Problem 4 - 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) \(570^{\circ}\) (b) \(\frac{2 \pi}{3}\) \((c)-\frac{5 \pi}{4}\)

Step-by-Step Solution

Answer

For 570°: positive 930°, 1290°; negative 210°, -150°. For \(\frac{2\pi}{3}\): positive \(\frac{8\pi}{3}\), \(\frac{14\pi}{3}\); negative \(-\frac{4\pi}{3}\), \(-\frac{10\pi}{3}\). For \(-\frac{5\pi}{4}\): positive \(\frac{3\pi}{4}\), \(\frac{11\pi}{4}\); negative \(-\frac{13\pi}{4}\), \(-\frac{21\pi}{4}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position on the coordinate plane. To find coterminal angles, we can add or subtract full rotations of the circle. In degrees, this means adding or subtracting multiples of 360°. In radians, this translates to adding or subtracting multiples of \(2\pi\).

Step 2: Finding Coterminal Angles in Degrees: 570°

To find coterminal angles for 570°, we can add and subtract 360° as needed:- Positive coterminal angle: \(570° + 360° = 930°\). Another positive coterminal angle: \(570° + 2 \times 360° = 1290°\).- Negative coterminal angle: \(570° - 360° = 210°\). Another negative coterminal angle: \(570° - 2 \times 360° = -150°\).

Step 3: Finding Coterminal Angles in Radians: \(\frac{2\pi}{3}\)

To find coterminal angles for \(\frac{2\pi}{3}\), add and subtract \(2\pi\):- Positive coterminal angle: \(\frac{2\pi}{3} + 2\pi = \frac{2\pi}{3} + \frac{6\pi}{3} = \frac{8\pi}{3}\). Another positive coterminal angle: \(\frac{8\pi}{3} + 2\pi = \frac{14\pi}{3}\).- Negative coterminal angle: \(\frac{2\pi}{3} - 2\pi = \frac{2\pi}{3} - \frac{6\pi}{3} = -\frac{4\pi}{3}\). Another negative coterminal angle: \(-\frac{4\pi}{3} - 2\pi = -\frac{10\pi}{3}\).

Step 4: Finding Coterminal Angles in Radians: \\(-\frac{5\pi}{4}\\)

To find coterminal angles for \(-\frac{5\pi}{4}\), add and subtract \2\pi\:- Positive coterminal angle: \(-\frac{5\pi}{4} + 2\pi = -\frac{5\pi}{4} + \frac{8\pi}{4} = \frac{3\pi}{4}\). Another positive coterminal angle: \(\frac{3\pi}{4} + 2\pi = \frac{11\pi}{4}\).- Negative coterminal angle: \(-\frac{5\pi}{4} - 2\pi = -\frac{5\pi}{4} - \frac{8\pi}{4} = -\frac{13\pi}{4}\). Another negative coterminal angle: \(-\frac{13\pi}{4} - 2\pi = -\frac{21\pi}{4}\).

Key Concepts

Measuring Angles

Measuring angles is a fundamental skill in geometry and trigonometry. An angle measures the rotation between two rays that share a common endpoint, known as the vertex. Angles are measured in two primary units:

• Degrees: A full circle rotation is 360 degrees (\(360^{\circ}\))
• Radians: The equivalent of one full circle is \(2\pi\) radians

Degrees are often used in everyday scenarios, like navigation and construction. Radians, however, are used extensively in mathematics due to their natural fit with trigonometric functions.
Using radians simplifies many mathematical expressions and equations, making them indispensable in calculus and physics. A simple way to convert between degrees and radians is: \[1^{\circ} = \frac{\pi}{180} \text{ radians and } 1 \text{ radian} = \frac{180}{\pi}^{\circ}\] Beginners can use this conversion to switch back and forth between the two, depending on the context.

Standard Position

Imagine a clock face where the minute and hour hands form angles. In mathematics, we often place angles in a **standard position** on a coordinate plane to simplify calculations and visualizations.
In standard position:

• The vertex of the angle is at the origin (0,0).
• The initial side of the angle lies along the positive x-axis.
• The angle is typically measured counterclockwise from this initial side.

For instance, if the angle originated from 3 o'clock on a clock face, and measured counterclockwise around the circle, it would be in standard position. This setup makes it easier to compare and calculate various angle properties, like coterminal angles, because everyone is working from the same starting point.

Degrees and Radians

Degrees and radians are two different units for measuring angles, each with unique benefits.
**Degrees: **

• Common in daily life for measuring angles
• Simple and easy to visualize
• Divides a circle into 360 equal parts

**Radians: **

• Essential for advanced mathematics
• Links directly with the radius of a circle
• Expresses angles as the length of the arc they subtend when matched with the circle's radius

Radians are incredibly useful in calculus, as they simplify derivatives and integrals involving trigonometric functions. In contrast, degrees are intuitive but less practical for computational purposes in higher math.

Angle Rotation

Angle rotation refers to the direction in which an angle is measured from its initial position. The direction of rotation can be either clockwise or counterclockwise.
**Clockwise Rotation:**

• Moves the terminal side in the same direction as a clock's hands.
• Results in negative angle measures.

**Counterclockwise Rotation:**

• Moves the terminal side opposite the direction of a clock's hands.
• Results in positive angle measures.

Recognizing the direction of rotation is crucial when determining the sign and measure of an angle. It defines whether an angle is positive or negative and affects the calculation of coterminal angles. Coterminal angles can be found by adding or subtracting full rotations (360° or \(2\pi\) radians) from a given angle. Each full circle added or subtracted results in a new angle positioned similarly on the coordinate plane.