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Magazine Chapter 8: Problem 42
Find all solutions of the equation in the interval \([0,2 \pi).\) $$3 \csc ^{2} x=4$$
Short Answer
Expert verified
The solutions in the interval \([0, 2\pi)\) are \(x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\).
Step by step solution
01
Understand the given equation
The equation given is \(3 \csc^2 x = 4\). We need to solve this equation for \(x\) within the interval \([0, 2\pi)\). The cosecant function is the reciprocal of sine, so \(\csc x = \frac{1}{\sin x}\).
02
Manipulate the equation into a simpler form
First, divide each side of the equation by 3 to isolate \(\csc^2 x\), resulting in \(\csc^2 x = \frac{4}{3}\).
03
Use the definition of cosecant
Since \(\csc x = \frac{1}{\sin x}\), we substitute to get \(\left(\frac{1}{\sin x}\right)^2 = \frac{4}{3}\). This simplifies to \(\sin^2 x = \frac{3}{4}\).
04
Solve for \(\sin x\)
Take the square root of both sides. \(\sin x\) can be either positive or negative: \(\sin x = \sqrt{\frac{3}{4}} = \pm\frac{\sqrt{3}}{2}\).
05
Find exact angles satisfying \(\sin x = \frac{\sqrt{3}}{2}\)
On the [unit circle](https://www.mathsisfun.com/unit-circle.html), \(\sin x = \frac{\sqrt{3}}{2}\) occurs at \(x = \frac{\pi}{3}\) and \(x = \frac{2\pi}{3}\).
06
Find exact angles satisfying \(\sin x = -\frac{\sqrt{3}}{2}\)
Similarly, \(\sin x = -\frac{\sqrt{3}}{2}\) occurs at \(x = \frac{4\pi}{3}\) and \(x = \frac{5\pi}{3}\).
07
State all solutions in the given interval
Thus, the solutions of the equation \(3 \csc^2 x = 4\) in the interval \([0, 2\pi)\) are \(x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\).
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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, often abbreviated as "csc," is a trigonometric function that is the reciprocal of the sine function. This means that for any angle \(x\), \( \csc x = \frac{1}{\sin x} \). Being a reciprocal function, it is undefined wherever the sine function equals zero, such as at angles like \(0, \pi, 2\pi\), etc.
- The cosecant function tends to infinity when the sine function approaches zero.
- It is important for equations involving sine because it transforms into simpler, more solveable forms often used in trigonometric identities and equations.
- Since it is based on sine, the cosecant function inherits many of sine's properties, such as periodicity.
Unit Circle
The unit circle is a fundamental concept in trigonometry, uniquely placing every angle in terms of coordinates on a circle of radius 1 centered at the origin. It's a simple tool, but it acts as an essential roadmap for understanding sine, cosine, and their reciprocals.
- Every point on the unit circle corresponds to an angle, whose measures range from 0 to \(2\pi\) radians (or \(0\) to 360 degrees).
- The \(x\)-coordinate corresponds to the cosine of the angle, and the \(y\)-coordinate corresponds to the sine of the angle.
- Angles in this setup allow us to pinpoint exact trigonometric values like \(\sin x = \frac{\sqrt{3}}{2}\) easily through visualization and reference.
- This tool also makes it easy to identify signs of sinusoidal functions in different quadrants, which is crucial in trigonometric problem-solving.
Sine Function
The sine function, one of the primary trigonometric functions, describes the \(y\)-coordinate of a point on the unit circle. For an angle \(x\), \( \sin x \) expresses how far above or below the horizontal axis a point lies.
- \( \sin x \) has a range of [-1, 1], reaching its maximum of 1 at \(\frac{\pi}{2} + 2k\pi\) and its minimum of -1 at \(\frac{3\pi}{2} + 2k\pi\), where \(k\) is an integer.
- The function is periodic with a period of \(2\pi\), meaning its values repeat every \(2\pi\) radians.
- Being at the heart of the problem, understanding \( \sin x \) is necessary to solve equations like \(3 \csc^2 x = 4\).
- In the given problem, transforming \( \csc^2 x = \frac{4}{3} \) to \( \sin^2 x = \frac{3}{4} \) allowed us to find the correct solutions for \( \sin x \).
