms is crucial for simplifying trigonometric expressions such as the given problem.

Tangent Function

The tangent function is another fundamental trigonometric function, and it is particularly significant in the context of angles and triangles.

The tangent of an angle \( \theta \) is defined as the ratio of the sine of the angle to the cosine of that angle: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This relationship helps in transformations where tangent is encountered in terms of sine and cosine.

Understanding the tangent function is essential because it represents the slope of the line in trigonometric graphs. It describes how steep or flat an angle is in trigonometry.

Through simplification, one can convert tangent into its equivalent sine and cosine terms, as seen in the initial steps of the problem. This conversion can often lead to easier manipulation and simplification of expressions involving multiple trigonometric relationships.