Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the quadrant in which each angle lies. (The angle measure is given in radians.) (a) \(\frac{5 \pi}{6}\) (b) \(-\frac{5 \pi}{3}\)

Step-by-Step Solution

Answer

The angle \(\frac{5 \pi}{6}\) lies in the second quadrant and the angle \(-\frac{5 \pi}{3}\) lies in the first quadrant.

Step 1: Translate radians into degrees

First we need to translate radians into degrees to get a better grasp on the angle location. For this, we use the formula \( \text{Degree} = \text{Radian} \times \frac{180}{\pi} \).

Step 2: Determine the quadrant for \(\frac{5 \pi}{6}\)

So we calculate it for \( \frac{5 \pi}{6} \): \[ \frac{5 \pi}{6} \times \frac{180}{\pi} = 150° \] Now that we have the degree measurement, we know that an angle of 150° is located in the second quadrant, because in this quadrant, a start from the 0° line (positive x-axis) to the 180° line (negative x-axis) is found.

Step 3: Determine the quadrant for \(-\frac{5 \pi}{3}\)

We follow the same steps for \( -\frac{5 \pi}{3} \): \[ -\frac{5 \pi}{3} \times \frac{180}{\pi} = -300° \] A negative angle measurement means we rotate clockwise from the positive x-axis, contrary to our initial counterclockwise rotation. When we consider a full rotation is 360°, the negative angle -300° would be equivalent to 60° in a standard positive rotation. Thus, \( -\frac{5 \pi}{3} \) lies in the first quadrant.

Key Concepts

Trigonometry Quadrants

Understanding trigonometry quadrants is crucial when working with angles in the Cartesian plane. Imagine the plane divided by the x and y axes into four sections, known as quadrants, each containing a range of angles that share certain sine, cosine, and tangent sign characteristics.

The first quadrant is where both x and y coordinates are positive, and it includes angles from 0° to 90°. Moving counter-clockwise, angles increase and pass through the second quadrant (90° to 180°) where sine is positive but cosine and tangent are negative. Continuing, the third quadrant (180° to 270°) contains angles with positive tangent but negative sine and cosine. Lastly, the fourth quadrant (270° to 360°) rounds off our complete circle with positive cosine, but negative sine and tangent.

By knowing these properties, we can locate any angle by its measure. In trigonometry, it's common to deal with angles larger than 360° or negative angles; these represent multiple rotations or clockwise rotations, respectively, but ultimately fall within the familiar four quadrants.

Negative Angle Measurement

Negative angle measurement might seem confusing at first, but it follows a straightforward principle. In mathematics, positive angles represent a counterclockwise rotation from the positive x-axis, while negative angles represent a clockwise rotation.

To visualize this, think of standing on a clock. A positive angle would be measured by moving against the direction of the clock's hands, while a negative angle would follow the direction of the clock's hands. It's common to convert negative angles to their positive equivalents to simplify their interpretation. This is done by adding or subtracting full rotations (360°) until the result is a positive value. For example, an angle of -90° can be thought of as a 270° angle because it is a three-quarter turn clockwise from the positive x-axis, which is equivalent to the three-quarter turn counterclockwise that defines 270°.

Angle in Radians

An angle measured in radians provides a different perspective compared to degrees. Radians are a way of measuring angles based on the arc length in a circle's radius. One full rotation around a circle, which is 360° in degrees, is equal to \(2\pi\) radians. One radian, therefore, is roughly 57.3° (since \(\frac{180}{\pi}\)).

Many students find it easier to work with degrees because it's a more common measurement in daily life. However, radians offer significant advantages in mathematical calculations, especially in calculus and other areas of higher mathematics.

To convert radians to degrees, we use the conversion factor \(\frac{180}{\pi}\). Conversely, to convert degrees to radians, we use the conversion factor \(\frac{\pi}{180}\). In the context of trigonometry, radians are often more convenient, as they simplify many formulas, making it essential to understand how to work with both degrees and radians.