Problem 52 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\sin \theta \cot \theta=\cos \theta$$

Step-by-Step Solution

Answer

Identity verified: \( \sin \theta \cot \theta = \cos \theta \).

Step 1: Understanding the Problem

We need to verify that the left-hand side of the equation, \( \sin \theta \cot \theta \), is equivalent to the right-hand side, \( \cos \theta \). This means transforming the expression on the left to match the right side.

Step 2: Substitute Cotangent

Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Substitute this into the equation on the left side: \( \sin \theta \cot \theta = \sin \theta \times \frac{\cos \theta}{\sin \theta} \).

Step 3: Simplify the Expression

Now simplify \( \sin \theta \times \frac{\cos \theta}{\sin \theta} \). The \( \sin \theta \) terms will cancel each other: \( \frac{\sin \theta \cdot \cos \theta}{\sin \theta} = \cos \theta \).

Step 4: Compare Both Sides

After simplifying, we get \( \cos \theta \), which is the same as the right-hand side of the original equation. Both sides are equal, verifying the identity.

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational in understanding various aspects of mathematics, especially when dealing with angles and periodic phenomena. The primary trigonometric functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). These functions are based on the ratios of sides of a right-angled triangle.

• Sine (\( \sin \theta \)) represents the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)) is the ratio of the opposite side to the adjacent side.

Another vital function is the cotangent (\( \cot \theta \)), which is not among the primary three but is quite essential. It can be expressed as the reciprocal of the tangent, which means \( \cot \theta = \frac{1}{\tan \theta} \), or alternatively, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Understanding these functions allows us to delve into angles and calculate transformations and identities accurately.

Cotangent

The cotangent function is one of the six main trigonometric functions and is often utilized in various mathematical transformations and proofs. As previously mentioned, it is defined as the ratio of the adjacent side to the opposite side in a right triangle.
It can also be expressed in terms of sine and cosine, an identity that is particularly useful in solving and simplifying trigonometric expressions. The identity is given as:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]This expression is exceptionally beneficial when simplifying trigonometric equations or verifying identities, as it allows for substitutions that lead to simpler forms. Using this identity, expressions involving cotangent can be transformed to contain only sine and cosine, which are often more straightforward to handle.

Simplifying Expressions

Simplifying expressions is a crucial aspect of mathematics that helps in reducing complexity and in solving equations efficiently. In the context of trigonometric identities, simplifying involves using known identities and operations like canceling terms.
When we simplify the expression \( \sin \theta \cot \theta \), the first step involves substitution based on known identities.

• Substitute \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) into the expression.
• Multiply: \( \sin \theta \times \frac{\cos \theta}{\sin \theta} \).
• Simplify by canceling the \( \sin \theta \) terms: resulting in \( \cos \theta \).

These steps illustrate how using trigonometric identities can transform a complex expression into a simpler form, thereby verifying an equation like \( \sin \theta \cot \theta = \cos \theta \). Simplification not only eases the solving process but also provides deeper insights into the relationship between different trigonometric functions.