Problem 53 - Precalculus : Building Concepts and Connections

Question

Find the exact values of the given expressions in radian measure. $$\csc ^{-1}(-2)$$

Step-by-Step Solution

Answer

The exact value of \(\csc ^{-1}(-2)\) in radian measure is \(- \frac{\pi}{6}\).

Step 1: Understanding the problem

The problem asks for the radian measure of the inverse cosecant of -2, written as \(\csc ^{-1}(-2)\). Take into account that the cosecant function is the reciprocal of the sine function, meaning that \(\csc \theta = \frac{1}{\sin \theta}\). Therefore, \(\csc ^{-1}(-2) = \sin ^{-1} \left(\frac{1}{-2}\right)\).

Step 2: Determining the appropriate quadrant

Notice that the inverse sine function \(\sin^{-1}(x)\) is defined for \(x \in [-1,1]\) and returns values in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\). However, because we are working with negative values (\(- \frac{1}{2}\)), the relevant quadrant for this problem is in the fourth quadrant, where the sine function yields negative values.

Step 3: Finding the radian value

When \(\sin \theta = \frac{1}{2}\), the value of \(\theta\) in the first quadrant is \(\theta = \frac{\pi}{6}\). Yet, because \(\sin \theta = - \frac{1}{2}\), the solution will be found in the fourth quadrant. Therefore, the radian measure of the angle is \(\theta = - \frac{\pi}{6}\).

Key Concepts

Radian Measure

Radian measure is a way to express angles using the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians measure angles based on the arc length. This makes calculations in trigonometry and calculus more natural and straightforward.

One full circle (360 degrees) is equal to \(2\pi\) radians. Therefore, half a circle is \(\pi\) radians, and a quarter circle is \(\frac{\pi}{2}\) radians. This form of measurement allows us to directly relate angles to the radius of the circle.

Key benefits of using radians include:

• Simpler formulation of trigonometric identities.
• Easier differentiation and integration in calculus.
• Direct relationship to the arc length of a circle.

Understanding radian measure is fundamental for working with trigonometric expressions, especially when dealing with inverse functions as shown in the problem solution.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions: sine, cosine, and tangent. They include cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). Each reciprocal function is the inverse of one of the primary trigonometric functions.

For instance:

• \(\csc \theta = \frac{1}{\sin \theta}\)
• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\cot \theta = \frac{1}{\tan \theta}\)

Understanding these functions is crucial when working with expressions involving their inverses. It allows you to convert between forms, making it easier to solve trigonometric equations.

In this exercise, we specifically dealt with the inverse cosecant function, \(\csc^{-1}\), which involves finding an angle whose cosecant is a given number. To solve it, converting to an inverse sine function helped determine the angle's radian measure.

Inverse Sine Function

The inverse sine function, denoted as \(\sin^{-1}(x)\) or \(\arcsin(x)\), is used to find the angle whose sine is \(x\). It provides an angle in radians within the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\). This function is especially useful when dealing with angles smaller than \(\pi/2\).

For example, \(\sin^{-1}(-0.5)\) gives us an angle in the fourth quadrant because it needs to satisfy that \(\sin(\theta) = -0.5\). The corresponding angle is \(-\frac{\pi}{6}\), which is the same as \(\sin^{-1}(-\frac{1}{2})\).

Key aspects of the inverse sine function include:

• Its output range is limited, making it predictable.
• Helps in determining angles using known sine values.

This function is vital in solving the original problem, where converting \(\csc^{-1}(-2)\) to \(\sin^{-1}(-\frac{1}{2})\) enabled us to find the correct angle in radians.