Problem 53 - Trigonometry

Question

The problems that follow review material we covered in Section 6.3. Find all solutions in radians using exact values only. $$ \sin 3 x=1 / 2 $$

Step-by-Step Solution

Answer

The solutions for \(x\) are \(\frac{\pi}{18} + \frac{2k\pi}{3}\) and \(\frac{5\pi}{18} + \frac{2k\pi}{3}\), where \(k\) is an integer.

Step 1: Identify the basic angle

The equation given is \(\sin 3x = \frac{1}{2}\). First, determine where the sine function equals \(\frac{1}{2}\) on the unit circle. The basic angles for this are \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\).

Step 2: Apply multiple angle conditions

Since we have \(3x\) instead of \(x\), we must consider the periodicity of the sine function. For sine, it is periodic every \(2\pi\). So, the solutions for \(3x\) are \(3x = \frac{\pi}{6} + 2k\pi\) and \(3x = \frac{5\pi}{6} + 2k\pi\), where \(k\) is an integer.

Step 3: Solve for x

Divide each equation by 3 to solve for \(x\). This gives us two sets of solutions: \(x = \frac{\pi}{18} + \frac{2k\pi}{3}\) and \(x = \frac{5\pi}{18} + \frac{2k\pi}{3}\).

Step 4: Generalize the solution

Considering both conditions, the complete solution set for \(x\) in terms of \(k\) is: \(x = \frac{\pi}{18} + \frac{2k\pi}{3}\) and \(x = \frac{5\pi}{18} + \frac{2k\pi}{3}\).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, particularly for understanding the trigonometric functions like sine and cosine. Every point on the unit circle has coordinates that can help identify the sine and cosine of an angle.

• The x-coordinate of a point on the unit circle gives the cosine of the angle, while the y-coordinate represents the sine of the angle.
• The angle is typically measured from the positive x-axis, counter-clockwise around the circle.

For example, at the angle \(\frac{\pi}{6}\) or 30 degrees, the coordinates are \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\). Here, the sine value, represented by the y-coordinate, is \(\frac{1}{2}\). Similarly, at \(\frac{5\pi}{6}\) or 150 degrees, the sine value is also \(\frac{1}{2}\). These points are crucial when solving trigonometric equations.

Radians

Radians are a unit of measure for angles used primarily in trigonometry and calculus. Unlike degrees, which divide a circle into 360 parts, radians measure the angle based on the radius of a circle.

• One full circle is \(2\pi\) radians, which is equivalent to 360 degrees.
• Therefore, \(\pi\) radians equals 180 degrees, and \(\frac{\pi}{6}\) radians equals 30 degrees.

Radians provide a more natural way of describing angles when dealing with trigonometric functions, as they simplify many formulas, including those for calculating arc length and sector area. In solving trigonometric equations, using radian measure is often required to find exact values.

Sine Function

The sine function is one of the primary trigonometric functions, and it expresses the y-coordinate of a unit circle as a function of the angle. It is periodic, meaning it repeats its values in regular intervals, which is a core aspect of solving trigonometric equations.

• Sine values range from -1 to 1.
• The sine of an angle \(x\) is noted as \(\sin(x)\).
• Basic angles include \(\sin(\frac{\pi}{6}) = \frac{1}{2}\) and \(\sin(\frac{5\pi}{6}) = \frac{1}{2}\).

These values help locate solutions on the unit circle. In the given exercise, solving \(\sin(3x) = \frac{1}{2}\) involves determining angles where the sine value equals \(\frac{1}{2}\) and later adjusting for the equation's specific circumstances.

Periodicity

The concept of periodicity in trigonometry refers to the repeating nature of trigonometric functions over specific intervals. For the sine function, this periodicity occurs every \(2\pi\) radians.

• For any angle \(x\), \(\sin(x + 2k\pi) = \sin(x)\), where \(k\) is any integer.
• This periodicity allows for multiple solutions when solving equations like \(\sin(3x) = \frac{1}{2}\).

In the solution of \(\sin(3x) = \frac{1}{2}\), the adjustment of angles by adding \(2k\pi\) accounts for the infinite number of angles that have the same sine value, leading to the general solutions given in terms of \(x\). Taking advantage of periodicity is vital for finding all possible solutions in trigonometric equations.