Problem 51 - Basic Technical Mathematics with Calculus

Question

The given angles are in standard position. Designate each angle by the quadrant in which the terminal side lies, or as a quadrantal angle. $$4 rad, \(\pi / 3\) rad$$

Step-by-Step Solution

Answer

\(4\) rad lies in the Fourth Quadrant, \(\pi/3\) rad lies in the First Quadrant.

Step 1: Understand the Problem

We need to determine the quadrant or identify if the terminal side of each given angle is a quadrantal angle. Angles are given in radians: the angles are \(4 \, \text{rad}\) and \(\frac{\pi}{3} \, \text{rad}\).

Step 2: Identify Full Circles

Recall that one full circle in radians is \(2\pi\). For an angle \(x\), the effective angle can be found by \(x \mod 2\pi\). Thus, we determine the position of \(4\) rad and \(\frac{\pi}{3}\) rad accordingly.

Step 3: Determine Quadrant for \(\pi / 3\) rad

Since \(\frac{\pi}{3}\) is approximately 1.047 rad, which is less than \(\frac{\pi}{2}\) (approx. 1.571 rad), it lies in the **First Quadrant**.

Step 4: Determine Effective Angle for 4 rad

To find the equivalent angle under \(2\pi\), calculate the remainder of \(4\) when divided by \(2\pi\). Since \(4\) rad is less than \(2\pi\) (~6.283 rad), it directly lies in this range. So, no need for additional computation in this step.

Step 5: Determine Quadrant for 4 rad

Now compare \(4\) rad with quadrant boundary angles: \(4\) rad is more than \(\frac{3\pi}{2}\) (~4.712 rad) but less than \(2\pi\). Therefore, it lies in the **Fourth Quadrant**.

Key Concepts

Angle Measurement

Angle measurement is a method to describe the direction of a ray or line segment from a given starting point, often the origin. In trigonometry, angles can be measured in degrees or radians.

• Degrees: A full circle is divided into 360 equal parts called degrees.
• Radians: A circle with a radius of 1 unit has a circumference of \(2\pi\) units, hence a full circle is divided by \(2\pi\) rad.

Understanding these two units is essential as they are often interchangeable. In many trigonometric functions and calculus problems, angles are typically expressed in radians for ease of calculation and standardization.

Quadrants

The Cartesian plane is divided into four sections known as quadrants. These quadrants help determine the location of an angle's terminal side when in standard position. Here’s how they are arranged:

• First Quadrant: Angles between 0 and \(\pi/2\) (0 to 90 degrees), where both x and y coordinates are positive.
• Second Quadrant: Angles between \(\pi/2\) and \(\pi\) (90 to 180 degrees), where x is negative and y is positive.
• Third Quadrant: Angles between \(\pi\) and \(3\pi/2\) (180 to 270 degrees), where both x and y coordinates are negative.
• Fourth Quadrant: Angles between \(3\pi/2\) and \(2\pi\) (270 to 360 degrees), where x is positive and y is negative.

Understanding which quadrant an angle lies in can provide insight into the sign and values of trigonometric functions for that angle.

Radian Measure

Radian measure is the standard unit of angular measure used in many areas of mathematics. It is based on the radius of the circle. Specifically, an angle's measure in radians is determined by the length of the arc on the unit circle subtended by that angle.

• Full circle: \(2\pi\) radians, equivalent to 360 degrees.
• Semi-circle: \(\pi\) radians, equivalent to 180 degrees.

Unlike degrees, radians provide a natural measure that connects arc length and angle size directly through the formula: \(\theta = \frac{\text{Arc Length}}{\text{Radius}}\). In many mathematical contexts, radians simplify the expressions for derivatives and integrals involving trigonometric functions, making it a preferred measure for calculus.

Standard Position

An angle is said to be in standard position when its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. This positioning forms a basis to measure the angle in an anti-clockwise direction.

• Initial Side: Always situated along the positive x-axis.
• Terminal Side: Rotates around the circle to define the angle's measure.

Using standard position helps in consistently applying trigonometric functions and evaluating angle measures within various quadrants. It is a convenient framework for understanding how angles interact with the coordinate plane, making calculations involving sine, cosine, and tangent more straightforward.