Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ 3 \cos 2 \theta-4 \cos ^{2} \theta+2=0 $$

Step-by-Step Solution

Answer

The answers are \(\theta = 45^\circ, 135^\circ, 225^\circ, \text{and}\ 315^\circ\).

Step 1: Use a Double Angle Formula

We start with the given equation \(3 \cos 2\theta - 4 \cos^2 \theta + 2 = 0\). Recall the double angle formula: \( \cos 2\theta = 2\cos^2 \theta - 1\). Substitute this into the equation: \(3(2\cos^2 \theta - 1) - 4 \cos^2 \theta + 2 = 0\).

Step 2: Simplify the Equation

Simplify the expression: \(6\cos^2 \theta - 3 - 4 \cos^2 \theta + 2 = 0\). This simplifies further to \(2 \cos^2 \theta - 1 = 0\).

Step 3: Solve for \(\cos^2 \theta\)

Set the equation \(2 \cos^2 \theta - 1 = 0\) and solve for \(\cos^2 \theta\). We add 1 to both sides to obtain \(2 \cos^2 \theta = 1\), which simplifies to \(\cos^2 \theta = \frac{1}{2}\).

Step 4: Solve for \(\cos \theta\)

Take the square root of both sides to solve for \(\cos \theta\): \(\cos \theta = \pm \frac{1}{\sqrt{2}}\). This can also be expressed as \(\cos \theta = \pm \frac{\sqrt{2}}{2}\).

Step 5: Find Angles in the Given Interval

Identify the angles \(\theta\) such that \(\cos \theta = \frac{\sqrt{2}}{2}\) or \(\cos \theta = -\frac{\sqrt{2}}{2}\). For \(\cos \theta = \frac{\sqrt{2}}{2}\), the solutions are \(\theta = 45^\circ, 315^\circ\). For \(\cos \theta = -\frac{\sqrt{2}}{2}\), the solutions are \(\theta = 135^\circ, 225^\circ\).

Step 6: Summarize All Solutions

Combine all valid angles from the interval \(0^\circ \leq \theta \leq 360^\circ\): \(\theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ\).

Key Concepts

Double Angle Formula

The double angle formula is a crucial tool in trigonometry which helps simplify expressions where the angle is doubled. Specifically, the cosine double angle formula is written as:

• \( \cos 2\theta = 2\cos^2 \theta - 1 \)
• This formula relates the cosine of a double angle \(2\theta\) to \(\cos \theta\).

By applying this formula, complex trigonometric equations can be rewritten in terms of a single angle, making them easier to solve.
In the original exercise, we used \(\cos 2\theta = 2\cos^2 \theta - 1\) to transform the given equation into a simpler form.
This step simplifies the process, allowing us to focus on solving for \(\cos \theta\) itself.

Cosine Function

The cosine function, denoted as \(\cos \theta\), is one of the fundamental trigonometric functions.

• It measures the adjacent side over the hypotenuse in a right-angled triangle.
• For the unit circle, \(\cos \theta\) represents the horizontal coordinate at the angle \(\theta\).

In our problem, finding \(\cos \theta\) is key to solving the equation. The solution is expressed as \(\cos \theta = \pm \frac{\sqrt{2}}{2}\), which represents specific points on the unit circle.
These specific values occur at common angles such as \(45^\circ\) and \(135^\circ\) for \(\cos \theta = \frac{\sqrt{2}}{2}\), and similar angles for the negative value.
By knowing these, we're able to efficiently find all possible solutions within the given interval.

Solving Equations

Solving trigonometric equations requires a step-by-step approach to find the values of the variable that satisfy the equation. Here, the goal was to find the angles \(\theta\) within the interval \(0^\circ \leq \theta \leq 360^\circ\).
The process typically involves:

• Simplifying the equation as much as possible.
• Applying known trigonometric identities, like the double angle formula, to reduce complexity.
• Solving for fundamental trigonometric values, such as \(\cos \theta\).
• Mapping the trigonometric solution to possible angle values within the specified interval.

In the exercise, through simple algebraic steps, we isolated \(\cos^2 \theta\) and then solved for \(\cos \theta\), leading us to explore the angles within the designated range.

Intervals in Trigonometry

Understanding and working within specific intervals in trigonometry is critical. By restricting the solutions to a set range, typically from \(0^\circ\) to \(360^\circ\), or potentially \(0\) to \(2\pi\) radians, problems become more manageable.

• These intervals help identify which angle solutions are valid for a problem.
• In our case, the interval \(0^\circ \leq \theta \leq 360^\circ\) encompasses a full rotation around the unit circle.
• This is crucial for ensuring all possible angles that satisfy the equation are considered, without redundancy beyond a complete cycle.

The angles found were \(45^\circ, 135^\circ, 225^\circ,\) and \(315^\circ\), which all fell neatly within our given interval.
By becoming familiar with these boundary limitations, solving trigonometric equations can be approached with greater confidence and accuracy.