Problem 16 - Algebra 2

Question

The measure \(\boldsymbol{\theta}\) of an angle in standard position is given. Find the exact values of \(\cos \theta\) and \(\sin \theta\) for each angle measure. \(\frac{\pi}{2}\) radians

Step-by-Step Solution

Answer

For the angle measure of \(\frac{\pi}{2}\) radians, the exact values are \(\cos \theta = 0\) and \(\sin \theta = 1\).

Step 1: Identify the angle in the unit circle

Angle \(\frac{\pi}{2}\) radians is equivalent to 90 degrees in the unit circle. This is the measure of an angle in standard position, which corresponds to the point on the unit circle where the terminal side of the angle intersects.

Step 2: Determine the coordinates on the unit circle

The coordinates of the point where the terminal side of an angle of \(\frac{\pi}{2}\) radians intersects the unit circle are (0,1).

Step 3: Match the coordinates with cosine and sine values

In the unit circle, the x-coordinate corresponds to \(\cos \theta\) and the y-coordinate corresponds to \(\sin \theta\). Therefore, when \(\theta = \frac{\pi}{2}\) radians, \(\cos \theta = 0\) and \(\sin \theta = 1\).

Key Concepts

Unit Circle

The unit circle is a crucial tool in trigonometry, offering a way to understand angles and trigonometric functions. Imagine a circle with a radius of one centered at the origin of a coordinate plane. This special circle allows us to define all trigonometric functions based on the coordinates of points along its circumference.

• Each point on the unit circle can be represented by coordinates \((x, y)\) where:
  • \(x\) represents the cosine of the angle (\(\cos \theta\))
  • \(y\) represents the sine of the angle (\(\sin \theta\))

Consider an angle in standard position, meaning its vertex is at the origin, and its initial side lies along the positive x-axis. As the angle's terminal side rotates counterclockwise from the initial side, the ending point of this side on the unit circle determines the cosine and sine for the angle's measure. For example, when an angle is \(\frac{\pi}{2}\), its terminal side stops at the point (0, 1) on the unit circle, leading to \(\cos \theta = 0\) and \(\sin \theta = 1\).

Radians to Degrees Conversion

Radians and degrees are two units for measuring angles, and converting between them is a common task. Here's how we can convert radians to degrees and vice versa. Knowing this conversion helps in understanding trigonometric problems effectively.

• To convert radians to degrees, we use the conversion factor \(\frac{180}{\pi}\).
• To convert degrees to radians, the conversion factor is \(\frac{\pi}{180}\).

For instance, to convert \(\frac{\pi}{2}\) radians to degrees, we calculate:\[\frac{\pi}{2} \times \frac{180}{\pi} = 90^\circ\]This conversion shows us that \(\frac{\pi}{2}\) radians equals 90 degrees. Understanding these conversions is essential because many real-world applications and some calculators operate in degrees while mathematical analysis often uses radians.

Standard Position Angle

The concept of "standard position" for angles is vital in trigonometry. An angle is in standard position if its vertex is at the origin of the coordinate plane, with its initial side along the positive x-axis. This configuration serves as a reference for measuring and working with angles in trigonometry.

• As the angle measures increase, the terminal side rotates counterclockwise around the origin.
• Negative angle measures indicate a clockwise rotation.

For the angle \(\frac{\pi}{2}\) radians, the terminal side of the angle is positioned at 90 degrees or \(\frac{\pi}{2}\) radians counterclockwise from the x-axis. This makes it lie on the y-axis, intersecting the unit circle at the point (0, 1). Understanding angles in standard position is crucial as it standardizes the definition of trigonometric functions and angles on the unit circle.