Problem 1 - Algebra 2

Question

Find all solutions of each equation for the given interval. \(4 \cos ^{2} \theta=1 ; 0^{\circ} \leq \theta<360^{\circ}\)

Step-by-Step Solution

Answer

\( \theta = 60^{\circ}, 120^{\circ}, 240^{\circ}, 300^{\circ} \)

Step 1: Simplify the Equation

The given equation is \( 4 \cos^2 \theta = 1 \). Divide both sides by 4 to isolate \( \cos^2 \theta \). This gives us:\[ \cos^2 \theta = \frac{1}{4} \]

Step 2: Solve for Cosine

Take the square root of both sides to solve for \( \cos \theta \):\[ \cos \theta = \pm \frac{1}{2} \]This means that \( \cos \theta \) can be \( \frac{1}{2} \) or \( -\frac{1}{2} \).

Step 3: Find Solutions for \( \cos \theta = \frac{1}{2} \)

Recall that \( \cos \theta = \frac{1}{2} \) occurs at angles \( \theta = 60^{\circ} \) and \( \theta = 300^{\circ} \) within the interval \( 0^{\circ} \leq \theta < 360^{\circ} \).

Step 4: Find Solutions for \( \cos \theta = -\frac{1}{2} \)

Similarly, \( \cos \theta = -\frac{1}{2} \) occurs at angles \( \theta = 120^{\circ} \) and \( \theta = 240^{\circ} \) within the interval \( 0^{\circ} \leq \theta < 360^{\circ} \).

Step 5: Compile All Solutions

Combine all solutions from the previous steps. The solutions to the equation \( 4 \cos^2 \theta = 1 \) in the given interval are \( \theta = 60^{\circ}, 120^{\circ}, 240^{\circ}, 300^{\circ} \).

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \theta \), is a fundamental trigonometric function. It relates the angle \( \theta \) of a right-angled triangle to the ratio of the adjacent side to the hypotenuse. Understanding the behavior of the cosine function is crucial for solving trigonometric equations.

• The cosine function varies between -1 and 1 as the angle \( \theta \) changes.
• It is periodic with a period of \( 360^{\circ} \) (or \( 2\pi \) radians).
• At \( \theta = 0^{\circ} \), the cosine value is 1. It decreases to 0 at \( \theta = 90^{\circ} \), -1 at \( 180^{\circ} \), 0 again at \( 270^{\circ} \), and back to 1 at \( 360^{\circ} \).

Such understanding of the cosine wave helps in recognizing the angles where specific values occur, as seen in the given exercise.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angles. These identities can simplify the process of solving trigonometric equations.

• One of the most essential identities is the Pythagorean identity: \( \cos^2 \theta + \sin^2 \theta = 1 \).
• This can be rearranged, as in the exercise, to isolate \( \cos^2 \theta \) and solve for cosine.
• The double-angle identities and sum-to-product identities are also pivotal in various problems.

By using these identities, equations involving squares of cosine or sine can be simplified, facilitating the process of finding angle solutions.

Solving Equations

Solving trigonometric equations involves finding all the angles that satisfy the equation within a given interval. This process can include several steps:

• First, simplify the equation using algebraic techniques. For example, dividing both sides by a constant to isolate the trigonometric function.
• Second, use trigonometric identities to further break down the equation.
• Finally, solve for the specific trigonometric function, as with finding that \( \cos \theta = \pm \frac{1}{2} \).

Each of these steps helps in narrowing down the possible angle solutions, which can then be identified precisely.

Angle Solutions

The final step in solving a trigonometric equation is to find all angle solutions within a specified interval. This requires knowledge of when particular trigonometric values occur:

• Given that \( \cos \theta = \frac{1}{2} \), the angles are \( 60^{\circ} \) or \( 300^{\circ} \).
• For \( \cos \theta = -\frac{1}{2} \), the angles are \( 120^{\circ} \) or \( 240^{\circ} \).

In trigonometry, knowing the range of angles for given cosine values is essential. The angles repeat every \( 360^{\circ} \), allowing predictions of solutions over larger intervals. Successfully determining these angles, as in the exercise, relies on understanding both the function's periodicity and its specific value points.