Problem 50 - Algebra 2

Question

Find the complete solution in radians of each equation. $$ \sin \theta \cos \theta=\frac{1}{2} \cos \theta $$

Step-by-Step Solution

Answer

The complete solution in radians is \(\theta = \frac{\pi}{6}\), \(\theta = \frac{5\pi}{6}\).

Step 1: Simplify the Equation

First, we will simplify the equation by dividing both sides by \(\cos \theta\) we have : \(\sin \theta = \frac{1}{2}\).

Step 2: Solving for \(\theta\)

We want to find the solutions in [0, 2\(\pi\)]. For \(\sin x = \frac{1}{2}\), we could use a unit circle or common values to determine \(\theta = \frac{\pi}{6}\) or \(\theta = \frac{5\pi}{6}\).

Step 3: Checking the Solution

Check the solutions in the original equation to ensure they are valid. In this case, both \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\) satisfy the original equation.

Key Concepts

Solving Equations

Solving equations involving trigonometric functions requires a firm grasp on algebraic manipulation and trigonometric identities. To solve the given equation \( \sin \theta \cos \theta = \frac{1}{2} \cos \theta \), the first step involves simplifying it by dividing both sides by \( \cos \theta \). This assumes that \( \cos \theta eq 0 \). So, the equation simplifies to \( \sin \theta = \frac{1}{2} \).
This reduction transforms the trigonometric equation into a basic trigonometric function problem. Solving allows us to find all possible angles that result in the sine of the angle equaling \( \frac{1}{2} \). Once simplified, solving such equations involves referencing known angle values or using a unit circle which provides a visual representation of sine and cosine values for different angles.

Unit Circle

The unit circle is a fundamental tool in trigonometry for understanding circular functions. It is a circle with a radius of one centered at the origin of a coordinate plane. The x- and y-coordinates of any point on this circle correspond to the cosine and sine values of an angle, respectively.
When solving the equation \( \sin \theta = \frac{1}{2} \), the unit circle helps identify the angles where the sine function equals \( \frac{1}{2} \). On the unit circle, these angles commonly include \( \frac{\pi}{6} \) and \( \frac{5\pi}{6} \), corresponding to 30° and 150° in degree measure. The unit circle method is beneficial as it directly relates trigonometric functions to geometric representations, providing an intuitive way to solve equations by visualization.

Radian Measure

Radian measure is an essential concept in trigonometry and calculus that defines angles in terms of a radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians offer a more natural approach as they relate angles directly to the length of arc on a unit circle.
To understand radians, consider that the total circumference of a unit circle is \( 2\pi \). Therefore, one full revolution around the circle is \( 2\pi \) radians. In the context of solving the equation \( \sin \theta = \frac{1}{2} \), finding \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{5\pi}{6} \) in radians is part of this approach. Radians provide a more seamless way to perform calculations in mathematical contexts because they inherently conform to the dimensionalities involved in calculus and physics.