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Chapter 6: Problem 34

In Exercises \(25-36\), use a calculator to evaluate each expression. Give the answer in degrees and round to two decimal places. $$ \csc ^{-1}(-6.1324) $$

Short Answer

Expert verified
\(\csc^{-1}(-6.1324) \approx -9.41^\circ\).

Step by step solution

01

Understand the Relationship

Recall that the cosecant function, written as \(\csc(x)\), is the reciprocal of the sine function: \(\csc(x) = \frac{1}{\sin(x)}\). The inverse function \(\csc^{-1}(x)\) gives the angle \(\theta\) such that \(\csc(\theta) = x\).
02

Calculate the Sine Value

Since \(\csc^{-1}(-6.1324) = \theta\), then \(\csc(\theta) = -6.1324\) and \(\sin(\theta) = -\frac{1}{6.1324}\). Calculate this using: \[\sin(\theta) = -0.16318\].
03

Evaluate the Angle

Use a calculator to find \(\theta\) for which \(\sin(\theta) = -0.16318\). Ensure the calculator is set to degree mode. The approximate value is \(\theta \approx -9.41\) degrees.
04

Adjust to the Cosecant Range

The principal value of \(\csc^{-1}(x)\) usually ranges from \(-90^\circ\) to \(0^\circ\) when the value is negative. Since \(-9.41\) degrees is within this range, \(\theta = -9.41\) degrees is the correct principal value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosecant Function
The cosecant function is an interesting part of trigonometry and is denoted as \( \csc(x) \). This function is the reciprocal of the sine function, meaning it is expressed as \( \csc(x) = \frac{1}{\sin(x)} \). This relationship allows us to explore and calculate values of angles based on the reciprocal properties of sine.

Some important things to note about the cosecant function are:
  • It is undefined for values of \( x \) where \( \sin(x) = 0 \), as division by zero is not possible.
  • The gaph of \( \csc(x) \) has an interesting pattern where it appears as a series of "U" and upside-down "U" shapes.
Understanding the backdrop of \( \csc(x) \) is crucial when solving problems that involve the inverse cosecant function as we delve deeper into trigonometry.
Inverse Trigonometric Functions
Inverse trigonometric functions help us find the angle when we know the value of a trigonometric function. When dealing with \( \csc^{-1}(x) \), it tells us the angle \( \theta \) such that \( \csc(\theta) = x \). These functions are often denoted with a "-1" exponent to signify the inverse.

Key characteristics of inverse trigonometric functions include:
  • The range of \( \csc^{-1}(x) \) is typically a specific restricted range, which in this case is from \( -90^\circ \) to \( 0^\circ \) for negative values of \( x \).
  • Inverse trigonometric functions work by reversing the "normal" functions; you input a ratio and get an angle instead of inputting an angle to get a ratio.
These concepts become evident when using expressions like \( \csc^{-1}(-6.1324) \), where the inverse helps us compute the actual angle that matches the given cosecant value.
Angle Calculation
Calculating angles using the inverse cosecant function often requires a few detailed steps. Knowing that \( \csc^{-1}(-6.1324) \) means finding an angle \( \theta \) such that the cosecant of \( \theta \) is \( -6.1324 \), we turn to the sine function for simplification.

Here's a breakdown of the process:
  • Find \( \sin(\theta) \) by taking the reciprocal: \( \sin(\theta) = -\frac{1}{6.1324} \).
  • This gives us the sine value of \( -0.16318 \).
  • Use the inverse sine function to calculate the angle \( \theta \), resulting in approximately \( -9.41^\circ \) in degrees.
This method is a reliable approach to angle calculation in problems involving reciprocal trigonometric values.
Trigonometric Calculator Use
Using a calculator effectively is essential in solving trigonometric problems, especially when determining inverse functions. To evaluate \( \csc^{-1}(-6.1324) \) and find \( \theta \), the calculator becomes a key player. Make sure it鈥檚 set to degree mode to get accurate angle measurements.

Follow these steps when using the calculator:
  • First, calculate the reciprocal to determine the sine value as \( -0.16318 \).
  • Then, input this value into the calculator using the sine inverse function \( \sin^{-1}(x) \).
  • Verify that the calculator returns the angle within the restricted range, in this case, \( -9.41^\circ \) for the principal value in degrees.
Being familiar with the calculator settings and functions makes solving trigonometric exercises like this straightforward and efficient.

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Most popular questions from this chapter

In Exercises 37-48, solve each of the trigonometric equations on the interval \(0^{\circ} \leq \theta<360^{\circ}\). Give answers in degrees and round to two decimal places. $$ \sin ^{2}(2 x)+\sin (2 x)-2=0 $$

Solve the inequality exactly: $$ -\frac{1}{4} \leq \sin x \cos x \leq \frac{1}{4} \text { on the interval }[0, \pi) $$

In Exercises 37-48, solve each of the trigonometric equations on the interval \(0^{\circ} \leq \theta<360^{\circ}\). Give answers in degrees and round to two decimal places. $$ \cos (2 x)+\frac{1}{2} \sin x=0 $$

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