Problem 79 - Precalculus

Question

Find all the values of \(\theta(\theta\) is any real number) for which the equation \(3 \csc \theta=4 \sin \theta\) is true.

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{3} + 2k\pi, \frac{2\pi}{3} + 2k\pi, \frac{4\pi}{3} + 2k\pi, \frac{5\pi}{3} + 2k\pi\).

Step 1: Recall Definitions and Pythagorean Identity

The equation given is \(3 \csc \theta = 4 \sin \theta\). Start by recalling the definition of cosecant: \( \csc \theta = \frac{1}{\sin \theta} \). Thus, the equation becomes \(3 \cdot \frac{1}{\sin \theta} = 4 \sin \theta\). Rewriting gives us \(3 = 4 \sin^2 \theta\).

Step 2: Rearrange and Solve the Equation

Rearrange the equation: \(4 \sin^2 \theta = 3\). Then, divide both sides by 4 to isolate \(\sin^2 \theta\): \(\sin^2 \theta = \frac{3}{4}\).

Step 3: Take the Square Root

Take the square root of both sides to solve for \(\sin \theta\): \(\sin \theta = \pm \sqrt{\frac{3}{4}}\). This simplifies to \(\sin \theta = \pm \frac{\sqrt{3}}{2}\).

Step 4: Find General Solutions

Recall the angles where \(\sin \theta = \pm \frac{\sqrt{3}}{2}\). From the unit circle, \(\sin \theta = \frac{\sqrt{3}}{2}\) at \(\theta = \frac{\pi}{3} + 2k\pi\) and \(\theta = \frac{2\pi}{3} + 2k\pi\), and \(\sin \theta = -\frac{\sqrt{3}}{2}\) at \(\theta = \frac{4\pi}{3} + 2k\pi\) and \(\theta = \frac{5\pi}{3} + 2k\pi\) for any integer \(k\).

Key Concepts

Cosecant

When working with trigonometric equations, understanding the cosecant function is important. Cosecant, denoted as \( \csc \theta \), is a trigonometric function equivalent to the reciprocal of the sine function.
This means that \( \csc \theta = \frac{1}{\sin \theta} \).
The cosecant function is undefined wherever the sine function is zero, which happens at angles like \( 0, \pi, 2\pi, \ldots \).

• Cosecant is useful for solving equations where a reciprocal is needed.
• It's important to remember that \( \csc \theta \) is undefined at points where \( \sin \theta = 0 \).

Understanding \( \csc \theta \) helps in tackling equations like \( 3 \csc \theta = 4 \sin \theta \), since we can convert it to a form involving only the sine function, making the equation easier to solve.

Unit Circle

The unit circle is a fundamental tool in trigonometry, especially when dealing with sine and its reciprocal, cosecant. It is a circle with a radius of 1, centered at the origin of a coordinate plane.
Each angle on the unit circle can provide valuable information about the sine, cosine, and other trigonometric functions.

• For any angle \( \theta \), \( \sin \theta \) is represented by the y-coordinate of the point where the terminal side of the angle intersects the unit circle.
• The x-coordinate at this intersection gives \( \cos \theta \).

For example:

• \( \sin \theta = \frac{\sqrt{3}}{2} \) happens at \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \).
• \( \sin \theta = -\frac{\sqrt{3}}{2} \) is found at \( \theta = \frac{4\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).

Knowing these values allows us to determine solutions for trigonometric equations using symmetry and periodicity of the circle.

Pythagorean Identity

The Pythagorean identity is a critical concept used in solving trigonometric equations. This identity is: \[ \sin^2 \theta + \cos^2 \theta = 1 \].
This fundamental identity stems from the Pythagorean theorem in a unit circle.

• It helps in expressing one trigonometric function in terms of another.
• This identity can simplify expressions and solve equations involving \( \sin \) or \( \cos \).

In the exercise \( 3 \csc \theta = 4 \sin \theta \):

• We use the identity indirectly by focusing on \( \sin \theta \) since \( \csc \theta \) is \( \frac{1}{\sin \theta} \).
• The rearranging step leads us to \( 4 \sin^2 \theta = 3 \), helping us isolate and solve for \( \sin \theta \).

It is essential to recognize how the Pythagorean identity underpins many trigonometric principles, including the relation between \( \sin \) and \( \cos \), and it's pivotal in breaking down more complex facets of trigonometry.