Problem 156 - Calculus Volume 1

Question

For the following exercises, solve the trigonometric equations on the interval \(0 \leq \theta<2 \pi.\) $$1+\cos \theta=\frac{1}{2}$$

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{2\pi}{3}\) and \(\theta = \frac{4\pi}{3}\).

Step 1: Isolate the Cosine Function

The given equation is \(1 + \cos \theta = \frac{1}{2}\). To solve for \( \cos \theta \), subtract 1 from both sides of the equation: \( \cos \theta = \frac{1}{2} - 1 \).

Step 2: Simplify the Equation

Simplify the right side of the equation: \( \cos \theta = -\frac{1}{2} \).

Step 3: Determine Relevant Angles for Cosine

Recall that \( \cos \theta = -\frac{1}{2} \) corresponds to angles in the second and third quadrants. The reference angle for \( \cos \theta = \frac{1}{2} \) is \( \frac{\pi}{3} \).

Step 4: Calculate Angles in Relevant Quadrants

For the second quadrant, the angle is \( \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \). For the third quadrant, the angle is \( \theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \).

Step 5: Verify the Solution

Check whether both \( \frac{2\pi}{3} \) and \( \frac{4\pi}{3} \) satisfy the original equation: \(1 + \cos \frac{2\pi}{3} = 1 - \frac{1}{2} = \frac{1}{2} \) and \(1 + \cos \frac{4\pi}{3} = 1 - \frac{1}{2} = \frac{1}{2}\). Both angles satisfy the equation.

Key Concepts

Cosine Function

The cosine function, noted as \( \cos \theta \), is a fundamental concept in trigonometry. It relates to the x-coordinate on the unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane. This function is periodic, meaning it repeats its values at regular intervals, specifically every \(2\pi\). It only ranges from -1 to 1.
For the cosine function, the complete cycle of angles through which it passes is known as its period. This cycle contains one full wave, starting and ending at the same function value. Whenever you work with equations involving \( \cos \theta \), consider the positive and negative values of cosine:

• \( \cos \theta = 1 \) at \(0\) and \(2\pi\) (completes one full circle).
• \( \cos \theta = 0 \) at \( \pi/2 \) and \(3\pi/2 \).
• \( \cos \theta = -1 \) at \( \pi \).
• \( \cos \theta = \frac{1}{2} \) or \(-\frac{1}{2}\) leads to specific reference angles dependent on the quadrant.

This function is crucial in solving trigonometric equations, as knowing the cosine's positive and negative ranges helps determine possible angle solutions in various quadrants.

Reference Angles

Reference angles are a powerful tool in trigonometry that help us find actual angles based on a specific known value of the cosine. They are always positive and are the smallest angle between the terminal side of the given angle and the horizontal axis.
For example, consider a situation where \( \cos \theta = -\frac{1}{2} \). The reference angle for the positive value \( \cos \theta = \frac{1}{2} \) is \( \frac{\pi}{3} \). This angle is common, and knowing it helps quickly determine our target angles based on the given cosine value. Depending on where within the coordinate circle (known as quadrants) the angle lies, different calculations are used to find it:

• In the second quadrant, \( \theta = \pi - \text{reference angle} \).
• In the third quadrant, \( \theta = \pi + \text{reference angle} \).

Reference angles essentially simplify the process of solving trigonometric equations, making them invaluable in both straightforward and complex calculations.

Unit Circle

The unit circle is a central concept in trigonometry, representing a circle with a radius of exactly one, centered at the origin \((0,0)\) of a coordinate system. This circle provides a straightforward way to visually understand trigonometric functions and their relationships.
When we talk about angles along the unit circle, they are measured in radians. Here's how it functions in relation to the cosine function:

• Along the x-axis, \( \cos(\theta) \) aligns with the x-coordinate, varying from 1 (at 0 radians) through 0 (at \( \pi/2 \)), down to -1 (at \( \pi \)), back to 0 (at \( 3\pi/2 \)) and returning to 1 (at \( 2\pi \)).
• The circle is divided into four quadrants, each affecting the sign of trigonometric functions differently.

The utility of the unit circle is its simplicity in determining the trigonometric values of many angles, making it a foundational tool for solving trigonometric equations.Its role in problems similar to the one given lies in recognizing angles like \( \theta = \frac{2\pi}{3} \) and \( \theta = \frac{4\pi}{3} \), where \( \cos(\theta)=-\frac{1}{2} \). These are derived using the circle's quadrant rules and reference angles.