Problem 2 - Algebra 2

Question

If the given point \(P\) is located on the unit circle, find \(\sin \theta\) and \(\cos \theta\) $$ P\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) $$

Step-by-Step Solution

Answer

\(\sin \theta = \frac{\sqrt{2}}{2}\), \(\cos \theta = \frac{\sqrt{2}}{2}\).

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1. Coordinates of any point on the unit circle are represented as \((x, y)\) and are equal to \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle in radians measured from the positive x-axis.

Step 2: Identify Coordinates on the Unit Circle

Since the given point is \(P\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\), this point is identified as \((\cos \theta, \sin \theta)\). Thus, \(x = \cos \theta = \frac{\sqrt{2}}{2}\) and \(y = \sin \theta = \frac{\sqrt{2}}{2}\).

Step 3: Assign Values to Sin and Cos

For the given point \(P\), \(\cos \theta = \frac{\sqrt{2}}{2}\) and \(\sin \theta = \frac{\sqrt{2}}{2}\). These values correspond to the coordinates of point \(P\) on the unit circle.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding the relationship between angles and sides of a triangle, especially in the context of circles. In a unit circle, these functions help us relate the position of a point on the circle to the angle it forms with the x-axis. The primary trigonometric functions are sine, cosine, and tangent, which explore the ratios of different sides of a right triangle. Since the unit circle has a radius of 1, it simplifies calculations because the hypotenuse of any such right triangle is 1, making it easier to directly interpret the sine and cosine values as coordinates.

• Sine (\( \sin \theta \)): Represents the y-coordinate of the point on the unit circle.
• Cosine (\( \cos \theta \)): Represents the x-coordinate of the point on the unit circle.

Both of these functions oscillate between -1 and 1 as the angle \( \theta \) varies from 0 to \( 2\pi \) radians, forming the basic waves we often see in graphs of these functions.

Sine and Cosine

The sine and cosine functions are critical for determining and interpreting locations on the unit circle. They describe how far a point is from the horizontal and vertical axes, respectively, as the circle is traversed.

• Sine (\( \sin \theta \)): For any given angle \( \theta \), the sine function provides the height of the point above the x-axis, corresponding to the y-coordinate of the circle.
• Cosine (\( \cos \theta \)): This function gives the horizontal distance from the origin to the point, which corresponds to the x-coordinate on the unit circle.

For example, at 0 radians, \( \sin(0) = 0 \) (on the x-axis) and \( \cos(0) = 1 \) (distance from origin to point). Such cyclical nature is why sine and cosine are pivotal in oscillatory processes, like waves and harmonic motion.

Coordinates

On the unit circle, the point determined by any angle \( \theta \) is denoted as \( (\cos \theta, \sin \theta) \). These coordinates are derived from drawing an imaginary line from the origin (center of the circle) to the circumference where it intersects.

• The x-coordinate is the cosine of the angle - it shows how far to the left or right the point lies.
• The y-coordinate is the sine of the angle - it indicates the upwards or downwards position related to the origin.

In the case of \( P\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \), it lies exactly on the line \( y = x \), showing symmetry between the two values, both equal to \( \frac{\sqrt{2}}{2} \). This creates a diagonal line at \( \pi/4 \) radians or 45 degrees from the positive x-axis.

Angle in Radians

Angles can be measured in various units, but radians are the standard in mathematics due to their direct relationship with the unit circle. One full rotation around the circle is \( 2\pi \) radians, equivalent to 360 degrees. Therefore, \( \pi \) radians is 180 degrees, indicating a half circle.

• \( \frac{\pi}{2} \) radians is a quarter turn, or 90 degrees, resulting in reaching the top of the circle where sine is at its maximum and cosine is zero.
• \( \pi \) radians represents a straight angle pointing left, with sine at zero again and cosine reaching \(-1\).

Using radians connects angles directly with arc lengths, allowing simpler calculations in calculus and other areas. In our example, the point coordinates suggest the angle \( \theta = \pi/4 \) radians, as both cosine and sine are equal, showing the radial symmetry of the 45-degree angle.