Problem 51 - Algebra 2

Question

Explain why the number of solutions to the equation \(\sin \theta=\frac{\sqrt{3}}{2}\) is infinite.

Step-by-Step Solution

Answer

The periodic nature of the sine function means solutions repeat every \(2\pi\), giving infinite solutions.

Step 1: Understand the Sine Function

The sine function, \(\sin \theta\), is periodic with a period of \(2\pi\). This means that the function repeats its values every \(2\pi\) radians. Hence, any angle \(\theta\) that satisfies the equation will have equivalently infinite solutions by adding or subtracting integer multiples of \(2\pi\).

Step 2: Identify Basic Solutions in One Period

Within one period \([0, 2\pi]\), \(\sin \theta = \frac{\sqrt{3}}{2}\) is satisfied at two specific angles: \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{2\pi}{3}\). These are the basic solutions before considering periodicity.

Step 3: Generalize the Solutions Using Periodicity

Given the periodicity of sine, you can generate all solutions using the basic solutions found in Step 2. For \(\theta = \frac{\pi}{3}\), the general solution is \(\theta = \frac{\pi}{3} + 2k\pi\), where \(k\) is any integer. Similarly, for \(\theta = \frac{2\pi}{3}\), the general solution is \(\theta = \frac{2\pi}{3} + 2k\pi\).

Step 4: Conclude Infinite Solutions

Since \(k\) can be any integer, each basic solution generates an infinite sequence of angles satisfying \(\sin \theta = \frac{\sqrt{3}}{2}\), thus there are infinitely many solutions.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It is denoted as \(\sin \theta\), where \(\theta\) represents an angle. The sine function relates the angle to the ratio of the side opposite the angle to the hypotenuse in a right triangle. In simpler terms, for any right triangle, the sine of an angle gives us a way to determine lengths of sides if we know one.
Key properties of the sine function include:

• It oscillates between -1 and 1 for all values of \(\theta\).
• It is an odd function, which means \(\sin(-\theta) = -\sin(\theta)\).
• It has certain symmetry properties, specifically odd symmetry.

Understanding the sine function is crucial when solving equations involving it, like \(\sin \theta = \frac{\sqrt{3}}{2}\), as it helps identify potential solutions.

Periodicity

Periodicity refers to how a function repeats its values in a regular fashion over a specific interval. For the sine function, its period is \(2\pi\). This means that the function repeats every \(2\pi\) radians.

• Every complete rotation (\(2\pi\)) brings the sine function back to the same value.
• This periodic nature allows us to generalize solutions for sine equations across multiple cycles.

The periodicity of the sine function is what allows for essentially infinite solutions to be derived from a basic equation. For example, solving \(\sin \theta = \frac{\sqrt{3}}{2}\) yields solutions that can be repeated by adding multiples of the period, \(2k\pi\), where \(k\) is an integer.

General Solutions

When solving trigonometric equations, finding general solutions allows us to account for all possible angles that satisfy the equation. The general solution for a sine equation \(\sin \theta = c\) has the form \(\theta = \theta_0 + 2k\pi\) for all basic solutions \(\theta_0\) and any integer \(k\).
General solutions are necessary because of the periodicity of trigonometric functions. Since these functions repeat their values, specifying only one solution may not encompass all possibilities. For instance, in the case of \(\sin \theta = \frac{\sqrt{3}}{2}\), the basic solutions within one period are \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{2\pi}{3}\). By using the formula, \(\theta = \frac{\pi}{3} + 2k\pi\) and \(\theta = \frac{2\pi}{3} + 2k\pi\), every possible solution can be found.

Angle

In trigonometry, angles are pivotal for understanding and solving equations involving trig functions. An angle can be defined as the measure of rotation between two rays with a common endpoint, called the vertex. Angles are typically measured in degrees or radians.
Angles determine the exact sine values for trigonometric functions as they specify the position on the unit circle. For example, the angle \(\frac{\pi}{3}\) lands a point on the unit circle where the sine value is \(\frac{\sqrt{3}}{2}\). Understanding angles helps to solve trigonometric equations by identifying principal values and determining general solutions for these equations.

Radians

Radians are the standard unit of angular measurement in trigonometry, and understanding them is crucial for solving trigonometric equations. One radian is the angle formed when the radius of a circle is wrapped along its circumference.
Radians are preferred in mathematical contexts for several reasons:

• The calculus of trigonometric functions is often simpler in radians than in degrees.
• Trigonometric identities and relationships express more naturally in radian measure.
• Radians provide a direct connection between the angle measure and the unit circle's arc.

In terms of converting, \(2\pi\) radians equal 360 degrees. So, the periodic properties of trigonometric functions like sine are conveniently expressed in radians.