Problem 17 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 2 \cos \theta-\sqrt{3}=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(2 \cos \theta - \sqrt{3} = 0\) for \(0 \leq \theta < 2 \pi\) is \(\theta = \frac{\pi}{6}\) or \(\theta = \frac{11\pi}{6}\)

Step 1: Make \(\cos(\theta)\) the Subject

The equation to solve is given as \(2 \cos \theta - \sqrt{3} = 0\). To isolate \(\cos(\theta)\), we add \(\sqrt{3}\) to both sides, then divide by 2. This gives us: \(\cos(\theta) = \frac{\sqrt{3}}{2}\) which is the standard value of cosine for a specific angle on the unit circle.

Step 2: Solve for \(\theta\)

The expression \(\cos(\theta) = \frac{\sqrt{3}}{2}\) corresponds to angles \(\theta\) on the unit circle where the x-coordinate is \(\frac{\sqrt{3}}{2}\). Thus, \(\theta = \frac{\pi}{6}\) in the first quadrant, because cosine output is positive in the first quadrant.

Step 3: Find Symmetric Solution based on the Cosine Function's Property

For the cosine function, the value of cosine is positive in both the first and the fourth quadrants. Hence, another angle exists symmetric to \(\theta = \frac{\pi}{6}\) about the x-axis that will yield the same cosine value. That angle is \(\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\).

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions, closely related to the angle and the sides of a right triangle. It is written as \( ext{cos}( heta) \), where \( \theta \) is the angle of interest.

The cosine function helps in determining the relationship between an angle and the adjacent side divided by the hypotenuse of a right triangle. In the context of the unit circle, the cosine of an angle represents the x-coordinate of a point on the unit circle. This function exhibits specific periodic properties, such as being periodic over \(2\pi\), meaning it repeats the same values every \(2\pi\) radians.

The equation \( 2 \cos \theta - \sqrt{3} = 0 \) can be linked to finding exact angles, as seen by simplifying to \( \cos \theta = \frac{\sqrt{3}}{2} \). For angles \( \theta \), this value signifies a point where the cosine output matches the known value \( \frac{\sqrt{3}}{2} \) common in trigonometric ratios for special angles.

Unit Circle

The unit circle is a fundamental concept in trigonometry, serving as a valuable tool for understanding the behavior of trigonometric functions like sine and cosine. It is a circle with a radius of 1, centered at the origin (0,0) of a coordinate plane.

Every point on the unit circle corresponds to an angle \( \theta \), measured from the positive x-axis:

• The x-coordinate of a point is given by \( \cos \theta \).
• The y-coordinate is given by \( \sin \theta \).

The point \( \left( \cos \theta, \sin \theta \right) \) signifies the terminal point of the angle \( \theta \).

The unit circle not only helps solve equations such as \( \cos \theta = \frac{\sqrt{3}}{2} \) but also allows you to visually confirm solutions. As the cosine of an angle equals the x-coordinate, finding angles where \( \cos \theta = \frac{\sqrt{3}}{2} \) (at \( \frac{\pi}{6} \) and \( \frac{11\pi}{6} \) radians) becomes straightforward.

Angle Measurement

Angle measurement is central to trigonometry and is expressed in degrees or radians. Radians are a natural measure of angles related directly to the unit circle.

A full circle is \(2\pi\) radians, which equals 360 degrees. This conversion is key when working through problems dealing with angular concepts in trigonometry.

In solving \( 2 \cos \theta - \sqrt{3} = 0 \) for \( \theta \) in the range of \(0 \leq \theta < 2\pi \), it helps to understand where each angle falls within the circle.

• The first quadrant ranges from \(0\) to \(\frac{\pi}{2}\) radians, where both sine and cosine are positive.
• The second quadrant spans from \(\frac{\pi}{2}\) to \(\pi\), where sine remains positive and cosine becomes negative.

Understanding these quadrants aids in identifying that the cosine function maintains its unique characteristics like signs based on the quadrant, facilitating the solution of the given trigonometric equation.