Question

Convert each angle in radians to degrees. \(-\frac{3 \pi}{4}\)

Short answer

-135 degrees

Step-by-step solution

- Understand the Conversion Formula

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

- Substitute the Radian Value

Substitute \(-\frac{3 \pi}{4}\) into the formula: \[ \text{Degrees} = -\frac{3 \pi}{4} \times \frac{180}{\pi} \]

- Simplify the Expression

Cancel out the \pi in the numerator and denominator, then perform the multiplication: \[ \text{Degrees} = -3 \times \frac{180}{4} = -3 \times 45 = -135 \]

- State the Result

The angle \(-\frac{3 \pi}{4}\) in radians is equivalent to \-135\ degrees.

Key concepts

degrees and radians

Understanding degrees and radians is crucial for angle conversions. Degrees and radians are units used to measure angles. A circle is divided into 360 degrees, which means a full rotation around the circle equals 360 degrees.
Radians, on the other hand, are based on the radius of the circle. One radian is the angle created when the radius length is wrapped along the circumference. There are approximately 6.28 radians in a full circle, which corresponds to \(2\text{π}\) radians, because the circumference of a circle is \(2\text{πr}\) where \(r\) is the radius.
Simply put, 360 degrees is the same as \(2\text{π}\) radians. This relationship is essential when converting between the two measurement systems.

angle conversion

Angle conversion involves changing an angle from one unit of measurement to another, such as from radians to degrees or vice versa. This process depends on a simple formula.
To convert radians to degrees, multiply the radian value by \(\frac{180}{\text{π}}\). This formula comes from the fact that 180 degrees is equivalent to \(\text{π}\) radians. To convert degrees to radians, multiply the degree value by \(\frac{\text{π}}{180}\).
For example, to convert \(-\frac{3 \text{π}}{4}\) radians to degrees, you use the conversion formula:

mathematical formulas

Mathematical formulas are key to quick and accurate calculations. For converting radians to degrees, use the formula:
\[ \text{Degrees} = \text{Radians} \times \frac{180}{\text{π}} \]
In the given problem, we substitute \(-\frac{3 \text{π}}{4}\):
\[ \text{Degrees} = -\frac{3 \text{π}}{4} \times \frac{180}{\text{π}} \]
Cancel out \(\text{π}\) and simplify the math:
\[ -3 \times \frac{180}{4} = -135 \]
This final result shows that \(-\frac{3 \text{π}}{4}\) radians equals -135 degrees.

trigonometric functions

Trigonometric functions often require converting between degrees and radians. These conversions are necessary when working with functions like sine, cosine, and tangent, which are fundamental in trigonometry and calculus.
For instance, if you need to find the sine of \(-\frac{3 \text{π}}{4}\), it is easier to understand in degrees as -135 degrees. Recognizing that -135 degrees is in the third quadrant helps when determining the sign and value of the trigonometric functions.
Radian to degree conversion simplifies the use of standard trigonometric values and enables better comprehension of angles and their measures in different contexts.