Problem 27 - Technical Mathematics with Calculus

Question

Find \(f\) The sine of an angle \(\theta\) (written \(\sin \theta\) ) is equal to the reciprocal of the cosecant of \(\theta(\csc \theta) .\) Find \(\sin \theta\) if \(\csc \theta=3.58\)

Step-by-Step Solution

Answer

\(\sin(\theta) = \frac{1}{3.58}\approx 0.2793\)

Step 1: Understand Reciprocal Relationship

Recognize that the sine of an angle is the reciprocal of the cosecant of that angle. This means \(\sin(\theta) = \frac{1}{\csc(\theta)}\).

Step 2: Substitute the Given Value of Cosecant

Since it is given that \(\csc(\theta) = 3.58\), substitute this value into the reciprocal relationship to find \(\sin(\theta)\): \(\sin(\theta) = \frac{1}{3.58}\).

Step 3: Calculate Sine

Use a calculator to find the value of \(\sin(\theta)\) by dividing 1 by 3.58.

Key Concepts

Trigonometric Identities

Trigonometric identities are like the very foundation upon which the land of trigonometry is built. They are equations that involve trigonometric functions and are true for every value of the variable within their domain. An understanding of these identities helps in solving various trigonometry-related problems. Commonly used trigonometric identities include Pythagorean identities, angle sum and difference identities, double angle identities, and reciprocal identities.

For our exercise, we're dealing with a reciprocal identity. Specifically, the identity connects the sine function (\textbackslash(sin)) and the cosecant function (\textbackslash(csc)) and is written as \textbackslash(sin(\theta) = \frac{1}{\csc(\theta)}). This means that for any angle \(\theta\), the sine of \(\theta\) is the multiplicative inverse (or reciprocal) of the cosecant of \(\theta\). Simply put, sine and cosecant are like best friends that share everything inversely!

Having this relationship in our toolkit makes it simpler to find the value of one function when given the other, as seen in the step-by-step solution provided for \(\csc(\theta) = 3.58\).

Reciprocal Trigonometric Functions

In trigonometry, the reciprocal functions provide a dual perspective of the primary functions. The three main trigonometric functions have their counterparts: sine (\textbackslash(sin)) and cosecant (\textbackslash(csc)), cosine (\textbackslash(cos)) and secant (\textbackslash(sec)), and tangent (\textbackslash(tan)) and cotangent (\textbackslash(cot)). These reciprocal relationships are essential when it comes to understanding how these functions interrelate.

To be a pro at trigonometry, it helps to be comfortable with these connections. For example, knowing that \(\csc(\theta) = \frac{1}{\sin(\theta)}\) and \(\sin(\theta) = \frac{1}{\csc(\theta)}\) can be truly handy. This theme of mutual reciprocity isn't just a neat party trick – it’s a powerful tool that can simplify complex problems, particularly when we're given one function and asked to find the other, which we seamlessly did for our sine-cosecant problem.

Solving Trigonometric Equations

Working with trigonometric equations can sometimes feel like navigating a maze – but with the right strategies, it becomes an enlightening journey. Whether we’re hunting for angles or function values, our approach focuses on understanding and applying fundamental identities and relationships.

Let's apply this to our original exercise. We knew that \(\csc(\theta) = 3.58\) and were challenged to find \(\sin(\theta)\). In this case, our first step was understanding the reciprocal relationship between sine and cosecant. Once this relationship was clear, we used the known value of \(\csc(\theta)\) to solve for \(\sin(\theta)\).

Using the reciprocal identity, we express \(\sin(\theta)\) as the reciprocal of the given cosecant value, turning the equation into a simple division problem (\(\sin(\theta) = \frac{1}{3.58}\)). Such methods underscore the value of algebraic manipulation and the use of a calculator for precision, leading to successful solutions in trigonometry.