Problem 21 - 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. $$210^{\circ}$$

Step-by-Step Solution

Answer

The reference angle for \(210^\circ\) is \(30^\circ\).

Step 1: Identify the Quadrant

The first step is to identify the quadrant in which the given angle resides. An angle of \(210^\circ\) lies in the third quadrant (Since it is more than \(180^\circ\) and less than \(270^\circ\).)

Step 2: Find the Reference Angle

Next, to find the reference angle, take the absolute difference between the given angle and the multiple of \(180^\circ\) nearest to it. In this case, you need to subtract \(180^\circ\) from \(210^\circ\).

Step 3: Subtract to Get the Result

Subtract \(180^\circ\) from \(210^\circ\) to get \(30^\circ\). Therefore, the reference angle for \(210^\circ\) is \(30^\circ\).

Key Concepts

Angle Measurement

Angles are one of the fundamental components in trigonometry and are measured in degrees. A full circle is composed of 360 degrees. When you measure an angle, you are essentially starting from a fixed line and looking to see where another line succeeds in rotation. There are a few key terms related to angle measurement that are helpful to understand:

• Positive Angles: These are measured counterclockwise from the positive x-axis.
• Negative Angles: These are measured clockwise from the positive x-axis.
• Standard Position: An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis.

When dealing with trigonometric calculations, proper angle measurement is crucial. Understanding how to measurably define the position of an angle paves the way for deeper exploration into other concepts like reference angles.

Quadrants in Trigonometry

The coordinate plane is divided into four quadrants, which play a major role in understanding the values and signs of trigonometric functions. Each quadrant corresponds to a specific range of angle measures:

• First Quadrant (0° to 90°): All trigonometric functions (sine, cosine, tangent) are positive.
• Second Quadrant (90° to 180°): Only sine and cosecant are positive.
• Third Quadrant (180° to 270°): Only tangent and cotangent are positive.
• Fourth Quadrant (270° to 360°): Only cosine and secant are positive.

When solving problems in trigonometry, knowing the quadrant in which an angle lies helps determine the sign of trigonometric functions. This is essential not only for identifying reference angles but also for ensuring calculations reflect real-world scenarios.

Third Quadrant

The third quadrant is a unique segment of the coordinate plane where angles range from 180° to 270°. In this quadrant, the sine and cosine values of angles are negative, but tangent and cotangent maintain positive values because both sine and cosine are negative, and a negative divided by a negative results in a positive. Understanding the characteristics of this quadrant aids in identifying the correct reference angle and in adjusting calculations accordingly.

When an angle falls into the third quadrant, like our example of 210°, its reference angle is found by taking the difference between the angle and 180°. The reference angle is the same acute angle that the terminal side of the distribution forms with the x-axis, which simplifies many calculations in trigonometry. For 210°, the reference angle is calculated as 210° - 180° = 30°. This helps maintain consistency across various trigonometric assessments where specific values for sine, cosine, and tangent of the reference angle are used. Knowing this ensures solutions align with expected trigonometric functions.