Problem 26 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sin \theta\left(1+\cot ^{2} \theta\right) $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(1/sin(\theta)\) or \(csc(\theta)\).

Step 1: Identity Recognition

Recognize that cotangent squared is part of a Pythagorean identity. The Pythagorean identity states that \(1 + cot^2(\theta) = csc^2(\theta)\). This is derived from the pythagorean theorem, \(1 = sin^2(\theta) + cos^2(\theta)\), and dividing all parts by \(sin^2(\theta)\).

Step 2: Substitute Cotangent Squared

Substitute \(1 + cot^2(\theta)\) with \(csc^2(\theta)\) according to the Pythagorean identity recognized in Step 1. This results in an expression: \(sin(\theta) * csc^2(\theta)\).

Step 3: Simplify the expression

Recognize that the cosecant function is the reciprocal of the sine function, therefore \(csc(\theta) = 1/sin(\theta)\). Substitute \(csc(\theta)\) with \(1/sin(\theta)\) which simplifies the expression to: \(sin(\theta) * (1/sin^2(\theta))\) or \(1/sin(\theta)\).

Key Concepts

Pythagorean identities

Trigonometric identities are vital tools in simplifying expressions and solving equations. One of the most significant sets of these identities is the Pythagorean identities. These are named after the Pythagorean theorem and they involve the square of trigonometric functions. The primary Pythagorean identity is: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] From this primary identity, we can derive two other related identities by dividing through by either \( \sin^2(\theta) \) or \( \cos^2(\theta) \). When divided by \( \sin^2(\theta) \), we get: \[ 1 + \cot^2(\theta) = \csc^2(\theta) \] This particular identity was used in simplifying the expression \( \sin \theta(1 + \cot^2 \theta) \). Recognizing these relationships helps in exchanging complex trigonometric terms with simpler ones, facilitating smoother problem-solving.

simplifying trigonometric expressions

Simplifying trigonometric expressions often involves using identities to reduce complexity. Let's take the problem of simplifying \( \sin \theta(1 + \cot^2 \theta) \).

• First, identify that \( \cot^2(\theta) \) is involved in a Pythagorean identity.
• Substitute \( 1 + \cot^2(\theta) \) with \( \csc^2(\theta) \) to make the expression simpler.

With the substitution, the expression becomes \( \sin(\theta) \cdot \csc^2(\theta) \). To further simplify, remember the relationship between sine and its reciprocal, cosecant. Finally, this allows the entire expression to be reduced to just \( 1/ \sin(\theta) \). Recognizing identities and reciprocal relationships is key to simplification.

reciprocal trigonometric functions

Reciprocal trigonometric functions are directly related to the primary trigonometric functions. They can be immensely helpful in simplifying expressions. The main reciprocal functions include:

• The cosecant function \( csc(\theta) = \frac{1}{\sin(\theta)} \)
• The secant function \( sec(\theta) = \frac{1}{\cos(\theta)} \)
• The cotangent function \( cot(\theta) = \frac{1}{\tan(\theta)} \)

In the task of simplifying \( \sin \theta(1 + \cot^2 \theta) \), after substituting with Pythagorean identities, it was crucial to use the reciprocal nature of cosecant. By knowing that \( \csc^2(\theta) = \frac{1}{\sin^2(\theta)} \), we effectively reduced the expression step by step. Using reciprocals can often turn more complex expressions into simpler, easier-to-handle forms.