Problem 7 - Precalculus

Question

Find the exact value of the cosine and sine of the given angle. $$ \theta=\pi $$

Step-by-Step Solution

Answer

\( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \).

Step 1: Understanding the Angle

First, let's understand the given angle \( \theta = \pi \) radians. In the unit circle, \( \pi \) radians corresponds to half the circle, which places the angle on the negative x-axis. The degree equivalent of \( \pi \) radians is \( 180^\circ \).

Step 2: Analyzing the Reference Point on the Unit Circle

On the unit circle, each angle corresponds to a point \((x, y)\). For \( \theta = \pi \), this point is located directly on the negative x-axis, and its coordinates are \((-1, 0)\).

Step 3: Applying the Definitions of Sine and Cosine

The cosine of an angle \( \theta \) corresponds to the x-coordinate of the point on the unit circle, while the sine of \( \theta \) corresponds to the y-coordinate. At \( \theta = \pi \), the x-coordinate is \(-1\) and the y-coordinate is \(0\).

Step 4: Writing the Exact Values

Therefore, \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \) based on the coordinates of the point on the unit circle corresponding to \( \theta = \pi \).

Key Concepts

Understanding Radian Measure

Radian measure is a way to express angles using the natural geometry of circles. One radian is defined based on the radius of the circle. A full circle has an angle of \[2\pi\] radians. This arises because the circumference of a circle, which is the distance around it, is \[2\pi\] times the radius. Unlike degrees, which divide the circle into 360 equal parts, radians relate directly to circle geometry. This makes radian measure especially useful in advanced mathematics and physics.

To convert between degrees and radians, remember that \[180^\circ\] is equivalent to \(\pi\) radians. Therefore:

• To convert from degrees to radians, multiply by \(\frac{\pi}{180}\).
• To convert from radians to degrees, multiply by \(\frac{180}{\pi}\).

If you have an angle of \(\pi\) radians, like in the exercise, this translates to \(180^\circ\). This connection is foundational for working comfortably with angles in different units.

Exploring Sine and Cosine

The sine and cosine functions are fundamental in trigonometry and relate to points on the unit circle. The unit circle is a circle with a radius of 1 centered at the origin \((0,0)\) in the coordinate plane. The special property of this circle allows us to analyze angles and their corresponding coordinates easily.

For any angle placed in standard position, the endpoints of the angle's terminal side on the unit circle define coordinates \((x,y)\). Here:

• Cosine of the angle corresponds to the x-coordinate \((\cos(\theta) = x)\).
• Sine of the angle corresponds to the y-coordinate \((\sin(\theta) = y)\).

In the specific example of \(\theta=\pi\), the coordinates on the unit circle are \((-1, 0)\). Thus, \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\). These values illustrate how sine and cosine link an angle to positions on the unit circle, aiding in various calculations involving periodic and oscillatory behavior in physics and engineering.

Angles in Radians

Angles measured in radians are common in many mathematical contexts, especially those involving trigonometric functions and calculus. Radians provide a more natural measure of angle size relative to the circle, which makes them ideal for mathematical calculations compared to degrees. They simplify many formulas and lead to more elegant solutions and theories.

When examining angles in radians, the position of the angle on the unit circle is critical. A full circle is \(2\pi\) radians, and each semicircle is \(\pi\) radians. The location of an angle in radians directly affects its sine and cosine values due to their geometric interpretations linked to the unit circle.

For example, at \(\theta = \pi\), which is half of a full revolution, the angle aligns perfectly with the negative x-axis. This positioning explains why \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\). Understanding these relationships allows deeper insights into the harmonics of waves and signals, essential for fields like engineering and sound design.