Problem 50 - Basic Technical Mathematics with Calculus

Question

Designate each angle by the quadrant in which the terminal side lies, or as a quadrantal angle. $$3 \mathrm{rad},-3 \pi \mathrm{rad}$$

Step-by-Step Solution

Answer

3 radians is in Quadrant II; -3π radians is a quadrantal angle at π.

Step 1: Understanding Radians and Quadrants

In a coordinate plane, the angles are generally measured from the positive x-axis in a counter-clockwise direction. One complete rotation equals \(2\pi\) radians, which is equivalent to 360 degrees. The four quadrants are divided as follows: \(0\) to \(\frac{\pi}{2}\) in Quadrant I, \(\frac{\pi}{2}\) to \(\pi\) in Quadrant II, \(\pi\) to \(\frac{3\pi}{2}\) in Quadrant III, and \(\frac{3\pi}{2}\) to \(2\pi\) in Quadrant IV. Quadrantal angles lie on the axes and include angles such as \(0, \frac{\pi}{2}, \pi,\) and \(\frac{3\pi}{2}\).

Step 2: Locate 3 radians

The angle \(3\) radians needs to be identified on the unit circle. Since \(\pi\approx3.14\), \(3\) is slightly less than \(\pi\), which places it in Quadrant II, where radians range from \(\frac{\pi}{2}\) to \(\pi\).

Step 3: Locate -3π radians

-\(3\pi\) radians means that the angle is measured in the clockwise direction starting from the positive x-axis. One full rotation clockwise is \(2\pi\), which lands back at \(0\), and another \(\pi\) rotation lands at \(\pi\), which lies on the negative x-axis. Therefore, -\(3\pi\) is a quadrantal angle at \(\pi\).

Key Concepts

Radian Measure

Radian measure is a way to express angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians work by relating the angle directly to the circle's radius. Specifically, one radian equals the angle created when the arc of the circle is equal in length to the radius. This makes radian a natural measure for mathematics.

• One full circle (360 degrees) is equivalent to \(2\pi\) radians.
• Half a circle (180 degrees) equals \(\pi\) radians.
• A quarter circle (90 degrees) equals \(\frac{\pi}{2}\) radians.

Understanding radian measure helps when navigating the unit circle, a fundamental tool in trigonometry.

Coordinate Plane

The coordinate plane is a two-dimensional space divided by an x-axis and a y-axis intersecting at a point called the origin. This system is key to visualizing angles and geometric figures, including circles. The quadrants of the coordinate plane are formed by these axes. Here's how they're structured:

• Quadrant I: both x and y are positive.
• Quadrant II: x is negative and y is positive.
• Quadrant III: both x and y are negative.
• Quadrant IV: x is positive and y is negative.

In trigonometry, angles are commonly measured counterclockwise from the positive x-axis, a convention that's crucial for understanding and working with radians and degrees.

Quadrantal Angles

Quadrantal angles are those that lie directly on the x or y axes within the coordinate plane. They are particularly significant because they simplify the calculation and understanding of trigonometric functions and their values. The quadrantal angles are like the cardinal directions on a compass:

• \(0\) radians: Positive x-axis direction.
• \(\frac{\pi}{2}\) radians: Positive y-axis direction.
• \(\pi\) radians: Negative x-axis direction.
• \(\frac{3\pi}{2}\) radians: Negative y-axis direction.

These angles correspond to special values for sine, cosine, and tangent, often making calculations simpler in trigonometric equations or problems involving the unit circle.