Problem 5 - Algebra and Trigonometry

Question

\(5-16=\) Solve the given equation. $$ \sin \theta=\frac{\sqrt{3}}{2} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Trigonometric Equation

The equation we're working with is \( \sin \theta = \frac{\sqrt{3}}{2} \). This is a basic trigonometric equation requiring us to find the angle \( \theta \) whose sine value equals \( \frac{\sqrt{3}}{2} \).

Step 2: Identify Key Angles from Unit Circle

Recall common angles where sine values are known. Sine of \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \) equals \( \frac{\sqrt{3}}{2} \). These angles correspond to 60º and 120º, respectively.

Step 3: List Possible Solutions

The general solution for \( \sin \theta = \frac{\sqrt{3}}{2} \) in radians is \( \theta = \frac{\pi}{3} + 2n\pi \) or \( \theta = \frac{2\pi}{3} + 2n\pi \), where \( n \) is any integer, reflecting periodic properties of the sine function.

Step 4: Convert General Solution to Degrees

If possible, convert the general solution into degrees. So \( \theta = 60^ ext{o} + 360^ ext{o}n \) or \( \theta = 120^ ext{o} + 360^ ext{o}n \), where \( n \) is any integer.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one, centered at the origin of the coordinate plane. The unit circle helps us understand the relationships between angles and trigonometric functions like sine, cosine, and tangent.
This circle is divided into four quadrants, and each point on the circle corresponds to an angle measured in radians or degrees.

• Angles increase counterclockwise from the positive x-axis.
• The coordinates of each point represent the cosine and sine of the angle.

For example, the angle \( \frac{\pi}{3} \), or 60 degrees, results in a point on the unit circle where the sine value is \( \frac{\sqrt{3}}{2} \). This link between angles and sine values is critical for solving trigonometric equations.

Sine Function

The sine function is one of the core trigonometric functions. It relates an angle in a right triangle to the ratio of the opposite side to the hypotenuse. In the context of the unit circle, it represents the y-coordinate of the corresponding point on the circle.

• The sine function is periodic with a period of \( 2\pi \) radians (360 degrees).
• It varies between -1 and 1, making it useful for describing oscillating systems like waves.

In our exercise, we are looking for all angles \( \theta \) such that \( \sin \theta = \frac{\sqrt{3}}{2} \). This value is found at angles \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \) on the unit circle. Because of its periodicity, these values repeat every \( 2\pi \) radians, creating multiple solutions.

General Solution

The general solution in trigonometric equations gives all possible angles that satisfy the equation. This is important because trigonometric functions like sine have repetitive values due to their periodic nature.For the equation \( \sin \theta = \frac{\sqrt{3}}{2} \), we derive two sets of solutions:

• \( \theta = \frac{\pi}{3} + 2n\pi \)
• \( \theta = \frac{2\pi}{3} + 2n\pi \)

Here, \( n \) is any integer, which allows us to account for the infinite repetition of these angles in the unit circle's full rotations. It captures every occurrence of the solution as it reemerges with each complete circle (or cycle) around the unit circle.

Degree-Radian Conversion

Converting between degrees and radians is essential in trigonometry, as angles can be measured in either unit. This conversion relies on the relationship \( 180^\circ = \pi \) radians.

• To convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \).
• To convert radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \).

In our exercise, the general solution in degrees looks like:

• \( \theta = 60^\circ + 360^\circ n \)
• \( \theta = 120^\circ + 360^\circ n \)

These conversions make it easier to understand the solutions in contexts where degrees are more intuitive, such as navigation or some engineering applications.