Problem 26 - Precalculus

Question

For the following exercises, convert angles in radians to degrees. \(\frac{3 \pi}{4}\) radians

Step-by-Step Solution

Answer

135 degrees

Step 1: Understand the Relationship

First, recall that the conversion factor between radians and degrees is that \(\pi\) radians is equivalent to 180 degrees. This means that to convert from radians to degrees, you multiply by \(\frac{180}{\pi}\).

Step 2: Identify the Given Angle

The angle given in this problem is \(\frac{3 \pi}{4}\) radians. We need to convert this angle into degrees using the conversion factor from Step 1.

Step 3: Apply the Conversion Factor

Multiply the given radians \(\frac{3 \pi}{4}\) by \(\frac{180}{\pi}\) to convert it to degrees:\[\frac{3 \pi}{4} \times \frac{180}{\pi} = \frac{3 \times 180}{4}\]

Step 4: Simplify the Expression

Now, calculate the multiplication in the numerator: \(3 \times 180 = 540\).Then, divide by 4:\[ \frac{540}{4} = 135 \]

Step 5: State the Result

The angle in degrees is 135 degrees. Thus, \(\frac{3 \pi}{4}\) radians is equal to 135 degrees.

Key Concepts

Angle Measurement

When you study trigonometry, one of the first things you'll learn is how angles can be measured. Angles are the space between two intersecting lines, usually measured in degrees or radians.
- **Degrees**: Most of us are familiar with degrees—it’s often what we use in our day-to-day experiences, like giving directions or measuring turns. There are 360 degrees in a full circle.
- **Radians**: Radians are a bit less intuitive because they relate the angle to the radius of a circle. In fact, a full circle in radians is equal to the circumference of a circle divided by its radius, which is approximately 6.283 or, more commonly, noted as \(2\pi\).
Each measurement has its use case and can be useful in different mathematical contexts. Understanding both can greatly enhance your understanding of geometry and trigonometry.

Radian to Degree Conversion

Converting radians to degrees involves a simple and straightforward formula. Given that \(\pi\) radians is equal to 180 degrees, you can convert an angle from radians to degrees by multiplying it by \((\frac{180}{\pi})\). Let's break down the steps for clarity.
- **Step 1**: Determine the angle in radians you need to convert. For example, \(\frac{3\pi}{4}\) radians.
- **Step 2**: Use the conversion factor \((\frac{180}{\pi})\). Multiply the radian measure by this factor:
\[\frac{3\pi}{4} \times \frac{180}{\pi}\]
- **Step 3**: Simplify the expression by performing multiplication and division:
First, multiply 3 by 180, which gives 540.Then, divide 540 by 4 to get 135.So, \(\frac{3\pi}{4}\) radians equals 135 degrees.
This formula can be applied to any radian measure to convert it to its equivalent in degrees.

Trigonometry Basics

Trigonometry is a branch of mathematics that deals with the study of triangles and the relationships between their angles and sides. A crucial part of trigonometry is understanding how to measure angles, as this will allow you to calculate distances, heights, and even model real-world phenomena.
- Understanding **Sine, Cosine, and Tangent**: These are the primary functions in trigonometry, deriving from the ratios within a right-angled triangle. - **Sine** is the ratio of the opposite side to the hypotenuse. - **Cosine** is the adjacent side to the hypotenuse. - **Tangent** is the opposite side to the adjacent side.
These functions entail a lot more than just basic triangle calculations; they allow us to make sense of periodic phenomena, wave functions, and more.
Additionally, understanding how to switch between radians and degrees is foundational because:

• Radians make calculus easier to work with in trigonometric contexts.
• Degrees are easier for visualization and everyday use.

Mastering these conversions and functions gives you a strong base in trigonometry, allowing you to tackle more advanced problems efficiently.