Problem 21 - Basic Technical Mathematics with Calculus

Question

Find the values of the indicated functions. In Exercises \(17-20,\) give answers in exact form. In Exercises \(21-24,\) the values are approximate. Given \(\sin \theta=0.750,\) find \(\cot \theta\) and \(\csc \theta\).

Step-by-Step Solution

Answer

\(\cot \theta \approx 0.882\) and \(\csc \theta = \frac{4}{3}.\)

Step 1: Understanding the Problem

We are provided with \(\sin \theta = 0.750\) and need to find the exact values of \(\cot \theta\) and \(\csc \theta\).

Step 2: Calculate \(\csc \theta\)

The cosecant function, \(\csc \theta\), is the reciprocal of the sine function. Therefore, \(\csc \theta = \frac{1}{\sin \theta}\). With \(\sin \theta = 0.750\), we have: \[\csc \theta = \frac{1}{0.750} = \frac{4}{3}.\]

Step 3: Determine \(\cos \theta\) Using Pythagorean Identity

Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can find \(\cos \theta\). \(\sin^2 \theta = 0.750^2 = 0.5625\). Hence, \(\cos^2 \theta = 1 - 0.5625 = 0.4375\).Then, \(\cos \theta = \sqrt{0.4375}\).

Step 4: Find \(\cot \theta\)

The cotangent function, \(\cot \theta\), is the ratio of \(\cos \theta\) to \(\sin \theta\). Therefore, \[\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\sqrt{0.4375}}{0.750}.\]Simplifying, \(\cot \theta = \frac{0.6614}{0.750} \approx 0.8818.\)

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often represented as \( \sin \theta \). It measures the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse in a right-angled triangle. This means:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

In trigonometric terms, the sine function helps us find relationships between angles and sides of triangles. For angles beyond the typical triangle scenario, the sine function still holds meaning within the unit circle. In this application, the range of the sine function is from \(-1\) to \(+1\).
When given \( \sin \theta = 0.750 \), it implies that the side opposite angle \( \theta \) is 0.75 times the length of the hypotenuse. Understanding this ratio helps in finding other trigonometric functions, which stems from knowing all you need is one value – and, in the context of the unit circle, a lot becomes apparent about the coordinate where the terminal side of angle \( \theta \) lands.

Cosecant Function

The cosecant function, symbolized as \( \csc \theta \), is the reciprocal of the sine function. Therefore, the formula for calculating the cosecant from the sine is:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This makes \( \csc \theta \) a statement about the relationship between the hypotenuse and the side opposite \( \theta \), effectively reversing the ratio of the sine function.
For example, with \( \sin \theta = 0.750 \), you can compute \( \csc \theta \) as:

• \( \csc \theta = \frac{1}{0.750} = \frac{4}{3} \)

This exact value expresses how many times the side opposite \( \theta \) fits along the hypotenuse, flipped from the context of sine.

Cotangent Function

The cotangent function, represented as \( \cot \theta \), is another trigonometric function which can be defined as the reciprocal of the tangent function, or alternatively, as the ratio of the cosine to the sine of an angle:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

With \( \sin \theta \) given as 0.750 and the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we find \( \cos^2 \theta = 0.4375 \). Taking the square root gives us \( \cos \theta \). Thus:

• \( \cos \theta = \sqrt{0.4375} \approx 0.6614 \)

Finally, the cotangent is:

• \( \cot \theta = \frac{0.6614}{0.750} \approx 0.8818 \)

This shows how \( \cot \theta \) reflects the balance between these two sides: the adjacent over the opposite (from \( \sin \theta \)), fully defining the trigonometric relationships for \( \theta \).