Problem 21 - Precalculus with Calculus Previews

Question

Use the given information to find the values of the remaining five trigonometric functions. $$ \csc x=\frac{4}{3}, \quad 0

Step-by-Step Solution

Answer

\( \sin(x) = \frac{3}{4}, \cos(x) = \frac{\sqrt{7}}{4}, \tan(x) = \frac{3}{\sqrt{7}}, \cot(x) = \frac{\sqrt{7}}{3}, \sec(x) = \frac{4}{\sqrt{7}} \).

Step 1: Understand csc Function

Given \( \csc(x) = \frac{4}{3} \), the cosecant is defined as \( \csc(x) = \frac{1}{\sin(x)} \). Therefore, \( \sin(x) = \frac{3}{4} \).

Step 2: Find Cosine Using Pythagorean Identity

Using the Pythagorean identity, \( \sin^2(x) + \cos^2(x) = 1 \), substitute \( \sin(x) = \frac{3}{4} \):\[\left(\frac{3}{4}\right)^2 + \cos^2(x) = 1\]\[\frac{9}{16} + \cos^2(x) = 1\]\[\cos^2(x) = \frac{7}{16}\]Taking the square root, \( \cos(x) = \frac{\sqrt{7}}{4} \), since \( x \in (0, \frac{\pi}{2}) \), the cosine value is positive.

Step 3: Calculate Remaining Functions Using Basic Definitions

Now that we have \( \sin(x) = \frac{3}{4} \) and \( \cos(x) = \frac{\sqrt{7}}{4} \), calculate the remaining trigonometric functions:- Tangent: \( \tan(x) = \frac{\sin(x)}{\cos(x)} = \frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}} = \frac{3}{\sqrt{7}} \)- Cotangent: \( \cot(x) = \frac{1}{\tan(x)} = \frac{\sqrt{7}}{3} \)- Secant: \( \sec(x) = \frac{1}{\cos(x)} = \frac{4}{\sqrt{7}} \)

Key Concepts

Cosecant Function

The cosecant function, often abbreviated as \( \csc(x) \), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function, which means that \( \csc(x) = \frac{1}{\sin(x)} \). In this exercise, knowing that \( \csc(x) = \frac{4}{3} \) helps us find the value of \( \sin(x) \), which is a key step to unlocking other trigonometric functions. Since \( \csc(x) = \frac{4}{3} \), we can solve for the sine by flipping the fraction to get \( \sin(x) = \frac{3}{4} \). This reflects the basic reciprocal relationship between these two functions. Understanding this relationship is essential because it allows for the transition between the sine and cosecant values, which are crucial for working with right triangles and angles.

Pythagorean Identity

The Pythagorean Identity is a fundamental equation in trigonometry given by \( \sin^2(x) + \cos^2(x) = 1 \). It is derived from the Pythagorean Theorem and applies to any angle \( x \). When we know one trigonometric function, such as sine, this identity allows us to find another, like cosine. In this problem, since \( \sin(x) = \frac{3}{4} \), we use the Pythagorean Identity to find \( \cos(x) \).

• Substitute: \( \left(\frac{3}{4}\right)^2 + \cos^2(x) = 1 \)
• Calculate \( \sin^2(x) \): \( \frac{9}{16} \)
• Solve for \( \cos^2(x) \): \( \cos^2(x) = \frac{7}{16} \)
• Extract \( \cos(x) \): \( \cos(x) = \frac{\sqrt{7}}{4} \), considering the positive square root since \( x \) is in the first quadrant.

Understanding the identity helps solve for unknowns efficiently and is fundamental to mastering trigonometric relationships.

Tangent Function

The tangent function, expressed as \( \tan(x) \), is defined as the ratio of the sine and cosine of an angle. Specifically, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). In our exercise, now that we know \( \sin(x) = \frac{3}{4} \) and \( \cos(x) = \frac{\sqrt{7}}{4} \), we can calculate \( \tan(x) \).

• Substitute sine and cosine values: \( \tan(x) = \frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}} \)
• Simplify the expression: \( \tan(x) = \frac{3}{\sqrt{7}} \)

The tangent function often represents the slope of a line represented by an angle. It is useful in various applications, including determining angles and distances in geometry and physics.

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is the reciprocal of the tangent function, meaning \( \cot(x) = \frac{1}{\tan(x)} \). It can also be expressed as the ratio of the cosine to the sine of an angle, \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). Given our calculations, we found:

• Use the reciprocal definition: \( \cot(x) = \frac{1}{\tan(x)} = \frac{\sqrt{7}}{3} \)
• This can also be checked through \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) using previously calculated values.

The cotangent often provides a measure of the steepness in an angle, much like the inverted concept of the tangent, and is widely used in trigonometry for solving equations and modeling periodic phenomena.

Secant Function

The secant function, indicated by \( \sec(x) \), is the reciprocal of the cosine function, which can be written as \( \sec(x) = \frac{1}{\cos(x)} \). In this exercise:

• Knowing \( \cos(x) = \frac{\sqrt{7}}{4} \), compute \( \sec(x) = \frac{1}{\cos(x)} = \frac{4}{\sqrt{7}} \).

The secant function emerges frequently in problems dealing with angles and distances, providing a consistent method to relate diagonal distances and projections in various fields including engineering and meteorology. Understanding the secant as a reciprocal of the cosine underscores the interconnected nature of trigonometric functions, making it easier to shift perspectives between different trigonometric concepts.