Problem 86 - Precalculus

Question

For the following exercises, find the exact value of each trigonometric function. $$ \cos \frac{\pi}{2} $$

Step-by-Step Solution

Answer

\( \cos \frac{\pi}{2} = 0 \).

Step 1: Identify the Angle

The angle given is \( \frac{\pi}{2} \), which is equivalent to 90 degrees. In trigonometry, angles can be expressed in either radians or degrees, and \( \pi \) radians is equal to 180 degrees. Therefore, \( \frac{\pi}{2} \) corresponds to 90 degrees.

Step 2: Determine Location on the Unit Circle

The angle \( 90^\circ \) or \( \frac{\pi}{2} \) is located at the top of the unit circle along the positive y-axis. The coordinates of this point on the unit circle are (0, 1). The unit circle helps us find exact trigonometric function values.

Step 3: Find the Cosine Value

On the unit circle, the cosine of an angle is represented by the x-coordinate of the point where the terminal side of the angle intersects the unit circle. For \( \frac{\pi}{2} \), the point is (0, 1), so the x-coordinate is 0. Thus, \( \cos \frac{\pi}{2} = 0 \).

Key Concepts

unit circle

The unit circle is a fundamental concept in trigonometry. Imagine a circle with a radius of one unit centered at the origin of a coordinate plane. This circle is essential because it helps to calculate the values of trigonometric functions such as sine and cosine.

Each point on the unit circle corresponds to an angle, typically measured from the positive x-axis. The coordinates of any point on the unit circle are \(x, y\), where \(x\) equals the cosine of the angle, and \(y\) equals the sine of the angle.

The simplicity of the unit circle makes it easier to visualize and solve trigonometric functions. For instance, the angle of \(\frac{\pi}{2}\) radians, or 90 degrees, falls directly on the positive y-axis of the unit circle, with the coordinates (0, 1). This positioning immediately tells us the values of sine and cosine for that angle.

cosine function

The cosine function is a principal trigonometric function important for calculating angles and lengths in various applications. Mathematically, the cosine of an angle is the x-coordinate of a point on the unit circle corresponding to that angle.

For example, \(\cos \frac{\pi}{2}\) is determined by locating this angle on the unit circle. At \(\frac{\pi}{2}\) radians (90 degrees), we find the point (0, 1). Here, the x-coordinate is 0, indicating that \(\cos \frac{\pi}{2} = 0\).

This function has various real-world applications, including analyzing waves, sound, and light. The cosine function smoothly varies between -1 and 1 as the angle changes from zero to 360 degrees (or \( 2\pi \) radians), following a periodic wave pattern, often used in physics and engineering.

radian measure

Radian measure is a way of expressing angles, using the radius of a circle as a reference. One radian is the angle at which the arc length is equal to the radius of the circle. This method of measuring angles is more natural in mathematical calculations than degrees because it directly relates angle sizes to the properties of the circles.

The complete circle in radian measure is \(2\pi\), equivalent to 360 degrees. Therefore, \(\pi\) radians equal half of the circle, or 180 degrees. Converting between radians and degrees is essential when solving trigonometry problems.

To illustrate, the angle \(\frac{\pi}{2}\) radians is a quarter of the circle, translating to 90 degrees. This conversion helps identify where an angle lies on the circle, simplifying the computation of trigonometric functions like cosine and sine.