Problem 27 - Algebra and Trigonometry

Question

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ 50^{\circ} $$

Step-by-Step Solution

Answer

Positive: 410°, 770°; Negative: -310°, -670°.

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but differ by a full circle (360°). To find coterminal angles, you add or subtract multiples of 360° from the given angle.

Step 2: Finding Positive Coterminal Angles

To find positive coterminal angles, add 360° to the given angle of 50°.\[ 50° + 360° = 410° \]Additionally, add 360° again to find another positive coterminal angle:\[ 410° + 360° = 770° \]Thus, 410° and 770° are positive angles coterminal with 50°.

Step 3: Finding Negative Coterminal Angles

To find negative coterminal angles, subtract 360° from the given angle of 50°.\[ 50° - 360° = -310° \]Next, subtract 360° again to find another negative coterminal angle:\[ -310° - 360° = -670° \]Therefore, -310° and -670° are negative angles coterminal with 50°.

Key Concepts

Positive Angles

Positive angles are angles measured in a counterclockwise direction from the initial side to the terminal side of the angle in the coordinate plane. When you are given an angle, such as 50°, which is already positive, you can always find other positive angles that are coterminal by adding full rotations of a circle, which is 360°, to it.

For example, to find two positive angles coterminal with 50°, simply add 360° to it:

• Start with the given angle: 50°
• Add 360° to get: 50° + 360° = 410°
• Add 360° again for another coterminal angle: 410° + 360° = 770°

This way, 410° and 770° are positive angles that share the same terminal side as 50°, making them coterminal.

Negative Angles

Negative angles are measured in the clockwise direction from the initial side to the terminal side of the angle. Finding negative coterminal angles involves subtracting full circles (360°) from a given angle.

Starting with 50°, a positive angle, you can find negative coterminal angles by subtracting 360°:

• Begin at 50°
• Subtract 360°: 50° - 360° = -310°
• Subtract 360° again to find yet another coterminal angle: -310° - 360° = -670°

Thus, -310° and -670° are negative angles that end at the same position as the starting angle of 50°.

Standard Position

The concept of standard position is crucial when working with angles, especially in trigonometry. An angle is said to be in standard position if its vertex is at the origin of the coordinate plane — a point denoted as (0,0) — with its initial side positioned along the positive x-axis.

Understanding an angle in standard position makes it easier to define coterminal angles because the terminal side can rotate around the origin, staying consistent with the definition. When an angle is placed in standard position, it helps in visualizing how angles share the same terminal side but have different measures due to the full rotations added or subtracted from the initial angle.

Angle Measurement

Angle measurement is a fundamental concept that allows us to describe the size of an angle in degrees or radians. In many practical applications, such as this exercise involving coterminal angles, the unit of degrees is used.

One complete rotation of a circle corresponds to a measure of 360°. This is the cornerstone of understanding coterminal angles since adding or subtracting 360° from an angle will not alter its terminal side's position. When we say that angles are coterminal, they have the same terminal position despite having different numeric measurements.

Learning to easily add or subtract 360° can simplify the process of finding angles that share the same fundamental position and illustrate cycles and repetitions in angles.