Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\sin \theta, \text { given that } \csc \theta=\sqrt{2}$$

Step-by-Step Solution

Answer

\( \sin \theta = \frac{\sqrt{2}}{2} \)

Step 1: Understand the Reciprocal Identity

Recall that the cosecant function is the reciprocal of the sine function. This means if \( \csc \theta = \sqrt{2} \), then \( \sin \theta = \frac{1}{\csc \theta} \).

Step 2: Substitute the Given Value

We are given that \( \csc \theta = \sqrt{2} \). Substituting this into the reciprocal identity, we have: \( \sin \theta = \frac{1}{\sqrt{2}} \).

Step 3: Rationalize the Denominator

To rationalize the denominator of \( \frac{1}{\sqrt{2}} \), multiply the numerator and the denominator by \( \sqrt{2} \). This gives us:\[\sin \theta = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}\]

Step 4: Final Expression for \(\sin \theta\)

The rationalized and simplified value for \( \sin \theta \) is \( \frac{\sqrt{2}}{2} \).

Key Concepts

Reciprocal Identities

Reciprocal identities are a key concept in trigonometry that simplify the relationship between various trigonometric functions. In essence, a reciprocal identity explains how one trigonometric function is the "inverse" of another. For example, the reciprocal identity for sine and cosecant states that cosecant is the reciprocal of sine.

• If you know one function value, you can easily find the other by taking the reciprocal. In mathematical terms, it means that if you have a value for \( \csc \theta \), the equivalent \( \sin \theta \) can be found using \( \sin \theta = \frac{1}{\csc \theta} \).
• This relationship is useful because it allows you to switch between functions, especially when only one of them is given.

It is crucial to remember how these relationships work since they form the backbone of solving many trigonometric equations. For example, given \( \csc \theta = \sqrt{2} \), using the reciprocal identity directly provides \( \sin \theta = \frac{1}{\sqrt{2}} \). Understanding reciprocal identities not only simplifies calculations but also deepens your comprehension of trigonometric functions.

Rationalizing the Denominator

Rationalizing the denominator is a mathematical process used to eliminate radicals from the bottom part of a fraction. In trigonometry, we often find expressions involving square roots in the denominators, and simplifying these is essential for cleaner expressions and easier further calculations.

• When presented with a fraction like \( \frac{1}{\sqrt{2}} \), it is often preferred to express it without a radical in the denominator.
• To do this, multiply both the numerator and the denominator by the radical you wish to remove. In this case, multiply by \( \sqrt{2} \) to obtain \( \frac{\sqrt{2}}{2} \).

Rationalizing the denominator makes the expression \( \frac{\sqrt{2}}{2} \) more standard and easier to interpret, especially in harmonic and wave-related trigonometric applications. This practice also facilitates checking and verifying results, as most trigonometric tables and software tools display output in rationalized forms.

Cosecant and Sine Relationship

In trigonometry, understanding the relationship between different trigonometric functions is vital. The sine and cosecant relationship is one such example, where the two functions are directly dependent on one another through a reciprocal connection.

• Mathematically, \( \csc \theta = \frac{1}{\sin \theta} \) and conversely, \( \sin \theta = \frac{1}{\csc \theta} \). This means if you know one, you can find the other easily.
• So, when you are given that \( \csc \theta = \sqrt{2} \), you can find \( \sin \theta \) by computing \( \sin \theta = \frac{1}{\sqrt{2}} \), and then rationalizing this gives \( \sin \theta = \frac{\sqrt{2}}{2} \).

Understanding how these functions are interrelated allows for solving many trigonometric problems more efficiently. The sine and cosecant relationship is particularly important in real-world applications like physics and engineering, where angles and their respective function values play a critical role in constructions and calculations.