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Magazine Chapter 5: Problem 73
Find all solutions of the equation that lie in the interval \([0, \pi] .\) State each answer correct to two decimal places. $$\csc x=3$$
Short Answer
Expert verified
The solutions are x 鈮� 0.34 and x 鈮� 2.80 radians.
Step by step solution
01
Understand the Relationship
Recall that the cosecant function is the reciprocal of the sine function. Therefore, if \( \csc x = 3 \), it follows that \( \sin x = \frac{1}{3} \).
02
Determine the Interval
We are asked to find solutions in the interval \([0, \pi]\). This interval includes the domain for which the sine function is positive, between 0 and \( \pi \).
03
Use Inverse Sine Function
To solve for \( x \), use the inverse sine function: \( x = \arcsin\left(\frac{1}{3}\right) \). Compute this using a calculator to ensure accuracy.
04
Calculate the Principal Solution
Calculate \( x = \arcsin\left(\frac{1}{3}\right) \). Enter \( \frac{1}{3} \) into the calculator and find the angle whose sine is \( \frac{1}{3} \). This gives \( x \approx 0.34 \) radians.
05
Consider Symmetry in the Interval
Since the interval is \([0, \pi]\), check for other possible solutions. The sine function is symmetric about \( \pi/2 \). Therefore, another solution can be found as \( x = \pi - \arcsin\left(\frac{1}{3}\right) \).
06
Calculate the Symmetric Solution
Compute the symmetric solution: \( x = \pi - 0.34 \), which yields \( x \approx 2.80 \) radians.
07
Verify Solutions
Ensure that both solutions \( x \approx 0.34 \) and \( x \approx 2.80 \) are within the interval \([0, \pi]\) and have \( \csc x = 3 \).
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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, denoted as \( \csc x \), is an important concept in trigonometry. It is defined as the reciprocal of the sine function, which means that \( \csc x = \frac{1}{\sin x} \). Because of this reciprocal relationship, the cosecant function is undefined whenever the sine function is zero. Understanding the cosecant function is crucial when solving trigonometric equations. It often requires converting a cosecant equation to a sine equation, as demonstrated in the exercise where we start with \( \csc x = 3 \) and rewrite it as \( \sin x = \frac{1}{3} \). This step is vital for finding the solutions within a specified interval.
Sine Function
The sine function, represented as \( \sin x \), is one of the foundational trigonometric functions. It describes the relationship between an angle in a right triangle and the ratio of the length of the opposite side to the hypotenuse.
- Sine function values range between -1 and 1.
- The sine wave starts at zero, rises to its maximum at \( \pi/2 \), returns to zero by \( \pi \), and is symmetric over an interval from \( 0 \) to \( 2\pi \).
Inverse Trigonometric Functions
Inverse trigonometric functions are essential in finding angles from given trigonometric values. The inverse sine function, denoted as \( \arcsin \) or \( \sin^{-1} \), provides the angle whose sine is a specific number.
- \( \arcsin \) returns a value between \(-\pi/2\) and \(\pi/2\).
- In our exercise, we calculated \( x = \arcsin\left(\frac{1}{3}\right) \).
- This angle-specific approach becomes critical when dealing with trigonometric equations, helping convert a problem into a calculable format.
Symmetry in Trigonometric Functions
Symmetry plays a significant role in trigonometric functions, aiding in finding all potential solutions in specified intervals. The sine function is symmetric about the line \( x = \pi/2 \), and this characteristic helps identify additional solutions that aren't immediately obvious from straightforward calculations.
- In the interval \([0, \pi]\), the symmetry property of sine suggests complementary angles that also satisfy the equation.
- For a given angle \( x \), another potential solution could be \( \pi - x \).
- This principle was employed to discover the second solution, \( x \approx 2.80 \) radians, from the symmetric equation \( \pi - \arcsin\left(\frac{1}{3}\right) \).
