Problem 1 - Precalculus

Question

Find the exact value of the cosine and sine of the given angle. $$ \theta=0 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Unit Circle

Recall that the unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. Each point on the circumference of the unit circle corresponds to an angle (in radians) measured from the positive x-axis.

Step 2: Identify Angle on the Unit Circle

The angle \( \theta = 0 \) is the point where the unit circle intersects the positive x-axis. This point is at the coordinates (1, 0).

Step 3: Recall Definitions of Cosine and Sine

For any angle \( \theta \) on the unit circle, the \( x \)-coordinate represents \( \cos(\theta) \) and the \( y \)-coordinate represents \( \sin(\theta) \).

Step 4: Determine Cosine of \( \theta \)

Since the \( x \)-coordinate at \( \theta = 0 \) is 1, we have \( \cos(0) = 1 \).

Step 5: Determine Sine of \( \theta \)

Since the \( y \)-coordinate at \( \theta = 0 \) is 0, we have \( \sin(0) = 0 \).

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry that relates an angle to the horizontal coordinate of a point on the unit circle. It describes how the angle's position affects the horizontal position on the unit circle.

• The unit circle is especially important in understanding cosine, as for any angle, its cosine is simply the x-coordinate at that point on the circle.
• When we say "cosine of 0," represented as \( \cos(0) \), we're referring to this x-coordinate when the angle is 0 degrees or radians on the unit circle.

At \( \theta = 0 \), the point is at (1, 0), thus \( \cos(0) = 1 \). This explains why the cosine value at zero is always 1, as it directly corresponds to the radius of the unit circle along the positive x-axis.

Sine Function

The sine function complements the cosine function and is crucial for understanding how an angle relates to the vertical coordinate of a point on the unit circle. It shows how the angle's orientation affects the vertical position on the unit circle.

• For any angle \( \theta \), the sine value is equivalent to the y-coordinate of the unit circle.
• At the angle \( \theta = 0 \), this point on the unit circle is reached at coordinates (1, 0), making \( \sin(0) = 0 \).

This is because at \( \theta = 0 \), there is no vertical displacement from the x-axis, keeping the vertical position at zero. Hence, the sine of 0 is always 0.

Angle Measurement

Angle measurement in trigonometry can be in degrees or radians, both of which quantify the size of an angle at the vertex.

• One complete revolution around a circle is 360 degrees, equivalent to \( 2\pi \) radians in radian measure.
• Therefore, an angle \( \theta = 0 \) is both 0 degrees and 0 radians.

Using radians is very practical in many mathematical contexts because they provide a direct relation to the arc length on the unit circle. Understanding both degrees and radians allows flexibility in solving various mathematical problems.

Coordinate System

In the context of trigonometry, understanding the coordinate system is key, as it helps in mapping the position of angles and points on the unit circle.

• The origin in a coordinate system divides the plane into four quadrants, but the unit circle is positioned such that the origin is the center, making the radius always 1.
• When determining the coordinates for angles, such as \( \theta = 0 \), the positive x-axis is the starting line for measuring angles in standard position.

The coordinate system's axes are also reference lines that define how angles and corresponding sine and cosine values are represented. This visualization helps in understanding complex trigonometric concepts with clarity.