Problem 80 - Algebra 2

Question

Solve \(2 \cos \theta=\sqrt{2}\) for \(0 \leq \theta<2 \pi\)

Step-by-Step Solution

Answer

The solutions to the given equation are \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{7\pi}{4}\)

Step 1: Solving 2 cos(θ) =√2

To find theta, first divide both sides of the equation by 2 to isolate the cosine term: \(\cos(\theta)= \frac{\sqrt{2}}{2}\)

Step 2: Obtain the angles

Cosine function takes the positive value for Quadrant I and Quadrant IV in the unit circle. So look for the cos^-1 of positive \(\frac{\sqrt{2}}{2}\), to find \(\theta\) values in [0, \(2\pi\) ) that satisfy the equation. \(\theta = \frac{\pi}{4}\), \(\theta = \frac{7\pi}{4}\)

Step 3: Check the results

Now, check that cosine of these angles is \(\frac{\sqrt{2}}{2}\), hence verifying our solutions are correct. We can substitute these values of \(\theta\) into our original equation \(2 \cos \theta=\sqrt{2}\) to ensure the equality stands true.

Key Concepts

Cosine Function

The cosine function, often denoted as \( \cos \theta \), is a fundamental trigonometric function that relates the angle \( \theta \) to the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. The cosine function is periodic with a period of \( 2\pi \), meaning that its values repeat every \( 2\pi \) radians.

Cosine values range from -1 to 1. This function is essential for solving trigonometric equations like \( 2 \cos \theta = \sqrt{2} \). By dividing both sides of this equation by 2, we isolate the cosine term: \( \cos(\theta) = \frac{\sqrt{2}}{2} \).

• This simplified form helps us determine specific angles where the cosine value matches \( \frac{\sqrt{2}}{2} \).
• Knowing the properties and behavior of the cosine function, including its range and periodicity, is crucial for understanding trigonometric equations.

Unit Circle

The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. It is a vital tool in trigonometry that helps visualize how angles correspond with coordinates on the circle. Each point on the unit circle can be expressed in terms of cosine and sine: \( (\cos \theta, \sin \theta) \).

The unit circle simplifies understanding how trigonometric functions behave, especially cosine and sine functions. At an angle \( \theta \), the x-coordinate is \( \cos(\theta) \) and the y-coordinate is \( \sin(\theta) \).

• This representation is instrumental in determining angles that yield specific cosine values, like \( \cos(\theta) = \frac{\sqrt{2}}{2} \).
• Using the unit circle, one can easily find the cosine values and corresponding angles by examining the positions along the circle.

Angle Measurement

Angle measurement is a critical concept in trigonometry. Angles can be measured in degrees or radians, with radians being the standard in many mathematical contexts. One complete circle is \( 360^\circ \) or \( 2\pi \) radians.

In trigonometric problems, angles are commonly given in radians. This measurement is based on the circumference of the unit circle, where \( 2\pi \) radians is the entire circumference. For example, a right angle is \( \frac{\pi}{2} \) radians.

• Converting between degrees and radians is often necessary when working with angles. Recall that \( \pi \) radians equal \( 180^\circ \).
• Understanding radians helps interpret trigonometric equations and solve problems involving the unit circle.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each representing a segment of \( 90^\circ \) or \( \frac{\pi}{2} \) radians. These quadrants help determine the sign and values of trigonometric functions like the cosine.

- **Quadrant I**: Both sine and cosine are positive.
- **Quadrant II**: Sine is positive, but cosine is negative.
- **Quadrant III**: Both sine and cosine are negative.
- **Quadrant IV**: Sine is negative, but cosine is positive.

When solving \( \cos(\theta) = \frac{\sqrt{2}}{2} \), the cosine is positive, indicating that \( \theta \) is in Quadrant I or IV. Therefore, the solutions \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{7\pi}{4} \) correspond to angles in these quadrants.

• The quadrant layout is crucial for determining valid angle solutions within a given interval, such as \([0, 2\pi)\).
• Recognizing which quadrants yield positive or negative trigonometric values aids in solving trig equations accurately.