Problem 77 - Precalculus

Question

Find all the values of \(\theta, 0 \leq \theta \leq 2 \pi,\) for which the equation \(2 \sin \theta=\csc \theta\) is true.

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\).

Step 1: Understand the Equation

The given equation is \(2 \sin \theta = \csc \theta\). We know that \(\csc \theta = \frac{1}{\sin \theta}\). The equation can thus be rewritten as \(2 \sin \theta = \frac{1}{\sin \theta}\).

Step 2: Multiply Both Sides by \(\sin \theta\)

To eliminate the fraction, multiply both sides of the equation by \(\sin \theta\). This gives us \(2 \sin^2 \theta = 1\).

Step 3: Solve for \(\sin^2 \theta\)

Divide both sides by 2 to solve for \(\sin^2 \theta\). We have \(\sin^2 \theta = \frac{1}{2}\).

Step 4: Solve for \(\sin \theta\)

Take the square root of both sides to find \(\sin \theta\). So, \(\sin \theta = \pm \frac{1}{\sqrt{2}}\). Since \(\frac{1}{\sqrt{2}}\) equals \(\frac{\sqrt{2}}{2}\), this simplifies to \(\sin \theta = \pm \frac{\sqrt{2}}{2}\).

Step 5: Find Angles for \(\sin \theta = \frac{\sqrt{2}}{2}\)

The angles \(\theta\) for which \(\sin \theta = \frac{\sqrt{2}}{2}\) are \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{4}\) within the interval \([0, 2\pi]\).

Step 6: Find Angles for \(\sin \theta = -\frac{\sqrt{2}}{2}\)

The angles \(\theta\) for which \(\sin \theta = -\frac{\sqrt{2}}{2}\) are \(\theta = \frac{5\pi}{4}\) and \(\theta = \frac{7\pi}{4}\) within the interval \([0, 2\pi]\).

Step 7: Combine Solutions

Combine all the angles found: \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\). These are the solutions to the original equation.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It relates the angle of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. In the unit circle, the sine of an angle \( \theta \) is the y-coordinate of the point where the terminal side of the angle intersects the circle.

• For an angle \( \theta \) in standard position, \( \sin \theta \) is positive in the first and second quadrants.
• It becomes negative in the third and fourth quadrants.
• It repeats every \( 2\pi \) as it is a periodic function.

Understanding how the sine function works is crucial in solving trigonometric equations, and it helps in determining potential solutions based on its range from -1 to 1.

Cosecant Function

The cosecant function, \( \csc \theta \), is the reciprocal of the sine function. This means that \( \csc \theta = \frac{1}{\sin \theta} \). Unlike the sine function, cosecant is undefined for angles where \( \sin \theta = 0 \), since division by zero is not possible.

The key points of the cosecant function are that it becomes very large (positively or negatively) when \( \sin \theta \) is near zero, which happens near the two angles \( 0 \) and \( \pi \).

• Cosecant is positive when sine is positive, and negative when sine is negative.
• It does not exist for angle values where the sine function hits zero.

When working with equations involving cosecant, it's essential to remember these restrictions to avoid errors in determining valid solutions.

Unit Circle

The unit circle is a circle with a radius of one unit centered at the origin of a coordinate plane. It serves as a fundamental tool in trigonometry.

Understanding the Unit Circle

The unit circle allows us to define trigonometric functions for all real numbers. On this circle:

• The x-coordinate corresponds to \( \cos \theta \), representing the adjacent side.
• The y-coordinate represents \( \sin \theta \), the opposite side in our unit radius.
• This structure simplifies understanding how each trigonometric function behaves over different intervals.

Points Around the Unit Circle

As \( \theta \) varies from \( 0 \) to \( 2\pi \), the point moves around the circle and defines the angle in both degrees and radians. Using the unit circle, you can easily visualize why particular solutions exist in specific quadrants for any trigonometric equation.

Angle Solutions

Solving trigonometric equations like \( 2 \sin \theta = \csc \theta \) involves finding angle solutions within a specified interval, typically from \( 0 \) to \( 2\pi \). The key here is identifying angles where the equation holds true.

Approach to Finding Solutions

Firstly, rewrite the given equation in understandable terms, such as rearranging terms or eliminating fractions by multiplication. Next, solve step-by-step by isolating trigonometric terms, simplifying them, and checking which angles satisfy the equation.

• For \( \sin \theta = \frac{\sqrt{2}}{2} \), potential angles are \( \frac{\pi}{4} \) and \( \frac{3\pi}{4} \).
• For \( \sin \theta = -\frac{\sqrt{2}}{2} \), the solutions are \( \frac{5\pi}{4} \) and \( \frac{7\pi}{4} \).

Once all solutions within the interval are noted, it's vital to verify each one to ensure they don't violate any boundary conditions of the original equation.