Problem 10 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the exact values for \(\theta\) in the interval \(0 \leq \theta<2 \pi\) $$ 5 \cos \theta+3=3 \cos \theta+5 $$

Step-by-Step Solution

Answer

\(\theta = 0\)

Step 1: Simplify the Equation

First, we start with the given equation: \[ 5\cos\theta + 3 = 3\cos\theta + 5 \] Subtract \(3\cos\theta\) from both sides to isolate terms involving \(\cos\theta\): \[ 5\cos\theta - 3\cos\theta + 3 = 5 \] This simplifies to: \[ 2\cos\theta + 3 = 5 \]

Step 2: Isolate Cosine Term

Now, subtract 3 from both sides of the equation to further isolate the cosine term:\[ 2\cos\theta = 5 - 3 \] Which simplifies to:\[ 2\cos\theta = 2 \]

Step 3: Solve for Cosine

Divide both sides by 2 to solve for \(\cos\theta\):\[ \cos\theta = \frac{2}{2} \] Thus, we find:\[ \cos\theta = 1 \]

Step 4: Find Values of \(\theta\)

Recall that \(\cos\theta = 1\) at specific points in the unit circle. Specifically, it occurs when \(\theta = 0\) within the interval \(0 \leq \theta < 2\pi\). Because the interval ends just before \(2\pi\), we do not include \(2\pi\) itself.

Key Concepts

Unit Circle

The unit circle is an essential concept in trigonometry. It is a circle centered at the origin of a coordinate plane with a radius of 1. Understanding the unit circle helps us easily identify the values of sine and cosine for specific angles.

• The positive x-axis represents 0 radians or 0 degrees.
• A full rotation around the circle means 2π radians or 360 degrees.
• At the top, where it intersects the y-axis, the angle is π/2 radians (90 degrees).
• The leftmost point equates to π radians (180 degrees).
• At the bottom, it’s 3π/2 radians (270 degrees).

Using these standard positions on the unit circle, we can deduce important trigonometric values, such as \( \cos \theta = 1 \) only at \( \theta = 0 \) or multiples of 2π.

Cosine Function

The cosine function is a fundamental trigonometric function, defined from the unit circle. It maps the angle \( \theta \) to the x-coordinate of the point on the unit circle.

• This means for each \( \theta \) on the unit circle, \( \cos \theta \) represents the horizontal distance from the origin.
• The value of \( \cos \theta \) ranges from -1 to 1.

Key characteristics include:

• Periodicity: \( \cos \theta \) repeats its values every 2π.
• Symmetry: It is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• Graphically: The waveform of \( \cos \theta \) forms a smooth, continuous curve known as a cosine wave.

In this exercise, determining \( \cos \theta = 1 \) plays a crucial role in finding the angle \( \theta \).

Solving Equations

Solving equations in trigonometry requires isolating the trigonometric functions and using known values from concepts like the unit circle. Here's a general approach:1. **Simplify the Equation** The first step involves simplifying the equation as much as possible, which often means regrouping similar terms.
For instance, starting with \( 5\cos\theta + 3 = 3\cos\theta + 5 \), subtracting \( 3\cos\theta \) from both sides simplifies the problem to: \( 2\cos\theta = 2 \).2. **Isolate the Trigonometric Function** Focus on isolating the trigonometric function by getting it alone on one side of the equation.
For example, dividing by the coefficient of \( \cos\theta \) leads to \( \cos\theta = 1 \).3. **Find Specific Angle Values** Use trigonometric properties or the unit circle to find possible angle solutions.
Given \( \cos\theta = 1 \), recognizing that \( \theta = 0 \) is a solution within the given interval \( 0 \leq \theta < 2\pi \).

Angle in Radians

Radians are a unit of angle measure based on the radius of the circle. The entire circle is 2π radians, correlating to 360 degrees. Understanding angles in radians is crucial for solving trigonometric equations as they are often more useful in calculations than degrees.

• One radian is the angle formed when the arc length equals the radius length.
• Key conversions include: π radians = 180 degrees and \( \frac{\pi}{2} \) radians = 90 degrees.

In many trigonometric problems, angles are given in radians, providing smoother calculations. For example, in the problem presented, \( \theta = 0 \) is an angle in radians that lies within the interval from 0 to \( 2\pi \, \) representing one complete counterclockwise rotation around the unit circle.