Problem 4 - Algebra and Trigonometry

Question

Find the reference angle for the given angle. $$ \begin{array}{llll}{\text { (a) } 120^{\circ}} & {\text { (b) }-210^{\circ}} & {\text { (c) } 780^{\circ}}\end{array} $$

Step-by-Step Solution

Answer

Reference angles: (a) 60° (b) 30° (c) 60°.

Step 1: Understanding Reference Angles

The reference angle for any angle is the smallest angle that the given angle makes with the x-axis. It's always between 0 and 90 degrees and is positive.

Step 2: Find Reference Angle for 120°

Since 120° is in the second quadrant, we find the reference angle by subtracting it from 180°. So, the reference angle is \( 180° - 120° = 60° \).

Step 3: Find Reference Angle for -210°

First, find a coterminal angle by adding 360°: \(-210° + 360° = 150°\). Since 150° lies in the second quadrant, the reference angle is \(180° - 150° = 30°\).

Step 4: Find Reference Angle for 780°

First, reduce 780° by subtracting 360° twice: \(780° - 360° - 360° = 60°\). Since 60° is already a reference angle, the reference angle for 780° is simply 60°.

Key Concepts

Understanding Angles in Quadrants

When you're learning about angles, it's important to know how they fit into the four different quadrants of a coordinate plane. Think of the quadrants as four sections of a graph divided by the x and y axes. Here's how they are laid out:

• Quadrant I: Both x and y coordinates are positive, where angles range from 0° to 90°.
• Quadrant II: The x coordinate is negative, and the y coordinate is positive, with angles ranging from 90° to 180°.
• Quadrant III: Both x and y coordinates are negative, for angles from 180° to 270°.
• Quadrant IV: The x coordinate is positive, and the y coordinate is negative, ranging from 270° to 360°.

Remember, these quadrants help to determine the properties of an angle, including its reference angle.

Exploring Coterminal Angles

Coterminal angles are fascinating because they share the same terminal side or end point on the coordinate plane. You can find these angles by adding or subtracting full rotations of 360° from your given angle.

For example, the angle \( -210^{\circ} \) in our exercise needs to be adjusted since it's negative. By adding \( 360^{\circ} \), you reach \( 150^{\circ} \), a positive coterminal angle.

Another example is \( 780^{\circ} \), which is greater than 360°. Subtracting \( 360^{\circ} \) twice results in \( 60^{\circ} \), another coterminal angle. This coterminal nature makes understanding and using angles far easier.

Mastering Angle Reduction

Angle reduction is an essential technique for simplifying the process of finding reference angles. When dealing with angles larger than 360° or negative angles, angle reduction comes into play.

• For positive angles much larger than 360°, like \( 780^{\circ} \), you repeatedly subtract 360° until you find an equivalent angle within the 0° to 360° range.
• For negative angles, you add 360° until the angle becomes positive and falls within the standard range.

Through angle reduction, you simplify the problem and can easily proceed to find the reference angle, helping clarify the position and size of the angle on the coordinate plane.