Problem 19 - Precalculus : Building Concepts and Connections

Question

Skills This set of exercises will reinforce the skills illustrated in this section. In Exercises \(9-22,\) find the reference angle for each of the angles given. $$300^{\circ}$$

Step-by-Step Solution

Answer

The reference angle of \(300^{\circ}\) is \(60^{\circ}\).

Step 1: Identify the position of the angle.

Identify which quadrant the angle is in. An angle of \(300^{\circ}\) is in the fourth quadrant since it's greater than \(270^{\circ}\) but less than \(360^{\circ}\).

Step 2: Calculate the reference angle.

Use the formula to calculate the reference angle for an angle in the fourth quadrant: \(360^{\circ}\) - original angle. For this case, the reference angle will be \(360^{\circ}\) - \(300^{\circ}\).

Step 3: Calculate the result.

Carry out the subtraction to get the reference angle: \(360^{\circ}\) - \(300^{\circ}\) = \(60^{\circ}\).

Key Concepts

Trigonometry

Understanding trigonometry is fundamental when we deal with angles and their relationships. Central to trigonometry is the concept of the reference angle, which is the acute angle (<90 degrees) that your given angle makes with the x-axis. Reference angles are helpful because they allow us to evaluate trigonometric functions for angles outside the first quadrant, using the corresponding acute angle in the first quadrant.

For example, when you’re given an angle like 300 degrees, you can determine that the trigonometric functions for this angle will have the same values as for its reference angle, just with different signs depending on which quadrant the angle lies in. In practice, this means you can apply your knowledge of the sine, cosine, and tangent of the reference angle, which in this case is 60 degrees, to find the values of the functions for the 300-degree angle.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants by the x and y axes. The quadrants are labeled counterclockwise starting from the upper right: Quadrant I (where both x and y are positive), Quadrant II (x is negative, y is positive), Quadrant III (both x and y are negative), and Quadrant IV (x is positive, y is negative). This ordering is essential in identifying the signs of trigonometric functions.

Each quadrant imparts different signs to the trigonometric functions. For instance, in the fourth quadrant where the 300-degree angle lies, sine values are negative while the cosine and tangent values are positive. Recognizing the quadrant where an angle is located is the first step in finding its reference angle, which can reveal the accurate value and sign of a trigonometric function.

Angle Measurement

Angles are measured in terms of degrees or radians, and understanding how to measure and classify them is crucial for solving trigonometry problems. Angles greater than 0 degrees but less than 90 degrees are considered acute and share the first quadrant. Angles that range from 90 to 180 degrees are obtuse and occupy the second quadrant, whereas angles from 180 to 270 degrees are called reflex angles and can be found in the third quadrant. Lastly, angles between 270 to just below 360 degrees belong to the fourth quadrant.

To find a reference angle, one must subtract the given angle from the nearest multiple of 90 or 180 degrees, depending on which quadrant it's in. For angles in the fourth quadrant, like our 300-degree example, you subtract the angle from 360 degrees to find the reference angle. This method simplifies complex angles back to their simplest, most usable form for evaluating trigonometric functions, making it a powerful tool in a mathematician's arsenal.