Problem 156 - Calculus Volume 1

Question

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 \(\theta = 2\pi/3\) and \(\theta = 4\pi/3\).

Step 1: Isolate Cosine Function

Start by isolating the cosine function in the equation. Given the equation \(1 + \cos \theta = \frac{1}{2}\), subtract 1 from both sides to get \(\cos \theta = \frac{1}{2} - 1\). Then simplify the right side to obtain \(\cos \theta = -\frac{1}{2}\).

Step 2: Determine Reference Angle

The cosine of an angle is \(-\frac{1}{2}\). To find the reference angle, consider where cosine has this value. For cosine, \(\cos \theta = \frac{1}{2}\) in quadrants I and IV. The reference angle is \(\pi/3\) (or 60 degrees) because \(\cos(\pi/3) = 1/2\).

Step 3: Identify Correct Quadrants for Cosine

Cosine is negative in quadrants II and III. Therefore, we need to find angles in these quadrants that have a reference angle of \(\pi/3\).

Step 4: Find Angles in Specified Quadrants

In quadrant II, the angle is \(\pi - \pi/3 = 2\pi/3\). In quadrant III, the angle is \(\pi + \pi/3 = 4\pi/3\). These are the solutions that satisfy \(\cos \theta = -\frac{1}{2}\) on the given interval.

Key Concepts

Cosine Function

The cosine function, denoted as \(\cos \theta\), is one of the fundamental trigonometric functions. It represents the ratio of the adjacent side to the hypotenuse in a right triangle. Cosine is an even function, which means it has symmetry about the y-axis. This symmetry plays a crucial role when solving equations.

• In the unit circle, the cosine of an angle represents the x-coordinate of a point.
• The function has a periodicity of \(2\pi\), repeating its values every full circle.
• Its range is from -1 to 1, meaning it never exceeds these values.

Understanding the properties of the cosine function is essential when solving trigonometric equations like \(1 + \cos \theta = \frac{1}{2}\). By subtracting 1, you isolate the cosine and determine it equals \(-\frac{1}{2}\). This means you need to find where on the unit circle the cosine value is \(-\frac{1}{2}\).

Reference Angle

The concept of a reference angle simplifies finding angles in any quadrant. A reference angle is the smallest angle between the terminal side of the given angle and the x-axis. It is always acute (less than \(90^\circ\) or \(\pi/2\) radians).

• Reference angles are critical because they help us determine related angles in different quadrants.
• They allow us to use known values, such as \(\cos(\pi/3) = 1/2\).

When the cosine equals \(-\frac{1}{2}\), its reference angle is \(\pi/3\) (or \(60^\circ\)). Even though the cosine value is negative, using the reference angle helps us find angles in the correct quadrants. Remember, the actual angle will differ based on which quadrant you are working in, as cosine signs change depending on that.

Quadrants

The unit circle is divided into four quadrants, each affecting the sign of the trigonometric functions.

• Quadrant I: Both sine and cosine values are positive.
• Quadrant II: Sine values are positive, cosine values are negative.
• Quadrant III: Both sine and cosine values are negative.
• Quadrant IV: Sine values are negative, cosine values are positive.

In our exercise, \(\cos \theta = -\frac{1}{2}\), meaning we focus on quadrants where cosine is negative, which are Quadrants II and III. In Quadrant II, the angle with a reference of \(\pi/3\) is \(\pi - \pi/3 = 2\pi/3\). In Quadrant III, it is \(\pi + \pi/3 = 4\pi/3\). Recognizing which quadrants fit these conditions is crucial for solving the equation accurately.