Problem 1 - Precalculus: Functions and Graphs

Question

Exer. 1-4: If the given angle is in standard position, find two positive coterminal angles and two negative coterminal angles. (a) \(120^{\circ}\) (b) \(135^{\circ} \quad\) (c) \(-30^{\circ}\)

Step-by-Step Solution

Answer

For (a) 120°: Positive angles 480°, 840°; Negative angles -240°, -600°. For (b) 135°: Positive angles 495°, 855°; Negative angles -225°, -585°. For (c) -30°: Positive angles 330°, 690°; Negative angles -390°, -750°.

Step 1: Understanding Coterminal Angles

Coterminal angles share the same initial and terminal sides. They can be found by adding or subtracting complete circles (or revolutions), which in degrees is 360°.

Step 2: Finding Coterminal Angles for 120°

To find two positive coterminal angles of 120°, add 360°: \[120° + 360° = 480°\]and add another 360°: \[120° + 720° = 840°\].To find two negative coterminal angles, subtract 360°: \[120° - 360° = -240°\]and subtract another 360°: \[120° - 720° = -600°\].

Step 3: Finding Coterminal Angles for 135°

For the angle 135°, add 360° to find a positive coterminal angle: \[135° + 360° = 495°\].Add another 360°: \[135° + 720° = 855°\].For negative angles, subtract 360°: \[135° - 360° = -225°\]and subtract another 360°: \[135° - 720° = -585°\].

Step 4: Finding Coterminal Angles for -30°

To find positive coterminal angles for -30°, add 360°: \[-30° + 360° = 330°\].Add another 360°: \[-30° + 720° = 690°\].For negative coterminal angles, subtract 360°: \[-30° - 360° = -390°\]and subtract another 360°: \[-30° - 720° = -750°\].

Key Concepts

Standard Position

When we talk about angles in their standard position, it’s all about where they start and how they’re oriented on the coordinate plane. An angle is in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis. This is like setting a baseline from which we measure the angle's rotation.

• The initial side is fixed on the positive x-axis.
• The vertex (the point where the two sides of the angle meet) is at the coordinate origin (0,0).
• The terminal side is where the angle stops after rotating counterclockwise or clockwise.

Having a consistent starting position for angles is important because it provides a standard way to measure and discuss angles with others. With this, every angle has a unique location, regardless of how much it’s rotated.

Angle Measurement

Angle measurement revolves around different units to quantify how much space an angle covers. The most common unit is degrees, which is a familiar concept introduced quite early in mathematics. A complete circle encompasses 360°, so 1° is 1/360th of a full turn.

• Angles are measured starting from the initial side in the standard position.
• Direction matters: counterclockwise increases the angle, while clockwise decreases it.
• Angle measures can be positive or negative, indicating direction of rotation.

When dealing with angles, it’s crucial to not only know the measure but also the direction of the rotation that distinguishes between angles like 120° and -240°. This system of measurement allows us to easily compare angles and work with them systematically.

Revolutions

Revolutions help in understanding coterminal angles, which are multiple angles that share the same position. When an angle completes one full turn, it is said to have made a full revolution. In degrees, a full revolution is 360°. Two angles are coterminal if they differ by a full revolution or multiples thereof:

• To find a coterminal angle, simply add or subtract 360°.
• Coterminal angles are not unique; there are infinitely many such angles for any given angle.
• Adding or subtracting multiples of 360° ensures the angle returns to the same terminal side.

Understanding revolutions aids in solving problems related to coterminal angles and gives the flexibility to convert between positive and negative angles, helping us to see how different angles relate geometrically.