Problem 16 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sin \theta \csc \theta $$

Step-by-Step Solution

Answer

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

Step 1: Utilize trigonometric identity

Write the trigonometric expression \(\sin \theta \csc \theta\) by replacing \(\csc \theta\) with its reciprocal identity, which is \(1/\sin \theta\). So, the expression becomes \(\sin \theta * \frac{1}{\sin \theta}\).

Step 2: Simplify the expression

Notice how \(\sin \theta\) is multiplied by its reciprocal \(1/\sin \theta\). Just like multiplying a number with its reciprocal gives 1, the result of this operation also comes out to be 1. Hence, \(\sin \theta * \frac{1}{\sin \theta}\) simplifies to 1.

Key Concepts

Reciprocal Identity

The concept of reciprocal identities in trigonometry is crucial for simplifying expressions. It's almost like the idea of flipping a fraction to find its "opposite" partner. For trigonometric functions, each function has a reciprocal counterpart. These counterparts allow us to break down and simplify complex expressions with greater ease.

Reciprocal identities play a fundamental role in evaluating trigonometric functions:

• The reciprocal of the sine function \(\sin \theta\) is the cosecant function \(\csc \theta = \frac{1}{\sin \theta}\).
• Similarly, the reciprocal of cosine \(\cos \theta\) is secant \(\sec \theta = \frac{1}{\cos \theta}\).
• The reciprocal of tangent \(\tan \theta\) is cotangent \(\cot \theta = \frac{1}{\tan \theta}\).

Whenever you see a trigonometric expression, it's helpful to think about its reciprocal identity. This can often lead you directly to a simplified form, as is the case with the expression \(\sin \theta \csc \theta\). Using the reciprocal identity of cosecant, it transforms to a simpler expression of 1. Understanding this gives you a powerful tool for tackling complex trigonometric problems.

Sine and Cosecant Relationship

The relationship between sine and cosecant is direct and straightforward due to their reciprocal nature. The sine function, \(\sin \theta\), and the cosecant function, \(\csc \theta\), are closely tied together - think of them as two sides of the same coin. This means they can directly influence each other in an expression.

Here's how it works:

• Since \(\csc \theta = \frac{1}{\sin \theta}\), multiplying \(\sin \theta\) by \(\csc \theta\) results in canceling out the sine function.
• The expression \(\sin \theta \times \csc \theta\) simplifies directly to 1. This happens because \(\sin \theta \) and \(\frac{1}{\sin \theta}\) are reciprocals and cancel each other out.

This reciprocal relationship makes it easy to simplify expressions that otherwise might look complicated.

Trigonometric Identities

Trigonometric identities are equations that are always true for any angle. They are like the rules and shortcuts of trigonometry, allowing us to simplify and evaluate expressions efficiently. These identities are incredibly useful and form the backbone of problem-solving in trigonometry.

Some of the most common trigonometric identities include:

• Pythagorean identities, such as \(\sin^2 \theta + \cos^2 \theta = 1\).
• Angle sum and difference identities, which help in calculating the sine, cosine, and tangent of an angle sum or difference.
• Reciprocal identities, which we used in the given problem, like \(\csc \theta = \frac{1}{\sin \theta}\).

By applying these identities, trigonometric expressions can often be reduced to much simpler forms. For example, when faced with an expression containing both \(\sin \theta\) and \(\csc \theta\), knowing the reciprocal identity allows you to quickly simplify it to 1. This not only makes calculations quicker but also enhances understanding of the relationships between different trigonometric functions.