Problem 37 - Precalculus: A Concise Course

Question

Evaluate the trigonometric function of the quadrant angle. $$ \sin \pi $$

Step-by-Step Solution

Answer

The value of \( \sin \pi \) is 0.

Step 1: Understand the Sine Function

The sine function, \(\sin(x)\), is one of the basic functions in trigonometry. The value of \(\sin(x)\) depends on the angle \(x\). Important values to remember are \(\sin(0) = 0\), \(\sin(\pi / 2) = 1\), \(\sin(\pi) = 0\), \(\sin(3\pi / 2) = -1\), and \(\sin(2\pi) = 0\). The function repeats every \(2\pi\) radians or every 360 degrees.

Step 2: Evaluate Sine of the Given Angle

We are asked to evaluate \(\sin(\pi)\). From our understanding of the sine function, we know that at \(\pi\) radians or 180 degrees, the value of \(\sin\) function is 0.

Key Concepts

Sine Function

The sine function, expressed as \( \sin(x) \), is a fundamental trigonometric function. It is widely used to describe relationships between angles and side lengths in right-angled triangles but also extends to unit circle concepts. The sine of an angle corresponds to the \(y\)-coordinate of a point on the unit circle—a circle with a radius of 1 centered at the origin of a coordinate plane.

This function is periodic, meaning it repeats its values in regular intervals. The sine of an angle repeats every \(2\pi\) radians or 360 degrees. Some critical values to remember include:

• \( \sin(0) = 0 \)
• \( \sin(\pi/2) = 1 \)
• \( \sin(\pi) = 0 \)
• \( \sin(3\pi/2) = -1 \)
• \( \sin(2\pi) = 0 \)

These values help when analyzing and predicting the behavior of the sine function over its cycle. The sine function is particularly useful in physics, engineering, and computer graphics for modeling oscillations and waves.

Quadrant Angles

Quadrant angles are special angles that lie on the axes of the standard coordinate plane. They are multiples of 90 degrees (or \( \pi/2 \) radians) and are key reference points in trigonometry.

When you look at the coordinate plane divided into four quadrants, each quadrant represents 90 degrees of rotation. Quadrant angles mark the boundaries between these quadrants:

• 0 degrees or \( 0 \) radians is on the positive x-axis.
• 90 degrees or \( \pi/2 \) radians is on the positive y-axis.
• 180 degrees or \( \pi \) radians is on the negative x-axis.
• 270 degrees or \( 3\pi/2 \) radians is on the negative y-axis.

In the context of the sine function, these quadrant angles often provide simplified values as we have distinct well-known results for \( \sin(0), \sin(\pi/2), \sin(\pi), \sin(3\pi/2) \), and \( \sin(2\pi) \). These angles are helpful, especially when working with circular motion and oscillatory systems.

Radians

Radians are a method of measuring angles, much like degrees. However, radians are based on the radius of a circle. When an angle's arc is equal in length to the radius of the circle, this angle is defined as 1 radian.

In one complete revolution around a circle, there are \( 2\pi \) radians, which corresponds to 360 degrees. The conversion between degrees and radians is crucial and often expressed as:

• 180 degrees = \( \pi \) radians
• 90 degrees = \( \pi/2 \) radians
• 60 degrees = \( \pi/3 \) radians

Radians provide a natural way of considering angles due to their direct relationship with a circle's radius, making them especially useful in higher mathematics and physics. Understanding radians is fundamental in trigonometry as it allows for a more seamless integration into formulas and expressions used in calculus and beyond.