Problem 26 - Algebra 2 and Trigonometry

Question

In \(25-28,\) find, to the nearest hundredth, the radian measures of all \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that make the equation true. $$ 9-2 \cos \theta=8-4 \cos \theta $$

Step-by-Step Solution

Answer

The radian measures are approximately 1.05 and 5.24.

Step 1: Set Up the Equation

Start with the equation given in the problem:\[ 9 - 2 \cos \theta = 8 - 4 \cos \theta \] Your task is to find the values of \(\theta\) between \(0\) and \(2\pi\) that satisfy this equation.

Step 2: Simplify the Equation

To simplify, first move all terms involving \(\cos \theta\) to one side and the constant terms to the other side. This yields:\[ 2 \cos \theta - 4 \cos \theta = 8 - 9 \]Simplified further, this gives:\[ -2 \cos \theta = -1 \]

Step 3: Solve for \(\cos \theta\)

Divide both sides of the equation by \(-2\) to solve for \(\cos \theta\):\[ \cos \theta = \frac{1}{2} \]

Step 4: Find \(\theta\) Values in the Specified Interval

Identify all angles \(\theta\) within the interval \([0, 2\pi)\) that have a cosine of \(\frac{1}{2}\). These angles are:\[ \theta = \frac{\pi}{3} \quad \text{and} \quad \theta = \frac{5\pi}{3} \] These are the angles where the cosine is \(\frac{1}{2}\) in the specified interval \([0, 2\pi)\).

Step 5: Round the Answers to the Nearest Hundredth

Convert the angles in radians to decimals and round to the nearest hundredth:- \( \frac{\pi}{3} \approx 1.05 \)- \( \frac{5\pi}{3} \approx 5.24 \)Thus, the solutions are approximately \(1.05\) and \(5.24\).

Key Concepts

Radian Measure

Unlike degrees, which are more intuitive and often used in everyday scenarios, radians provide a mathematical purity. In essence, radians relate the angle to the radius of a circle. One complete revolution around a circle corresponds to an angle of \(2\pi\) radians, equivalently \(360^\circ\). This is key when working with trigonometric equations on more advanced levels.

• One radian is the measure of an angle created when the length of the arc is equal to the radius of the circle.
• To convert from degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\).
• Similarly, to convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).

For example, in this problem, you are asked to find angles in radians that satisfy the equation within the interval \([0, 2\pi)\). Keeping this in mind aids in ensuring all solutions align with the radians measure.

Cosine Function

The cosine function, one of the core trigonometric functions, is quite valuable in mathematics, especially within the realm of trigonometric equations. It is defined using a right triangle or the unit circle approach.

• In the context of a right triangle, cosine of an angle \(\theta\) is the ratio of the adjacent side to the hypotenuse.
• Regarding the unit circle, \( \cos \theta \) represents the x-coordinate of the point where the terminal side of an angle \(\theta\) intersects the circle.

In solving the original equation, you simplified to find \( \cos \theta = \frac{1}{2} \). Knowing cosine values and their corresponding angles in standard positions on the unit circle helps solve such equations efficiently.

Interval Notation

Interval notation is a handy tool in representing sets of numbers, particularly useful in specifying domains and ranges in mathematics.

• The notation uses brackets \([\ and \)] to denote closed intervals, where endpoints are included, and parentheses \((\ and \)) for open intervals, meaning endpoints are not included.
• For instance, \([0, 2\pi)\) means all numbers starting from 0 up to but not including \(2\pi\).

In trigonometric equations like the one solved here, ensuring you're considering solutions within the given interval is essential. Because periodic functions repeat, restricting solutions to an interval such as \([0, 2\pi)\) simplifies identification of all unique solutions.