Problem 9 - Precalculus

Question

Find the exact values of the indicated trigonometric functions using the unit circle. $$(5 \pi)$$

Step-by-Step Solution

Answer

The exact values are \(\cos(5\pi) = -1\) and \(\sin(5\pi) = 0\).

Step 1: Understand the Angle

The given angle is \(5\pi\). Since \(2\pi\) is one full rotation around the unit circle, we have to simplify \(5\pi\) within the range of one rotation. This means we will calculate \(5\pi\) modulo \(2\pi\).

Step 2: Simplify the Angle

Perform the calculation: \(5\pi \mod 2\pi = 5\pi - 2\pi \times 2 = 5\pi - 4\pi = \pi\). Thus, \(5\pi\) is equivalent to \(\pi\) on the unit circle.

Step 3: Locate \(\pi\) on the Unit Circle

On the unit circle, \(\pi\) is located at the point \((-1, 0)\). It is directly opposite the positive x-axis, making half a full rotation from \(0\).

Step 4: Determine Trigonometric Values

The cosine of an angle corresponds to the x-coordinate, and the sine corresponds to the y-coordinate on the unit circle. At \(\pi\), these are \((-1, 0)\). Thus, \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles and sides of a triangle to each other. They are particularly important in the study of periodic phenomena and are used extensively in geometry, engineering, and physics. On the unit circle, these functions give the x and y coordinates for a point at a specific angle. The most basic trigonometric functions are sine (\(\sin\theta\)) and cosine (\(\cos\theta\)), and they can be used to derive other functions like tangent, cotangent, secant, and cosecant.

• Sine relates to the y-coordinate on the unit circle at any given angle.
• Cosine relates to the x-coordinate on the unit circle at that angle.

Understanding these basics helps in simplifying angles and calculating trigonometric values, which is essential for graphing functions and solving trigonometric equations.

Angle Simplification

Angle simplification is the process of reducing an angle to an equivalent angle within a specific range, commonly between 0 and \(2\pi\) for unit circle applications. This is important because the unit circle is a complete rotation of \(2\pi\) radians, equivalent to 360 degrees, so any angle can be translated to an equivalent within this range.
For instance, simplifying \(5\pi\) involves finding its equivalent angle within one rotation of the unit circle. This is done using modular arithmetic by calculating \(5\pi \mod 2\pi\), which brings us back to \(\pi\). This practice makes it easier to locate angles on the unit circle and compute their trigonometric functions effectively.

• Helps in identifying angles within one rotation of the unit circle.
• Utilizes modular arithmetic to bring angles into a manageable range.
• Simplifies computation of trigonometric functions for repeated cycles.

Cosine and Sine Values

The values of cosine and sine for an angle can easily be found using the unit circle. When dealing with an angle like \(\pi\) on the unit circle, it's crucial to determine these values for locating the point accurately.
At angle \(\pi\), the unit circle's corresponding point is \((-1,0)\). Here:

• The x-coordinate \(-1\) indicates the cosine value, \(\cos(\pi) = -1\).
• The y-coordinate \(0\) represents the sine value, \(\sin(\pi) = 0\).

These values are straightforward on the unit circle. Remember, the cosine corresponds to how far to the left or right the angle takes you on the x-axis, while the sine indicates the distance up or down the y-axis. This understanding helps in visualizing and calculating trigonometric values efficiently.