Problem 35 - Precalculus

Question

Use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$\cos \theta=\frac{\sqrt{3}}{2}, 0 \leq \theta \leq 2 \pi$$

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{11\pi}{6}\).

Step 1: Understand the Problem

We are given the equation \(\cos \theta = \frac{\sqrt{3}}{2}\) and we need to find the values of \(\theta\) within the interval \([0, 2\pi]\). This means we need to determine where the cosine function equals \(\frac{\sqrt{3}}{2}\) on the unit circle.

Step 2: Recall the Unit Circle

On the unit circle, the cosine of an angle \(\theta\) represents the x-coordinate of the point where the terminal side of \(\theta\) intersects the circle. Knowing that \(\cos \theta = \frac{\sqrt{3}}{2}\), we look for corresponding angles on the unit circle.

Step 3: Identify the Standard Angles

The values of \(\cos \theta = \frac{\sqrt{3}}{2}\) are known at the angles \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{11\pi}{6}\) within the interval \([0, 2\pi]\).

Step 4: Consider the Symmetry

Since cosine is positive in the first and fourth quadrants and \(\cos \theta = \frac{\sqrt{3}}{2}\) at the reference angle \(\frac{\pi}{6}\), the angles \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{11\pi}{6}\) meet the condition efficiently.

Step 5: Check Validity of Solutions

Verify that both \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{11\pi}{6}\) give \(\cos \theta = \frac{\sqrt{3}}{2}\). Since they are within our interval, these are valid solutions.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. On the unit circle, which is a circle with radius 1 centered at the origin, the value of cosine represents the x-coordinate of a point.
This means for any angle \( \theta \) measured from the positive x-axis, the length of the horizontal leg of the right triangle formed from the radius is given by cosine.

• The cosine of an angle can have values ranging from -1 to 1.
• Angles in the unit circle are typically measured in radians.
• On the unit circle, cosine is positive when the x-coordinate is positive, hence in the first and fourth quadrants.

Understanding these properties helps quickly find \( \theta \) values with a known cosine.

Trigonometric Angles

Trigonometric angles are specific angles that are commonly used in trigonometry and they can take values in degrees or radians. These angles are fundamentally important in analyzing the unit circle. In trigonometry, commonly used trigonometric angles include \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \) and their counterparts in other quadrants.

• These angles yield known sine and cosine values, allowing quick calculations.
• They repeat every \( 2\pi \) or 360 degrees, as the circle completes a full rotation.

For instance, the cosine of \( \theta = \frac{\pi}{6} \) is known to be \( \frac{\sqrt{3}}{2} \, \) which occurs in the first and fourth quadrants on the unit circle. This simplifies the process of solving trigonometric equations by providing exact values for certain angles.

Reference Angle

The concept of a reference angle is crucial for understanding the positions of angles on the unit circle. A reference angle is the smallest angle made by the terminal side of a given angle and the x-axis. It is always positive and lies between 0 and \( \frac{\pi}{2} \) radians.

• Reference angles allow us to express an angle's trigonometric function in terms of acute angles.
• They simplify finding trigonometric function values for angles outside of the first quadrant.
• For example, an angle \( \theta = \frac{11\pi}{6} \) in the fourth quadrant has a reference angle of \( \frac{\pi}{6} \).

By using reference angles, one can quickly deduce trigonometric values in different quadrants based on their behavior in the first quadrant, streamlining solving trigonometric equations.