Problem 41 - Algebra 2

Question

REVIEW Which of the following is equivalent to \(\sin \theta+\cot \theta \cos \theta ?\) $$ \begin{array}{l}{\text { F } 2 \sin \theta} \\ {\text { G } \frac{1}{\sin \theta}} \\ {\text { H } \cos ^{2} \theta} \\ {\text { J } \frac{\sin \theta+\cos \theta}{\sin ^{2} \theta}}\end{array} $$

Step-by-Step Solution

Answer

The expression is equivalent to \( \frac{1}{\sin \theta} \) (Option G).

Step 1: Rewrite the expression

Begin with the given expression: \( \sin \theta + \cot \theta \cos \theta \). We know that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), so we substitute this into the expression.

Step 2: Substitute and Simplify

Substitute \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) into the expression: \[\sin \theta + \frac{\cos \theta}{\sin \theta} \cdot \cos \theta\]This simplifies to:\[\sin \theta + \frac{\cos^2 \theta}{\sin \theta}\]

Step 3: Combine Terms

To combine the terms, express \( \sin \theta \) as \( \frac{\sin^2 \theta}{\sin \theta} \). The expression now becomes:\[\frac{\sin^2 \theta}{\sin \theta} + \frac{\cos^2 \theta}{\sin \theta}\]

Step 4: Simplify the combined expression

Combine the fractions:\[\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta}\]Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), the expression simplifies to:\[\frac{1}{\sin \theta}\]

Step 5: Verify with the given options

The simplified expression \( \frac{1}{\sin \theta} \) corresponds to option G in the given choices.Thus, \( \sin \theta + \cot \theta \cos \theta \) is equivalent to \( \frac{1}{\sin \theta} \).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the fundamental identities in trigonometry. It forms the foundation for many trigonometric simplifications and transformations. The most commonly used version of the identity is:

• \( an^2 \theta + 1 = \sec^2 \theta\)
• \( ext{and } 1 + \cot^2 \theta = \csc^2 \theta\)

In the step by step solution of our exercise, the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) was used to transform an expression into a simpler form. By recognizing this identity, we can replace the combination \( \sin^2 \theta + \cos^2 \theta \) with 1, which makes it easier to work with complex equations.
This identity is incredibly useful because it ties together the trigonometric functions \( \sin \theta \) and \( \cos \theta \) in a universal relationship that always holds true. Understanding and knowing when to use the Pythagorean Identity is crucial for solving trigonometric problems efficiently.

Trigonometric Simplification

Trigonometric Simplification involves manipulating trigonometric expressions to make them simpler and easier to solve or evaluate. Simplification can often involve numerous trigonometric identities and algebraic techniques, making it a vital skill in tackling complex trigonometric problems.
In the original exercise, trigonometric simplification was a critical step in solving the problem. We took the given expression \( \sin \theta + \cot \theta \cos \theta \) and used trigonometric identities to express the cotangent function in terms of sine and cosine:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

By substituting this definition into the original expression, we could further simplify it to \( \sin \theta + \frac{\cos^2 \theta}{\sin \theta} \). From there, using the Pythagorean Identity helped us simplify it to a much easier form, \( \frac{1}{\sin \theta} \).
Knowing how to identify and apply proper identities and techniques is essential for efficient problem solving in trigonometry.

Trigonometric Functions

Trigonometric functions, such as sine \(\sin\), cosine \(\cos\), and cotangent \(\cot\), are fundamental in mathematics and are widely used in geometry, physics, engineering, and various fields of science. Each function has unique properties and relationships with other trigonometric functions.
The trigonometric function cotangent is defined as the reciprocal of the tangent, expressed as:

• \(\cot \theta = \frac{1}{\tan \theta}\)
• Or, alternatively, \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

In the step by step solution of the exercise, understanding the property of \(\cot \theta\) was essential to express it in terms of sine and cosine. This allowed us to substitute and simplify expressions. Recognizing these properties and relationships can provide an intuitive understanding that aids in solving complex trigonometric problems.
Mastering trigonometric functions and their relationships is crucial for applying these concepts across numerous practical applications, from analyzing waves and oscillations to solving engineering problems.