Problem 32 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sin \theta \csc \theta-\cos ^{2} \theta $$

Step-by-Step Solution

Answer

So, the simplified form of the given expression is \(\sin^2(theta)\).

Step 1: Apply the Reciprocal Identity

The definition of \(\csc(\theta)\) is \(1/\sin(\theta)\). So multiply \(\sin(\theta)\) with \(\csc(\theta) = 1/\sin(\theta)\). This gives us 1. So our expression simplifies to \(1 - \cos^2(\theta)\).

Step 2: Apply the Pythagorean Identity

The Pythagorean identity is \(\sin^2(\theta) + \cos^2(\theta) = 1\). You can solve this equation for \(\sin^2(\theta)\): \(\sin^2(\theta) = 1 - \cos^2(\theta)\). Replace \(1 - \cos^2(\theta)\) in our expression from step 1, with \(\sin^2(\theta)\).

Key Concepts

Reciprocal Identity

The reciprocal identities in trigonometry are fundamental concepts that relate each trigonometric function to its reciprocal counterpart. These identities help simplify expressions and solve trigonometric equations.For example, the reciprocal identity for cosecant is given by \[ \csc(\theta) = \frac{1}{\sin(\theta)} \]This means that cosecant is the reciprocal or the inverse of sine. In the context of the problem, this identity lets us simplify the product \[ \sin(\theta) \times \csc(\theta) = \sin(\theta) \times \frac{1}{\sin(\theta)} = 1 \] This is because multiplying sine by its reciprocal results in 1. The reciprocal identities are handy tools for simplifying complex trigonometric expressions, as they allow you to cancel out terms efficiently.

Pythagorean Identity

The Pythagorean identities are central relationships in trigonometry that express critical connections between sine, cosine, and the number 1.The most essential Pythagorean identity is \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] This identity emerges from the Pythagorean theorem, reflecting the inherent relationship between the sides of a right triangle.For this exercise, you can rearrange the identity to solve for the sine squared: \[ \sin^2(\theta) = 1 - \cos^2(\theta) \] By identifying that \(1 - \cos^2(\theta)\) is equal to \(\sin^2(\theta)\), you can further simplify the expression resulting from the first step of the problem and deepen your understanding of trigonometric expressions. Thus, recognizing this identity is crucial when simplifying trigonometric expressions or solving equations involving these functions.

Expression Simplification

Simplifying trigonometric expressions involves rewriting them in a simpler and more concise form using various identities and algebraic rules.By breaking down the original complex expression into manageable parts, you can make comprehension and computations easier.The process typically involves:

• Identifying familiar identities or algebraic structures within the expression.
• Applying relevant trigonometric identities, such as reciprocal or Pythagorean identities.
• Rewriting the expression by substituting equivalent forms, like replacing \(1 - \cos^2(\theta)\) with \(\sin^2(\theta)\).

In this particular exercise, the initial step used the reciprocal identity to condense the product into the number 1. Subsequently, the Pythagorean identity provided a pathway to express the remaining component in a simpler form—ultimately simplifying the task to its core.Through such techniques, expression simplification not only reduces complexity but also enhances clarity and usability in mathematical problem-solving.