Problem 15 - Algebra 2

Question

Find all solutions of each equation for the given interval. \(4 \cos ^{2} \theta=3 ; 0^{\circ} \leq \theta<360^{\circ}\)

Step-by-Step Solution

Answer

\(\theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ\).

Step 1: Rearrange the Equation

The given equation is \(4 \cos^2 \theta = 3\). Start by solving for \(\cos^2 \theta\) by dividing both sides by 4 to get \(\cos^2 \theta = \frac{3}{4}\).

Step 2: Take the Square Root

Next, take the square root of both sides of the equation to find \(\cos \theta\). This gives us \(\cos \theta = \pm \sqrt{\frac{3}{4}}\), which simplifies to \(\cos \theta = \pm \frac{\sqrt{3}}{2}\).

Step 3: Find General Solutions

For \(\cos \theta = \frac{\sqrt{3}}{2}\), \(\theta\) can be \(30^\circ\) or \(330^\circ\) as these angles correspond to the positive cosine value in the interval \(0^\circ \leq \theta < 360^\circ\). For \(\cos \theta = -\frac{\sqrt{3}}{2}\), \(\theta\) can be \(150^\circ\) or \(210^\circ\) as these angles correspond to the negative cosine value in the given interval.

Step 4: Validate Solutions within Interval

All angles found: \(30^\circ, 150^\circ, 210^\circ,\) and \(330^\circ\) fall within the required interval \(0^\circ \leq \theta < 360^\circ\). Thus, these are all valid solutions.

Key Concepts

Understanding the Cosine Function

The cosine function is one of the fundamental trigonometric functions, often represented by the symbol \( \cos \). It relates the angle in a right triangle to the ratio of the length of the adjacent side to the hypotenuse. When working with angles on the unit circle, the cosine function maps the angle to the x-coordinate of the corresponding point on the circle. This function is periodic, meaning it repeats itself after a fixed interval, specifically every \(360^{\circ}\) or \(2\pi\) radians.Cosine values range from -1 to 1. You can visualize these values on the unit circle, where:

• A cosine value of 1 corresponds to \(0^{\circ}\) or \(360^{\circ}\), situated at the point (1, 0) on the circle.
• A cosine value of 0 corresponds to \(90^{\circ}\) and \(270^{\circ}\), situated at the points (0, 1) and (0, -1) respectively.
• A cosine value of -1 corresponds to \(180^{\circ}\), situated at the point (-1, 0).

Recognizing these standard positions helps in quickly finding angle solutions to equations involving the cosine function.

Exploring Angle Intervals

An angle interval defines a specific range of angles to consider when solving trigonometric equations. In this exercise, the interval is \(0^{\circ} \leq \theta < 360^{\circ}\). This specifies that we're only interested in angles starting from \(0^{\circ}\) up to, but not including, \(360^{\circ}\).To understand angle intervals effectively:

• Recognize that angles wrapping around the circle beyond \(360^{\circ}\) are equivalent to angles within it. For instance, \(720^{\circ}\) is equivalent to \(0^{\circ}\) because it completes two full circles.
• The notation \([0^{\circ}, \; 360^{\circ})\) is common in math, meaning the interval includes \(0^{\circ}\) but not \(360^{\circ}\) itself.
• This ensures there are no overlapping values at \(0^{\circ}\) and \(360^{\circ}\), which represent the same point on the circle.

Working within defined intervals helps eliminate redundant or unnecessary solutions, keeping the focus on relevant angles for the specific problem.

Understanding Quadrants in Trigonometry

The concept of quadrants is crucial when solving trigonometric equations because it helps in determining the signs of the trigonometric functions. The coordinate plane is divided into four quadrants, each representing a sector of \(90^{\circ}\).Here's a quick guide to understanding quadrants:

• First Quadrant: Both sine and cosine values are positive here. Angles range from \(0^{\circ}\) to \(90^{\circ}\).
• Second Quadrant: Sine is positive, but cosine is negative. Angles range from \(90^{\circ}\) to \(180^{\circ}\).
• Third Quadrant: Both sine and cosine values are negative. Angles range from \(180^{\circ}\) to \(270^{\circ}\).
• Fourth Quadrant: Cosine is positive, but sine is negative. Angles range from \(270^{\circ}\) to \(360^{\circ}\).

In the given exercise, after finding the possible cosine values, the angle solutions are mapped to the respective quadrants where cosine has the specific value:

• \( \cos \theta = \frac{\sqrt{3}}{2} \) gives angles in the first and fourth quadrants, \(30^{\circ}\) and \(330^{\circ}\) respectively.
• \( \cos \theta = -\frac{\sqrt{3}}{2} \) gives angles in the second and third quadrants, \(150^{\circ}\) and \(210^{\circ}\) respectively.

Understanding quadrants helps students determine which angles align with the required cosine values within the specified interval.