Problem 88 - Precalculus

Question

For the following exercises, find the exact value of each trigonometric function. $$ \sin \frac{\pi}{4} $$

Step-by-Step Solution

Answer

\( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 1: Understanding the Problem

The exercise asks us to find the exact value of the sine function at \( \frac{\pi}{4} \). We know that \( \frac{\pi}{4} \) corresponds to a 45-degree angle since \( \pi \) radians equals 180 degrees.

Step 2: Recognizing the Special Angle

\( \frac{\pi}{4} \) is a special angle where the sine and cosine values are well-known. In trigonometry, the angle of 45 degrees has specific sine and cosine values based on the properties of a 45-45-90 triangle (isosceles right triangle).

Step 3: Using the Unit Circle

On the unit circle, the coordinates of any point at angle \( \theta \) are given by \((\cos \theta, \sin \theta)\). For \( \theta = \frac{\pi}{4} \), the coordinates are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

Step 4: Identifying the Sine Value

From the unit circle, the sine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \), because it is the y-coordinate of the corresponding point on the unit circle.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in the study of mathematics, particularly in relation to triangles and circles. These functions include sine, cosine, and tangent, among others, and they help define the relationship between the angles and sides of triangles. In trigonometry, these functions are incredibly useful for solving problems involving angles, such as defining the position of a point on the unit circle.
For any given angle in a right triangle, we can attribute the following:

• Sine (\(\sin\theta\)): the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\(\cos\theta\)): the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(\tan\theta\)): the ratio of the length of the opposite side to the adjacent side.

Understanding these basic trigonometric functions is crucial, especially when working with the angles on the unit circle. The unit circle has a radius of one, simplifying the calculations and visualization of these functions. Once you comprehend this basic knowledge, solving trigonometric problems, like finding the values on the unit circle, becomes more straightforward.

Special Angles

In trigonometry, some angles are classified as "special." These special angles are typically 30° (or \(\frac{\pi}{6}\) radians), 45° (or \(\frac{\pi}{4}\) radians), and 60° (or \(\frac{\pi}{3}\) radians). Each of these angles has known sine, cosine, and tangent values that are frequently used in trigonometric calculations.
Special angles are significant because they simplify the process of finding exact values of trigonometric functions without using a calculator. For example:

• \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)
• \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)
• \(\tan \frac{\pi}{4} = 1\)

When using the unit circle, recognizing these special angles helps quickly identify the coordinates at those angles. For instance, at \(\frac{\pi}{4}\), the coordinates on the unit circle are \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\). It simplifies solving problems that involve finding sine, cosine, or tangent values.

Sine Function

The sine function is one of the most fundamental trigonometric functions. It is defined at any angle \(\theta\) as the y-coordinate of the corresponding point on the unit circle. For angles in a right triangle, it is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Its periodic nature makes it crucial in understanding wave patterns and oscillatory motions.
In the context of a unit circle, the sine of an angle \(\frac{\pi}{4}\) or 45 degrees is particularly interesting because it is one of the special angles. At this angle, the sine function takes an exact value of \(\frac{\sqrt{2}}{2}\). Why? Because on the unit circle, the coordinates at \(\frac{\pi}{4}\) are \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\) and the sine value corresponds to the second coordinate, or the y-value.
In practical applications, the sine function helps us understand and predict cyclical phenomena like sound waves, light waves, and tides, all of which rely on this simple yet powerful trigonometric function.