Problem 34 - Algebra and Trigonometry

Question

\(25-38\) . Find all solutions of the given equation. $$ 4 \sin ^{2} \theta-3=0 $$

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{\pi}{3} + 2n\pi, \frac{2\pi}{3} + 2n\pi, -\frac{\pi}{3} + 2n\pi, -\frac{2\pi}{3} + 2n\pi\), where \(n\) is an integer.

Step 1: Rearrange the Equation

Start by rearranging the given equation to isolate the trigonometric function. The equation is \(4 \sin^2 \theta - 3 = 0\). Add 3 to both sides to get:\[4 \sin^2 \theta = 3\]

Step 2: Divide by 4

Divide both sides of the equation by 4 to further isolate \(\sin^2 \theta\) on one side. This gives:\[\sin^2 \theta = \frac{3}{4}\]

Step 3: Take the Square Root

Take the square root of both sides to solve for \(\sin \theta\). Remember to consider both the positive and negative square roots:\[\sin \theta = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}\]

Step 4: Find the General Solutions for \(\sin \theta = \frac{\sqrt{3}}{2}\)

The angle \(\theta\) where \(\sin \theta = \frac{\sqrt{3}}{2}\) corresponds to \(\theta=\frac{\pi}{3} + 2n\pi\) and \(\theta=\frac{2\pi}{3} + 2n\pi\), where \(n\) is an integer.

Step 5: Find the General Solutions for \(\sin \theta = -\frac{\sqrt{3}}{2}\)

The angle \(\theta\) where \(\sin \theta = -\frac{\sqrt{3}}{2}\) corresponds to \(\theta = -\frac{\pi}{3} + 2n\pi\) and \(\theta = -\frac{2\pi}{3} + 2n\pi\), where \(n\) is an integer.

Key Concepts

Understanding the Sine Function

The sine function, often abbreviated as \(\sin\), is a fundamental trigonometric function that appears frequently in mathematics, especially in solving equations related to angles and triangles. In the context of a unit circle, \(\sin \theta\) represents the y-coordinate of a point that corresponds to an angle \(\theta\) measured from the positive x-axis. This function has a range of values from \(-1\) to \(1\), and it' s periodic, meaning that it repeats its values in regular intervals. For the sine function, this period is \(2\pi\), so \(\sin(\theta + 2\pi n) = \sin\theta\) for any integer \(n\).

Key properties of the sine function include:

• Amplitude: The maximum absolute value, always \(1\).
• Period: \(2\pi\), as mentioned.
• Symmetry: It's an odd function, so \(\sin(-\theta) = -\sin\theta\).

These properties are crucial when solving trigonometric equations, as they guide us in finding all possible solutions.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angles that satisfy a given trigonometric expression. The key steps generally involve:

• Isolate the trigonometric function on one side of the equation.
• Use inverse trigonometric functions or known values to determine the specific values of the angle that satisfy the equation.
• Consider the periodic nature of the trigonometric functions to find the general solution, which includes all possible angles.

For example, in solving \(4\sin^2\theta - 3 = 0\), our goal is to make \(\sin \theta\) the subject of the formula. This leads us to first rearrange the equation to \(4\sin^2\theta = 3\) and then \(\sin^2\theta = \frac{3}{4}\), and finally, by taking the square root, we arrive at \(\sin \theta = \pm \frac{\sqrt{3}}{2}\).

This example illustrates the necessity of considering both positive and negative solutions because the sine function can take both values due to its symmetry across different quadrants of the unit circle.

Connections to Quadratic Equations

Trigonometric equations often transform into quadratic equations, much like the exercise \(4\sin^2\theta - 3 = 0\). In this case, it resembles a quadratic equation in the format \(ax^2 + c = 0\), where \(x = \sin \theta\). Quadratic equations are a significant part of algebra, and they can be solved using various techniques such as the quadratic formula, factoring, or completing the square.

In the trigonometry realm, these were simplified into trigonometric expressions like the one in this exercise. By treating \(\sin^2 \theta\) analogously to \(x^2\), we simplify the trigonometric problem to a solvable format. Solving for \(x\), or in this context, \(\sin \theta\), requires a systematic approach, often involving rearranging the equation to set it in a solvable form and finding its roots, both positive and negative solutions.

Finding General Solutions

When we find solutions for trigonometric equations, especially those based on the sine function, we aim to discover the general solutions, which represent all possible angles that satisfy the equation. The general solution considers the periodic nature of sine, where solutions repeat every \(2\pi\) radians.

For the equation \(\sin \theta = \frac{\sqrt{3}}{2}\), the known angle solutions are \(\frac{\pi}{3}\) and \(\frac{2\pi}{3}\). However, due to the periodicity of sine, we express this in general solution form as \(\theta = \frac{\pi}{3} + 2n\pi\) and \(\theta = \frac{2\pi}{3} + 2n\pi\), where \(n\) is any integer.

Similarly, for the solution \(\sin \theta = -\frac{\sqrt{3}}{2}\), the angles \(-\frac{\pi}{3}\) and \(-\frac{2\pi}{3}\) translate into general solutions \(\theta = -\frac{\pi}{3} + 2n\pi\) and \(\theta = -\frac{2\pi}{3} + 2n\pi\). This complete solution ensures we capture every possible angle that satisfies the original trigonometric equation.