Problem 2 - Precalculus

Question

Find the exact values of the indicated trigonometric functions using the unit circle. $$\cos \left(\frac{5 \pi}{3}\right)$$

Step-by-Step Solution

Answer

\( \cos \left(\frac{5\pi}{3}\right) = \frac{1}{2} \)

Step 1: Identify the Reference Angle

The angle given is \( \frac{5\pi}{3} \). First, identify its reference angle within the unit circle. Recall that the unit circle covers angles from 0 to \( 2\pi \). Subtract \( 2\pi \) from \( \frac{5\pi}{3} \) if necessary to find an equivalent angle within this interval. Since \( \frac{5\pi}{3} > \pi \), it is still within the valid range \( 0 < \theta \leq 2\pi \). Thus, \( \frac{5\pi}{3} - 2\pi \) gives an equivalent angle, which is \( \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3} \). This angle is negative, so we add \( 2\pi \) to get \( \frac{5\pi}{3} \) itself is the angle considered.

Step 2: Locate Known Angle on Unit Circle

\( \frac{5\pi}{3} \) is located in the fourth quadrant of the unit circle. The reference angle is \( 2\pi - \frac{5\pi}{3} = \frac{\pi}{3} \). In the unit circle, common angles should be memorized: \( \frac{\pi}{3} \) has a cosine value of \( \frac{1}{2} \).

Step 3: Determine Correct Sign for Quadrant

In the fourth quadrant, the cosine value is positive, as both the x-coordinates (cosine values) in this quadrant are positive. Hence, the cosine of \( \frac{5\pi}{3} \) is positive.

Step 4: Confirm the Cosine Value

Since the reference angle \( \frac{\pi}{3} \) has a cosine of \( \frac{1}{2} \), and taking the positive sign in the fourth quadrant into account, we confirm that \( \cos \left(\frac{5\pi}{3}\right) = \frac{1}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are the mathematical functions of angles that relate the angles of a triangle to the lengths of its sides. When it comes to the unit circle, which is a circle with a radius of one, the trigonometric functions can be used to determine the coordinates of a point on the circle's circumference.

• Sine (\(\sin \theta\)): This function gives the y-coordinate of a point on the unit circle.
• Cosine (\(\cos \theta\)): This function provides the x-coordinate of a point on the unit circle.
• Tangent (\(\tan \theta\)): It is the ratio of sine to cosine, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Trigonometric functions are periodic, which means they repeat their values in regular intervals. Understanding the unit circle and these functions is crucial for solving equations and understanding wave patterns.

Reference Angle

The reference angle of a given angle is the smallest positive angle that can represent it in standard position. It is always between 0 and \(\frac{\pi}{2}\), or 0 and 90 degrees. Reference angles are helpful because they simplify the calculation of trigonometric functions.
To find a reference angle:

• If your angle is in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from \(\pi\) or 180 degrees.
• For the third quadrant, subtract \(\pi\) or 180 degrees from the angle.
• In the fourth quadrant, subtract the angle from \(2\pi\) or 360 degrees.

In our example with \(\frac{5\pi}{3}\), we find the reference angle by calculating \(2\pi - \frac{5\pi}{3} = \frac{\pi}{3}\), making problem-solving more straightforward.

Quadrants

The unit circle is divided into four quadrants, each one representing a range of angles:

• Quadrant I: Angles from 0 to \(\frac{\pi}{2}\) (0 to 90 degrees). Here, both sine and cosine values are positive.
• Quadrant II: Angles from \(\frac{\pi}{2}\) to \(\pi\) (90 to 180 degrees). Sine values are positive, cosine values are negative.
• Quadrant III: Angles from \(\pi\) to \(\frac{3\pi}{2}\) (180 to 270 degrees). Both sine and cosine values are negative.
• Quadrant IV: Angles from \(\frac{3\pi}{2}\) to \(2\pi\) (270 to 360 degrees). Cosine values are positive, sine values are negative.

In our example, \(\frac{5\pi}{3}\) is located in the fourth quadrant, where cosine values are positive.

Exact Values

In trigonometry, exact values refer to the precise values of trigonometric functions at common angles on the unit circle. These angles include \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2},\) etc., and their corresponding points.
For each fundamental angle, memorizing the sine, cosine, and tangent values helps quickly solve trigonometric problems. For example, for \(\frac{\pi}{3}\):

• \(\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\)
• \(\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}\)
• \(\tan \left(\frac{\pi}{3}\right) = \sqrt{3}\)

For \(\frac{5\pi}{3}\), the cosine value is thus \(\frac{1}{2}\) following the same logic as the reference angle and quadrant analysis.