Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-12\) , find the exact function value of each of the following if the measure of the angle is given in radians. $$ \sin \frac{\pi}{4} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We are asked to find the exact value of the sine function when the angle is given in radians. Specifically, we need \( \sin \frac{\pi}{4} \).

Step 2: Identify Key Angle

The angle \( \frac{\pi}{4} \) is a standard angle in trigonometry. It is the equivalent of 45 degrees in degree measure.

Step 3: Recall Trigonometric Value

For the angle \( \frac{\pi}{4} \), which is 45 degrees, the sine function value is \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). This is a commonly known trigonometric value.

Step 4: Verify Using Unit Circle

On the unit circle, the point corresponding to \( \frac{\pi}{4} \) radians has coordinates \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \). The \( y \)-coordinate gives the sine value, confirming that \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Key Concepts

Understanding the Sine Function

The sine function is a fundamental concept in trigonometry, often denoted as \( \sin \theta \), where \( \theta \) represents an angle. This function helps us determine the vertical position on the unit circle given an angle. It's important to realize that the sine function measures how high above or below the x-axis a point is.

A few key properties of the sine function include:

• It varies between -1 and 1 for all angles \( \theta \).
• It is periodic with a period of \(2\pi\) radians, meaning it repeats its values every \(2\pi\) radii.
• It is an odd function, implying that \( \sin(-\theta) = -\sin \theta \).

The sine of an angle when plotted on a graph, oscillates like a smooth wave, repeatedly rising to 1 and falling to -1. Understanding its behavior on the unit circle is crucial for solving problems involving this trigonometric function.

Navigating the Unit Circle

The unit circle is a vital tool in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Using the unit circle, we can easily find trigonometric function values for key angles.

When dealing with the unit circle:

• The horizontal axis is labeled the \( x \)-axis and the vertical the \( y \)-axis.
• The angle \( \theta \) is measured starting from the positive \( x \)-axis and moving counter-clockwise around the circle.
• Any point on the circle corresponds to \((\cos \theta, \sin \theta)\).

These coordinates mean that for any angle \( \theta \) on the unit circle, the cosine of the angle is the \( x \)-coordinate, while the sine is the \( y \)-coordinate. This understanding helps us find trigonometric function values efficiently as demonstrated with \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Recognizing Standard Angles

Standard angles in trigonometry are specific angles that have easily remembered sine, cosine, and tangent values. These angles, typically given in both degrees and radians, serve as reference points for various trigonometric calculations.

The key standard angles are:

• \(0^{\circ} = 0\) radians
• \(30^{\circ} = \frac{\pi}{6}\) radians
• \(45^{\circ} = \frac{\pi}{4}\) radians
• \(60^{\circ} = \frac{\pi}{3}\) radians
• \(90^{\circ} = \frac{\pi}{2}\) radians

These angles are crucial because they provide exact trigonometric values that are often required in various mathematical problems. Knowing the values for these standard angles, such as \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), can greatly simplify and speed up calculations.