Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\csc x=-2$$

Step-by-Step Solution

Answer

The solutions in the interval [0, 2\pi) are \(x = 7\pi /6\) and \(x = 11\pi /6\).

Step 1: Conversion

Convert \(\csc x =-2\) into terms of sine function. Thus, the equation becomes \(\sin x = -1/2\).

Step 2: Identify Related Acute Angle

Identify the acute angle \(\alpha\) for which \(\sin \alpha = 1/2\). There are two such angles within the interval \([0,2\pi)\), and they are at \(\alpha=\pi /6\) and \(\alpha = 5\pi/6 \). However, we know that one of our solutions should be negative, as \(\sin x = -1/2\). Therefore, we discard \(\alpha = \pi /6\) and retain \(\alpha = 5\pi/6\).

Step 3: Determine Quadrants for Solutions

Remember that sine is negative in the third and fourth quadrants. So, we should find an angle in each of those quadrants that has the same sine value as \(\alpha= 5\pi /6\). The corresponding angles are \(\pi + \alpha\) and \(2\pi - \alpha\). This gives the solutions \(x= 7\pi /6\) and \(x = 11\pi /6\).

Step 4: Check Values

To make sure the results are correct, substitute \(x = 7\pi /6\) and \(x = 11\pi /6\) into the original equation \( \csc x=-2\). Both values satisfy the equation.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is one of the primary functions in trigonometry. It is the reciprocal of the sine function, specifically defined as \( \csc x = \frac{1}{\sin x} \). This means that whenever you have a cosecant function, the sine value can never be zero, because division by zero is undefined.

• Understanding Reciprocal Functions: Reciprocal functions take the form of \( \frac{1}{f(x)} \). If you know the value of the sine function, you can easily find the cosecant by taking the reciprocal.
• Periodicity and Range: The cosecant function has the same periodicity as the sine function, repeating every \(2\pi\). Its range excludes the interval \([-1, 1]\), since you can never get a value between \(-1\) and \(1\) from \(\frac{1}{\text{something between -1 and 1}}\).

If \( \csc x = -2 \), then \( \sin x = -\frac{1}{2} \). This is because taking the reciprocal of \(-2\) gives you \(-\frac{1}{2}\).

Unit Circle

The unit circle is an essential concept in trigonometry, acting as a visual tool to understand angles and trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate system. Each point on the unit circle corresponds to a unique angle \( x \) and has coordinates \( (\cos x, \sin x) \).

• Visualizing Angles: As you move around the circle, angles are measured from the positive x-axis in radians. One full revolution around the circle equals \(2\pi\) radians.
• Quadrants: The circle is divided into four quadrants. The sine function is positive in the first and second quadrants and negative in the third and fourth quadrants, which is crucial in solving equations like \( \csc x = -2 \).

Knowing the sign of sine in each quadrant helps determine which angles correspond to a given sine or cosecant value.

Sine Function

The sine function is one of the foundational trigonometric functions, often abbreviated as \( \sin \). For an angle \( x \), \( \sin x \) represents the y-coordinate of the corresponding point on the unit circle.

• Properties of the Sine Function: The sine function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi \) radians. Its range is \([-1, 1]\), encompassing all possible y-values on the unit circle.
• Identifying Key Angles: Certain angles are commonly used in trigonometry due to their simple sine values, such as \(\pi/6\), \(\pi/4\), and \(\pi/3\), which correspond to sine values of \(1/2\), \(\sqrt{2}/2\), and \(\sqrt{3}/2\), respectively.

When solving the equation \( \sin x = -\frac{1}{2} \), it indicates that the angle \( x \) makes the y-coordinate of the point on the unit circle \(-\frac{1}{2}\). This happens in the third and fourth quadrants, as sine is negative there.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved angles. These identities are useful tools for simplifying trigonometric expressions and solving equations.

• Reciprocal Identities: These include identities like \( \csc x = \frac{1}{\sin x} \), which relate cosecant to sine.
• Pythagorean Identities: The most famous is \( \sin^2 x + \cos^2 x = 1 \), linking sine and cosine values for any angle \( x \).
• Angle Sum and Difference Identities: These identities, such as \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \), help in calculating the sine and cosine of sums of angles.

Leveraging these identities can help solve more complex trigonometric equations or simplify expressions. For the equation \( \csc x = -2 \), using the identity \( \csc x = \frac{1}{\sin x} \) allowed the conversion to \( \sin x = -\frac{1}{2} \).