Understanding Radians 📐
Radians are a way to measure angles, just like degrees. They are widely used in mathematics, physics, and engineering because of their natural relationship with circles and trigonometric functions.
What is a Radian? 🤔
A radian is the angle created when the length of an arc on a circle is equal to the radius of that circle.
Key Facts:
- 1 radian is approximately 57.3 degrees.
- A full circle has 2π radians, which is equivalent to 360 degrees.
- Half a circle (a straight angle) is π radians or 180 degrees.
Why Use Radians? 🧮
Radians simplify the relationship between an angle and the arc length of a circle. For example:
- The arc length is calculated as:
Arc length = Radius × Angle in radians
This makes radians the natural choice for mathematical calculations involving circles.
Converting Between Degrees and Radians 🔄
To convert degrees to radians:
Multiply the angle in degrees by π/180
.
To convert radians to degrees:
Multiply the angle in radians by 180/π
.
Examples:
Convert 90 degrees to radians:
90 × π / 180 = π/2 radians
- Approximately 1.57 radians.
Convert π radians to degrees:
π × 180 / π = 180 degrees
.
How Radians Relate to the Unit Circle 🎯
In the unit circle:
- The radius is 1.
- The circumference is
2π
, so a full rotation equals2π radians
. - Some common angles and their radian measures:
- 0 degrees = 0 radians
- 90 degrees =
π/2
radians (about 1.57 radians) - 180 degrees =
π
radians (about 3.14 radians) - 270 degrees =
3π/2
radians (about 4.71 radians) - 360 degrees =
2π
radians (about 6.28 radians)
Practice Problems 📝
Problem 1:
Convert the following angles to radians:
- 45 degrees
- 120 degrees
- 210 degrees
Problem 2:
Convert the following radians to degrees:
π/6
5π/4
2π
Problem 3:
A wheel with a radius of 10 cm rolls 20 cm along a flat surface. What angle (in radians) does the wheel rotate through?