Problem 41 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \csc \theta=3 $$

Step-by-Step Solution

Answer

The solutions are \(\theta \approx 0.34, 2.80\) in radians.

Step 1: Rewrite \(\csc \theta\) in terms of \(\sin \theta\)

Given \(\csc \theta = 3\), we know that \(\csc \theta = 1/\sin \theta\). Therefore, we get the equation \(1/\sin \theta = 3\). Now, cross multiplying, we get \(\sin \theta = 1/3\).

Step 2: Find the angles

We need to find the angles \(\theta\) that satisfy \(\sin \theta = 1/3\) within our given domain \([0, 2\pi)\). Using sine inverse function (asin or \(\sin^{-1}\)), we find that \(\sin^{-1}(1/3) = \theta_1\). However, sine function is positive in both 1st and 2nd quadrants, so we need to find another angle, \(\theta_2\), in the 2nd quadrant. This can be found using the relation \(\theta_2 = \pi - \theta_1\).

Step 3: Solve for \(\theta\)

Now we solve for \(\theta\). Using a calculator, we find that \(\sin^{-1}(1/3) = \theta_1 \approx 0.34\) radians. Then, \(\theta_2 = \pi - \theta_1 \approx 2.80\) radians. Therefore, \(\theta = \theta_1, \theta_2\).

Key Concepts

Cosecant Function

The cosecant function is an essential concept in trigonometry. It is the reciprocal of the sine function. If you are familiar with the sine function, the cosecant shouldn't be too difficult to understand. The relationship is simple:

• The sine function is typically denoted as \( \sin \theta \), where \( \theta \) is the angle.
• The cosecant function is \( \csc \theta \), computed as \( 1/\sin \theta \).

The key point to remember is that the cosecant is undefined for any angle where the sine value is zero, because division by zero is not possible. Therefore, when working with the cosecant function, ensure that the sine value is not zero to avoid undefined expressions.

Sine Inverse

Sine inverse, frequently denoted as \( \sin^{-1} \) or asin, is used when we want to find the angle whose sine is a given number. This process is essentially the reverse of finding the sine of an angle.

• The range of the sine inverse function is restricted to \([-\pi/2, \pi/2]\) or \([-90^\circ, 90^\circ]\) to ensure it is a function — single output for every input.
• When you use \( \sin^{-1} \), you're asking, "What angle gives me this particular sine value?"

In our specific example, the solution involves finding \( \theta \) such that \( \sin \theta = 1/3 \). Using the inverse sine function, we can determine \( \theta_1 \), the first angle.

Unit Circle

The unit circle is a powerful tool in understanding trigonometric functions and their properties. It is a circle with a radius of one, centered at the origin of a coordinate plane. This circle helps illustrate the angles and their corresponding sine, cosine, and other trigonometric values.

• Each angle \( \theta \) is measured from the positive x-axis.
• The coordinates of each point on the unit circle represent \( (\cos \theta, \sin \theta) \).
• Angles in the first quadrant are between \(0\) and \(\pi/2\), and in the second quadrant, they are between \(\pi/2\) and \(\pi\).

When dealing with the cosecant function \(\csc \theta = 3\), visualizing on the unit circle can clarify why we find solutions in the first and second quadrants. Since sine is positive in both, these correspond to different angles \(\theta_1\) and \(\theta_2\).

Angle Measurement

Angle measurement in trigonometry can be in degrees or radians, though radians are more common in calculus and advanced mathematics. Understanding both is crucial, but our focus here is on radians, as used in the exercise.

• One full circle, or \(360^\circ\), is equivalent to \(2\pi\) radians.
• Therefore, \( \pi \) radians is \(180^\circ\) and \( \pi/2 \) is \(90^\circ\).

Converting between these helps in interpreting trigonometric equations. In the exercise, the restrictions are set between \(0\) and \(2\pi\), meaning we are interested only in angles within one full circle, or one rotation around the origin. For a solution like \( \theta = 0.34 \), it's clear and simple to see where it lies on the unit circle, relying on the radian understanding.