Convert 45 degrees from degrees to radians. Give the answer in exact form in terms of pie .
The Correct Answer and Explanation is:
To convert 45 degrees from degrees to radians, we use the conversion factor:
[
\text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right)
]
Step-by-Step Process:
- Start with the given angle in degrees:
The given angle is 45 degrees. - Use the conversion factor:
We multiply the degree measure by (\frac{\pi}{180}), since there are (\pi) radians in 180 degrees. [
45^\circ \times \left(\frac{\pi}{180^\circ}\right)
] - Simplify the expression:
Perform the multiplication: [
45 \times \frac{\pi}{180} = \frac{45\pi}{180}
] - Simplify the fraction:
Now, simplify (\frac{45}{180}). Both 45 and 180 are divisible by 45: [
\frac{45}{180} = \frac{1}{4}
] Therefore, the expression becomes: [
\frac{1}{4} \pi
]
Final Answer:
The angle of 45 degrees is equivalent to (\frac{\pi}{4}) radians.
Explanation:
The conversion from degrees to radians is necessary because the two units measure angles differently. Degrees are based on dividing a circle into 360 equal parts, while radians are based on the radius of the circle. Specifically, 1 radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
In the conversion formula (\text{Radians} = \text{Degrees} \times \frac{\pi}{180}), we use the fact that 180 degrees corresponds to (\pi) radians, establishing the ratio between the two units. By multiplying the degree measure by this ratio, we convert it to radians. This is a standard formula used whenever you need to convert between degrees and radians.