Problem 2 - Algebra 2

Question

Verify each identity. $$ \sin \theta \cot \theta=\cos \theta $$

Step-by-Step Solution

Answer

Verified: the identity \( \sin \theta \cot \theta = \cos \theta \) is correct.

Step 1: Identify known trigonometric identities

Recall the definition of the cotangent function: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This will help to simplify the left-hand side of our equation.

Step 2: Substitute the definition of cotangent into the equation

Substituting the identity into our equation gives: \( \sin \theta * \frac{\cos \theta}{\sin \theta} \). This simplifies to \( \cos \theta \), which completes the verification.

Key Concepts

Cotangent Function

The cotangent function (\( \cot \theta \)) is a fundamental trigonometric function. It is defined as the reciprocal of the tangent function. In mathematical terms, this means:
- \( \cot \theta = \frac{1}{\tan \theta} \)Another way to represent the cotangent is by using sine and cosine:
- \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)This alternative definition is particularly helpful when simplifying trigonometric expressions. When you need to verify identities or solve equations, substituting \( \cot \theta \) using \( \frac{\cos \theta}{\sin \theta} \) can often make the process easier. It's essential for students to be comfortable switching between these representations for effective problem solving.
The cotangent function, like other trigonometric functions, is periodic. It repeats its values in regular intervals. For \( \cot \theta \), this period is \( \pi \) (180 degrees).

Trigonometric Functions

Trigonometric functions are at the heart of trigonometry. They describe relationships between the angles and sides of triangles. These functions are also essential for modeling periodic phenomena. The most common trigonometric functions are:

• Sine (\(\sin \theta\))
• Cosine (\(\cos \theta\))
• Tangent (\(\tan \theta\))
• Cotangent (\(\cot \theta\))
• Secant (\(\sec \theta\))
• Cosecant (\(\csc \theta\))

Each has a unique definition using ratios of the sides of a right triangle.
- \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)- \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)- \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)Understanding these definitions makes it easier to explore complex identities and functions. Practicing with these functions strengthens your problem-solving skills, crucial for advanced mathematics, calculus, and physics.
These functions help with verifying identities, solving equations, and modeling wave patterns, among other uses.

Verifying Identities

Verifying trigonometric identities is an important skill in mathematics. It involves proving that two different-looking expressions are actually equivalent. This can often simplify more complex problems.
The verification process starts with using known identities. These are standard equations in trigonometry that relate two functions. For instance, the identity:\( \cot \theta = \frac{\cos \theta}{\sin \theta} \)can simplify many expressions. When verifying, you:

• Substitute known identities into the expressions
• Simplify both the left-hand and right-hand sides
• Possibly use algebraic manipulation to demonstrate equivalence

Let's consider an example: verifying the identity \( \sin \theta \cot \theta = \cos \theta \).
Start with the definition \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
Substitute into the equation: \( \sin \theta \times \frac{\cos \theta}{\sin \theta} \).
The \( \sin \theta \) terms cancel each other, leaving:\( \cos \theta \).
Thus, the original identity holds true.Verifying identities gives a deeper understanding of the relationships between trigonometric functions and improves mathematical intuition.