Question

Convert each angle in radians to degrees. \(\frac{3 \pi}{20}\)

Short answer

27°

Step-by-step solution

- Understand the Conversion Formula

To convert an angle in radians to degrees, use the conversion formula: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\boldsymbol{\text{π}}} \]

- Substitute the Given Radians

Substitute \( \frac{3 \text{π}}{20} \) for radians in the formula: \[ \text{Degrees} = \frac{3 \text{π}}{20} \times \frac{180}{\text{π}} \]

- Simplify the Expression

Simplify the expression by canceling out the \(\text{π}\): \[ \text{Degrees} = \frac{3}{20} \times 180 \]

- Perform the Multiplication

Multiply the numbers: \[ \text{Degrees} = \frac{3 \times 180}{20} = \frac{540}{20} = 27 \]

- Write the Final Answer

The angle in degrees is 27°.

Key concepts

radians to degrees

Converting angles from radians to degrees is a fundamental concept in trigonometry. It helps in interpreting angles in a more familiar format, as degrees are commonly used in navigation, geometry, and other practical applications. To begin with, you should know the conversion formula:

The conversion formula from radians to degrees is given by: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\boldsymbol{\text{π}}} \]

This formula works because one complete revolution around a circle is 360 degrees or \[ 2 \text{π} \] radians, hence dividing 180 by π converts radians to degrees directly.

angle measurement

Understanding how angles are measured is key to mastering trigonometry. Angles can be measured in different units, the most common being degrees and radians. Here's a breakdown of how these measurements work:

- **Degrees**: A degree (\text{°} ) is 1/360th of a full circle. Using degrees is convenient for everyday applications like navigation and construction.
- **Radians**: A radian is based on the radius of a circle. Specifically, one radian is the angle formed when the arc length equals the radius of the circle. Since the circumference of a circle is \[ 2 \text{π} \] times the radius, there are \[ 2 \text{π} \] radians in a full circle, which is equivalent to 360 degrees.

These two measurement systems are interconvertible, and understanding how to switch between them is vital.

trigonometric conversion

Trigonometric conversion involves translating angles from one unit of measurement to another, and it's crucial for solving various mathematical problems. Let's go through an example, converting an angle given in radians to degrees using the formula we discussed earlier.

We'll use an example \[ \frac{3 \text{π}}{20} \] radians:

• **Step 1**: Apply the conversion formula:

\[ \text{Degrees} = \frac{3 \text{π}}{20} \times \frac{180}{\text{π}} \]

• **Step 2**: Cancel out π \(\text{π} \) to simplify the expression:

\[ \text{Degrees} = \frac{3}{20} \times 180 \]

• **Step 3**: Multiply the simplified fraction:

\[ \text{Degrees} = \frac{3 \times 180}{20} = \frac{540}{20} = 27° \]
This tells us that \[ \frac{3 π}{20} \] radians is equivalent to 27 degrees. Simple steps like these can help you handle trigonometric conversions with ease.