Problem 52 - Algebra and Trigonometry with Analytic Geometry

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: Rewrite Cotangent in Terms of Sine and Cosine

Start by expressing the cotangent function in terms of sine and cosine. The cotangent of an angle \( \theta \) is given by the formula:\[\cot \theta = \frac{\cos \theta}{\sin \theta}\]

Step 2: Substitute the Expression for Cotangent

Replace \( \cot \theta \) in the original equation with \( \frac{\cos \theta}{\sin \theta} \). The left-hand side of the identity becomes:\[\sin \theta \cdot \frac{\cos \theta}{\sin \theta}\]

Step 3: Simplify the Expression

Observe that the sine function \( \sin \theta \) in the numerator and the denominator cancel each other out, simplifying the expression to:\[\cos \theta\]

Step 4: Confirm the Identity

Having simplified the left-hand side to \( \cos \theta \), we see that it matches the right-hand side of the given identity. Thus, the identity is verified.

Key Concepts

Cotangent

The cotangent, often abbreviated as "cot," is one of the six primary trigonometric functions, used widely in different scenarios in mathematics, especially in dealing with angles and periodic functions. It’s defined specifically as the reciprocal of the tangent function. In mathematical terms, for an angle \( \theta \), the cotangent is given by:\[ \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \]Understanding this identity is crucial, as it helps us rewrite expressions more conveniently using sine and cosine.

• Reciprocal Nature: Because cotangent is the reciprocal of the tangent, whenever you see \( \cot \theta \), you can replace it with \( \frac{\cos \theta}{\sin \theta} \).
• Applications in Simplification and Proofs: This form is incredibly useful in simplifying expressions and verifying identities, as seen in the exercise above.
• Periodicity: Like sine and cosine, the cotangent function is periodic, recurring its values over a cycle.

Understanding cotangent as part of the basic trigonometric identities gives you a strong foundation to tackle complex trigonometric expressions.
It's also essential when solving triangles, evaluating limits, and analyzing periodic functions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a key skill in mathematics that involves transforming them into their simplest form. This process often makes complex problems much more manageable. Let’s illustrate this concept using our example:

• Start by Rewriting Functions: Begin simplification by expressing trigonometric functions in terms of sine and cosine. For example, \( \cot \theta \) can be rewritten as \( \frac{\cos \theta}{\sin \theta} \). This was done in Step 1 of our exercise and is often the first step in simplification.
• Cancel Common Factors: Identify and cancel out common factors in the numerator and denominator to simplify the expression, which was seen when \( \sin \theta \) canceled itself out in the example.
• Basic Identities: Using basic identities can help further simplify expressions. For example, the Pythagorean identities can be used where necessary.

By mastering how to simplify trigonometric expressions, you allow complex problems to become much simpler, paving the way for easier solutions and verifications of trigonometric identities.

Verifying Trigonometric Identities

Verifying trigonometric identities involves proving that two different trigonometric expressions are equivalent for any angle where the expressions are defined. This process relies on transforming one side of the equation to look like the other side using trigonometric identities and algebraic techniques.

• Understanding the Identity to Verify: In our exercise, the identity \( \sin \theta \cot \theta = \cos \theta \) needs verification. The key is to break down \( \cot \theta \) into sine and cosine, as shown in the step-by-step solution.
• Logical Transformation: This involves manipulation such as factoring, adding, subtracting, or substituting using known identities (like \( \sin \theta \cdot \frac{\cos \theta}{\sin \theta} = \cos \theta \) after cancellation), to convert one side to the other.
• Verify:“Matching Both Sides: The ultimate goal is for transformations on one side to match exactly with the other, confirming the identity is true, as demonstrated in the fourth step of our solution.

The verification process reinforces understanding of how trigonometric functions relate to one another. It also enhances problem-solving skills through logical reasoning and manipulation of expressions.