Problem 6 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Find \(\csc \theta\) if \(\cot \theta=\frac{\sqrt{5}}{2}\) and \(0 \leq \theta \leq \frac{\pi}{2}\).

Step-by-Step Solution

Answer

The value of \(\csc \theta\) is \(\frac{3}{2}\).

Step 1: Understand the Given Values

Recognize that \(\cot \theta\) is given as \(\frac{\sqrt{5}}{2}\) and that \(0 \leq \theta \leq \frac{\pi}{2}\). This means \(\theta\) is in the first quadrant where all trigonometric functions are positive.

Step 2: Express \(\cot \theta\) in Terms of \(\sin \theta\) and \(\cos \theta\)

Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Substituting the given value, we have \(\frac{\cos \theta}{\sin \theta} = \frac{\sqrt{5}}{2}\). So, \(\cos \theta = \frac{\sqrt{5}}{2} \sin \theta\).

Step 3: Use the Pythagorean Identity

The Pythagorean identity states \(\sin^2 \theta + \cos^2 \theta = 1\). Substituting \(\cos \theta = \frac{\sqrt{5}}{2} \sin \theta\), we get \(\sin^2 \theta + \left( \frac{\sqrt{5}}{2} \sin \theta \right)^2 = 1\).

Step 4: Simplify the Equation

Simplify the equation to \(\sin^2 \theta + \frac{5}{4} \sin^2 \theta = 1\). Combine like terms to get \((1 + \frac{5}{4}) \sin^2 \theta = 1\) which simplifies to \(\frac{9}{4} \sin^2 \theta = 1\).

Step 5: Solve for \(\sin \theta\)

Solve for \(\sin \theta\) by isolating \(\sin^2 \theta\), giving \(\sin^2 \theta = \frac{4}{9}\). Thus, \(\sin \theta = \frac{2}{3}\) since \(0 \leq \theta \leq \frac{\pi}{2}\).

Step 6: Find \(\csc \theta\)

Recall that \(\csc \theta = \frac{1}{\sin \theta}\). Substitute \(\sin \theta = \frac{2}{3}\) into the equation, yielding \(\csc \theta = \frac{1}{\frac{2}{3}} = \frac{3}{2}\).

Key Concepts

cotangent

The cotangent function, denoted as \(\romptheta\) or \(\text{cot} \theta\), is one of the six fundamental trigonometric functions. Cotangent is defined as the ratio of the adjacent side to the opposite side in a right triangle. Mathematically, this can be written as:
\(\text{cot} \theta = \frac{\cos \theta}{\sin \theta} \).
Since it is derived from the cosine and sine functions, it's important to understand these relationships fully. In our exercise:

• We begin with the value of \(\text{cot} \theta = \frac{\root{5}}{2} \).

Substituting this into its trigonometric identity, we need to find how it relates to sine and cosine.

Pythagorean identity

The Pythagorean identity is a fundamental relationship in trigonometry, expressing the relationship between the square of sine and cosine. It is written as:
\(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\).
This identity holds for all angles and is foundational for solving many trigonometric problems.
In our case, we use the given value of \(\text{cot} \theta \theta = \frac{\root{5}}{2} \) and the definition from the first section to substitute \(\text{cos}\theta\). Then the identity becomes:

• \(\text{sin}^2 \theta + \frac{5}{4}\text{sin}^2 \theta = 1 \)

. Simplifying this, we isolate \(\text{sin}\theta\). This relationship is important because it allows us to discover the value of one trigonometric function when we know another.

cosecant

Cosecant, written as \(\text{csc} \theta \), is the reciprocal of the sine function. It is defined as:
\(\text{csc} \theta = \frac{1}{\text{sin} \theta} \).
This function is useful because it helps to find the length of the hypotenuse relative to the opposite side in a right triangle. When we know \(\text{sin} \theta\), we can easily find \(\text{csc} \theta\) by taking the reciprocal.

• In our example, once we found \(\text{sin}\theta = \frac{2}{3}\).
• Therefore, \(\text{csc}\theta = \frac{3}{2}\).

This reciprocal relationship is a general way to switch between sine and cosecant and can be performed straightforwardly with any angle where the sine is known.