Problem 29 - Algebra 2

Question

Find the exact value of each trigonometric function. \(\sin \frac{3 \pi}{4}\)

Step-by-Step Solution

Answer

\( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 1: Identify the Reference Angle

The angle \( x = \frac{3\pi}{4} \) is in radians. To find its reference angle, we first need to know its relation to the unit circle. Since \( \frac{3\pi}{4} \) radians is equivalent to 135 degrees, it lies in the second quadrant.

Step 2: Determine the Reference Angle

In the second quadrant, the reference angle can be found by subtracting the given angle from \( \pi \): \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \). Therefore, the reference angle is \( \frac{\pi}{4} \).

Step 3: Evaluate the Sine of the Reference Angle

Knowing that \( \sin \) of \( \frac{\pi}{4} \) is a commonly known value, we recall that \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 4: Determine the Sign of the Sine Function

In the second quadrant, the sine function is positive. Therefore, \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is used to relate the angle of a right triangle to the ratio of the length of the opposite side over the hypotenuse. This function is essential in understanding wave patterns and oscillations.
In the context of the unit circle, the sine function gives the y-coordinate of a point on the circle. This is because the sine of an angle is associated with the vertical component. It is periodic, meaning it repeats its values in regular intervals, specifically every \( 2\pi \) radians or 360 degrees.

• The sine of 0 is 0, which means that at 0 radians (0 degrees), the y-coordinate is 0.
• As the angle increases, the sine function grows until it reaches 1 at \( \frac{\pi}{2} \) radians (90 degrees).
• It decreases back to 0 at \( \pi \) radians (180 degrees) and becomes negative after passing 180 degrees.

Understanding the sine function involves grasping not only its geometric interpretation but also its behavior through different angles. This knowledge is crucial for solving problems involving trigonometric functions.

Radians

Radians are a way to measure angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians express angles as the length of the arc created by the angle in a unit circle. One full revolution of a circle is \( 2\pi \) radians, which is equivalent to 360 degrees.

• One radian is the angle created when the arc length is equal to the radius of the circle.
• \( \pi \) radians are equivalent to 180 degrees. This conversion is useful for switching between degree and radian measurements.
• The use of radians often simplifies mathematical formulas, particularly those involving trigonometric functions.

For example, \( \frac{3\pi}{4} \) radians, as seen in the problem, corresponds to an angle of 135 degrees. This measurement can help locate the angle's position on the unit circle.

Understanding radians is fundamental in advanced mathematics, physics, and engineering, as they provide a natural way to discuss angular measurements.

Unit Circle

The unit circle is a crucial concept in trigonometry. It is a circle with a radius of 1 centered at the origin of a coordinate plane. This circle helps in visualizing and understanding the trigonometric functions sine, cosine, and tangent based on angles. Every point on the unit circle has coordinates \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed with the positive x-axis.
Some key aspects of the unit circle include:

• Any angle in radians can be used to find its corresponding point on the unit circle.
• The x-coordinate gives the cosine of the angle, while the y-coordinate gives the sine.
• It neatly maps angles to the trigonometric functions, offering a visual representation of their values.
• The circular nature helps understand the periodic behavior of these functions.

The unit circle is divided into four quadrants, which determine the signs of the sine and cosine functions. For instance, in the second quadrant, where \( \frac{3\pi}{4} \) radians lies, sine is positive while cosine is negative. Recognizing which quadrant an angle belongs to is key in evaluating trigonometric functions.

Reference Angle

A reference angle is the smallest angle that an original angle makes with the x-axis. It's an essential part of working with angles on the unit circle because reference angles help in finding the values of trigonometric functions. They are always positive and less than \( \pi/2 \) radians (90 degrees).

• To find a reference angle, subtract the given angle from \( \pi \) if it's in the second quadrant, as was done with \( \frac{3\pi}{4} \) in the problem.
• In the third quadrant, subtract \( \pi \) from the angle.
• In the fourth quadrant, subtract the angle from \( 2\pi \).

These rules help identify the reference angle regardless of the original angle's location.
The reference angle retains the same sine and cosine values as the original angle, except they must be adjusted for their signs depending on the specific quadrant. Understanding reference angles simplifies calculations and improves understanding of the behavior of trigonometric functions across different angles.