Problem 7 - 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. $$2 \cos x+5=6$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\).

Step 1: Isolate the Trigonometric Function

We start with the equation: \[ 2 \cos x + 5 = 6 \]First, subtract 5 from both sides to isolate the term with cosine:\[ 2 \cos x = 6 - 5 \]This simplifies to:\[ 2 \cos x = 1 \]

Step 2: Solve for Cosine Value

Next, divide both sides by 2 to solve for \(\cos x\):\[ \cos x = \frac{1}{2} \]

Step 3: Find General Solutions

We need to determine the values of \(x\) in the range \([0, 2\pi)\) that make \(\cos x = \frac{1}{2}\).\The cosine function equals \(\frac{1}{2}\) at specific standard angles in radians: \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\).

Step 4: Verify Solutions in Interval

Ensure the solutions found in Step 3 lie within the interval \([0, 2\pi)\).\Both \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\) are in this interval, so these are both valid solutions.

Key Concepts

Interval Notation

Interval notation is a way to describe a set of numbers between two endpoints. It uses brackets [ ] and parentheses ( ) to indicate whether the endpoints are included or not. For example:

• "[a, b]" means all numbers between a and b, including both a and b.
• "(a, b)" means all numbers between a and b, but not including a or b.
• When using "[a, b)", it includes a but not b, making it a half-open interval.

In trigonometric equations, interval notation helps us specify the range of solutions we are interested in.
This is important because trigonometric functions are periodic, meaning they repeat their values over certain intervals.

Radians

Radians are a unit of angular measurement used in trigonometry. Unlike degrees, which divide a full circle into 360 parts, radians relate to the π constant, where a full circle is 2π radians.

• One full circle = 2π radians
• Half a circle or a semicircle is π radians
• A right angle is \( \frac{\pi}{2} \) radians

Radians are effective for mathematical calculations involving periodic functions because they simplify many formulas, making them more concise. They are used in calculus and are preferred in higher-level mathematics due to their natural emergence in various formulas and functions.

Cosine Function

The cosine function is a fundamental trigonometric function represented as \( \cos x \). It takes an angle as input and returns a ratio, which is the adjacent side over the hypotenuse in a right triangle.

• The function is periodic with a period of \( 2\pi \), meaning it repeats every \( 2\pi \).
• It ranges between -1 and 1 for all input values.
• Important values occur at standard angles like \( \frac{\pi}{3} \) where \( \cos(\frac{\pi}{3}) = \frac{1}{2} \).

When solving an equation involving \( \cos x\), such as in our exercise, identifying this periodic nature helps in finding all possible solutions within a specified interval.

General Solutions

General solutions refer to the complete set of solutions to a trigonometric equation over all possible intervals. Since trigonometric functions are periodic, they repeat their values, meaning a solution will occur in multiple periods unless restricted to a certain interval.

• When you find a solution to \( \cos x = \frac{1}{2} \), it initially leads you to solutions like \( \frac{\pi}{3} \) and \( \frac{5\pi}{3} \) within \([0, 2\pi)\).
• Using these, general solutions are represented as: \( x = \frac{\pi}{3} + 2k\pi \) and \( x = \frac{5\pi}{3} + 2k\pi \) for any integer \( k \).
• These equations indicate that every full cycle ( \( 2\pi \)) repeats the solutions found.

Understanding general solutions is crucial for identifying how often solutions recur over extended intervals, which is essential in advanced math contexts.