Problem 15 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sin \theta \cot \theta $$

Step-by-Step Solution

Answer

The simplified form of \( \sin \theta \cot \theta \) is \( \cos \theta \).

Step 1: Understanding the relationship between trigonometric functions

Firstly, the cotangent is the reciprocal of the tangent, which is sin divided by cos. Therefore, the cotangent of theta (\( \cot \theta \)) can be expressed as cos theta divided by sin theta (\( \frac{\cos \theta}{\sin \theta} \)).

Step 2: Substituting cot with cos and sin

Replace \( \cot \theta \) with \( \frac{\cos \theta}{\sin \theta} \) in the given expression. After substituting, the given expression becomes:\( \sin \theta \times \frac{\cos \theta}{\sin \theta} \).

Step 3: Simplifying the expression

The sin theta in the numerator and the sin theta in the denominator will cancel each other out. This leaves us with cos theta as the simplified expression.

Key Concepts

Trigonometric Functions

Trigonometric functions are a crucial aspect of mathematics, especially in geometry and calculus. They help us understand the relationships between the angles and sides of triangles. The most common trigonometric functions are sine (\(\sin \theta\)), cosine (\(\cos \theta\)), and tangent (\(\tan \theta\)). These functions are used to calculate unknown angles and sides in right-angled triangles.

Each function has a specific definition based on a right triangle, where \(\theta\) is one of the angles:

• Sine (\(\sin\)): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Cosine (\(\cos\)): The ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(\tan\)): The ratio of the length of the opposite side to the adjacent side.

These functions are not only essential in solving and simplifying trigonometric expressions but also in various scientific fields such as physics and engineering.

Cotangent

Cotangent (\(\cot\)) is another trigonometric function less commonly encountered compared to sine and cosine but is quite useful nonetheless. It is defined as the reciprocal of the tangent function:\(\cot \theta = \frac{1}{\tan \theta}\). This means that cotangent is the ratio of the cosine of an angle to the sine of that angle:\(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

This relationship makes the cotangent extremely useful in mathematical expressions and problem-solving, particularly when you need to simplify expressions that involve multiple trigonometric functions. In the given problem, using the identity for cotangent allows us to break down the expression and simplify it by eliminating common factors.

Reciprocal Identities

Reciprocal identities are relationships in trigonometry where one function is the inverse, or reciprocal, of another. Understanding these identities is key to simplifying complex trigonometric expressions. For example:

• Cosecant (\(\csc\)): The reciprocal of sine, \(\csc \theta = \frac{1}{\sin \theta}\).
• Secant (\(\sec\)): The reciprocal of cosine, \(\sec \theta = \frac{1}{\cos \theta}\).
• Cotangent (\(\cot\)): As mentioned, the reciprocal of tangent, \(\cot \theta = \frac{1}{\tan \theta}\).

These reciprocal relationships can simplify trigonometric expressions by transforming complex ratios into straightforward terms. They are vital tools in calculus and other advanced mathematics courses where trigonometric simplifications are frequently required.

Cosine Function

The cosine function (\(\cos \theta\)) is one of the primary trigonometric functions, playing a central role in analyzing and simplifying expressions. Cosine defines the intrinsic relationship between an angle in a right triangle and the side lengths.

In the context of our exercise, cosine emerges as the simplified result of a more complex expression involving the sine and cotangent. This illustrates how fundamental trigonometric identities and simplifications can reveal the underlying simplicity of what initially appears to be intricate expressions.

Because cosine relates directly to a right triangle's adjacent side over hypotenuse ratio, it offers practical advantages in deriving angles and calculations involved in periodic functions, physics, and engineering applications. When simplifying expressions, achieving a final result in terms of sine or cosine is often the goal for clearer interpretation and application.