Problem 27 - Algebra and Trigonometry

Question

\(25-30\) me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ \frac{3 \pi}{4} $$

Step-by-Step Solution

Answer

Positive: \(\frac{11\pi}{4}\), \(\frac{19\pi}{4}\); Negative: \(-\frac{5\pi}{4}\), \(-\frac{13\pi}{4}\).

Step 1: Convert to degrees

The given angle is in radians. Start by converting it to degrees using the formula \( ext{degrees} = ext{radians} imes \frac{180}{ ext{pi}} \). For this angle, \( \frac{3\pi}{4} \) radians becomes \( \frac{3\pi}{4} \times \frac{180}{\pi} = 135 \) degrees.

Step 2: Find positive coterminal angles

To find positive coterminal angles, add \( 2\pi \) radians (equivalent to \( 360 \) degrees) to the original angle. Calculate \( \frac{3\pi}{4} + 2\pi = \frac{3\pi}{4} + \frac{8\pi}{4} = \frac{11\pi}{4} \). Another coterminal angle can be found by adding \( 4\pi \) (i.e., \( 720 \) degrees), resulting in \( \frac{3\pi}{4} + 4\pi = \frac{3\pi}{4} + \frac{16\pi}{4} = \frac{19\pi}{4} \). Thus, the positive angles are \( \frac{11\pi}{4} \) and \( \frac{19\pi}{4} \).

Step 3: Find negative coterminal angles

To find negative coterminal angles, subtract \( 2\pi \) radians from the original angle. Compute \( \frac{3\pi}{4} - 2\pi = \frac{3\pi}{4} - \frac{8\pi}{4} = -\frac{5\pi}{4} \). Subtracting another \( 2\pi \) gives \( \frac{3\pi}{4} - 4\pi = \frac{3\pi}{4} - \frac{16\pi}{4} = -\frac{13\pi}{4} \). Therefore, the negative angles are \(-\frac{5\pi}{4} \) and \(-\frac{13\pi}{4} \).

Key Concepts

Angle Conversion

When you are dealing with angle measurements, it's important to know how to convert angles between different units. Angles can be measured in degrees or radians, and understanding how to switch between these two is crucial in mathematics. Degrees are commonly used in everyday applications like navigation and construction, while radians are often used in calculus and trigonometry.
Converting angles helps in comparing and understanding various angle measures more effectively and is an essential skill for solving many geometric and trigonometric problems.

Radian to Degree Conversion

To convert an angle from radians to degrees, use the formula:

• degrees = radians × \( \frac{180}{\pi} \)

This formula comes from the relation that \( \pi \) radians is equal to 180 degrees. Let's take an example: the angle \( \frac{3\pi}{4} \) radians.
By multiplying \( \frac{3\pi}{4} \) by \( \frac{180}{\pi} \), you will cancel out \( \pi \) and calculate the angle in degrees as:

• \( \frac{3\pi}{4} \times \frac{180}{\pi} = 135 \text{ degrees} \)

This conversion is crucial as it allows you to work in a measurement system that's more intuitive for interpreting rotations and angles in various fields.

Positive Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position, starting from the positive x-axis. Finding positive coterminal angles involves adding \( 360 \) degrees (or equivalent to \( 2\pi \) radians) repeatedly. For an angle like \( \frac{3\pi}{4} \), to find two positive coterminal angles, perform these calculations:

• First angle: \( \frac{3\pi}{4} + 2\pi = \frac{11\pi}{4} \)
• Second angle: \( \frac{3\pi}{4} + 4\pi = \frac{19\pi}{4} \)

By adding multiples of \( 2\pi \), you can generate as many positive coterminal angles as needed, continuing the pattern as desired.

Negative Coterminal Angles

Just as you can find positive coterminal angles by adding full rotations (\( 2\pi \)), you find negative coterminal angles by subtracting them. This gives us angles that are coterminal but with a negative measure.
For the angle \( \frac{3\pi}{4} \), let's find negative coterminal angles by subtracting \( 2\pi \) and \( 4\pi \) as follows:

• First angle: \( \frac{3\pi}{4} - 2\pi = -\frac{5\pi}{4} \)
• Second angle: \( \frac{3\pi}{4} - 4\pi = -\frac{13\pi}{4} \)

These calculations help in understanding directional rotations and are often used in more advanced areas of math where angles can be negative.