Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \cos x+1=0$$

Step-by-Step Solution

Answer

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

Step 1: Isolate the Trigonometric Function

The equation given is \(2 \cos x + 1 = 0\). To isolate the trigonometric function \(\cos x\), subtract 1 from both sides to get \(2 \cos x = -1\).

Step 2: Solve for the Function Value

Next, divide both sides of the equation by 2 to solve for \(\cos x\). This will give \(\cos x = -\frac{1}{2}\).

Step 3: Find the Angle(s) Satisfying the Equation

We want to find all values of \(x\) that satisfy \(\cos x = -\frac{1}{2}\) in the interval \([0, 2\pi)\). The cosine function equals \(-\frac{1}{2}\) at angles \(x = \frac{2\pi}{3}\) and \(x = \frac{4\pi}{3}\). These values correspond to the second and third quadrants where the cosine is negative.

Key Concepts

Cosine Function

In trigonometry, the cosine function, often written as \( \cos \theta \), is essential for analyzing right triangles and periodic patterns. It relates the ratio of the adjacent side to the hypotenuse of a right-angled triangle. Like other trigonometric functions, cosine is periodic, repeating its values in every interval of \( 2\pi \), making it ideal for modeling cyclical phenomena.

Key characteristics of the cosine function include:

• It ranges between -1 and 1.
• The graph of \( \cos x \) is wave-like, starting at \( 1 \) when \( x = 0 \).
• It has a period of \( 2\pi \), meaning the graph repeats every \( 2\pi \) unit.
• The function is symmetric about the y-axis (even function).

These properties are useful when solving trigonometric equations, where recognizing cosine values can help determine the solutions.

Quadrants in Trigonometry

The unit circle is a fundamental concept in trigonometry, divided into four quadrants, each representing a specific range of angles.

Quadrants are vital when interpreting the signs of trigonometric functions at different angles.

• First Quadrant: Both sine and cosine values are positive.
• Second Quadrant: Sine is positive, but cosine is negative. For example, \( \cos x = -\frac{1}{2} \) can be found here at \( x = \frac{2\pi}{3} \).
• Third Quadrant: Both sine and cosine are negative. Another solution for \( \cos x = -\frac{1}{2} \) occurs here at \( x = \frac{4\pi}{3} \).
• Fourth Quadrant: Cosine is positive, sine is negative.

By understanding these quadrants, students can determine which angles satisfy a given trigonometric equation, as certain values only occur in specific quadrants.

Solving Techniques for Equations

Solving trigonometric equations involves a series of efficient steps to identify solutions accurately:

• Isolate the Trigonometric Function: Start by simplifying the equation to focus on the trigonometric part. For example, in the equation \( 2 \cos x + 1 = 0 \), subtract 1 and divide by 2 to isolate \( \cos x \), resulting in \( \cos x = -\frac{1}{2} \).
• Use Known Values: Use known values of the trigonometric functions, from tables or the unit circle, to determine angles satisfying the equation.
• Check Quadrants: Determine which quadrants the solution angles lie in, based on the function's positivity or negativity in those regions. In this instance, \( \cos x = -\frac{1}{2} \) results in angles \( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \).
• Verify Solutions: Verify that the solutions fall within the specified interval, \([0, 2\pi)\).

These techniques ensure that all solutions are identified correctly, and critical logical steps aren't missed, ensuring thorough understanding and application of trigonometry.