Definition of a Radian
A radian is a unit of angular measure defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. This definition provides a natural and fundamental way to measure angles, particularly useful in advanced mathematics and physics.
NoteOne radian is approximately equal to 57.2958 degrees.
The full circle, which measures 360° in degrees, is equal to $2\pi$ radians. This relationship forms the basis for conversion between degrees and radians.
Conversion Between Degrees and Radians
To convert between degrees and radians, we use the following relationships:
- 1 radian = $\frac{180°}{\pi}$ ≈ 57.2958°
- 1 degree = $\frac{\pi}{180}$ radians ≈ 0.0175 radians
To convert 45° to radians: 45° × $\frac{\pi}{180}$ = $\frac{\pi}{4}$ radians
To convert $\frac{\pi}{3}$ radians to degrees: $\frac{\pi}{3}$ × $\frac{180°}{\pi}$ = 60°
TipWhen working with radians, it's often more convenient to leave answers in terms of π rather than calculating decimal approximations.
Calculating Area of Sector and Length of Arc
Radians simplify formulas for calculating the area of a circular sector and the length of an arc.
Area of a Sector
The area of a sector with radius $r$ and angle $\theta$ in radians is given by:
$$ A = \frac{1}{2}r^2\theta $$
ExampleCalculate the area of a sector with radius 5 cm and central angle $\frac{\pi}{3}$ radians.
$A = \frac{1}{2} × 5^2 × \frac{\pi}{3} = \frac{25\pi}{6}$ cm²
Length of an Arc
The length of an arc $s$ with radius $r$ and angle $\theta$ in radians is:
$$ s = r\theta $$
ExampleFind the length of an arc with radius 10 m and central angle $\frac{\pi}{4}$ radians.
$s = 10 × \frac{\pi}{4} = \frac{5\pi}{2}$ m
Common MistakeStudents often forget to use radians in these formulas. If given an angle in degrees, always convert to radians first!
Expressing Radian Measure
Radian measures can be expressed in several forms:
- Exact multiples of π (e.g., $\frac{\pi}{6}$, $\frac{5\pi}{4}$)
- Decimal approximations (e.g., 0.5236, 3.9270)
In the IB HL curriculum, it's generally preferred to leave answers in terms of π when possible, as this maintains exactness.
NoteOn HL examination papers, radian measure is assumed unless otherwise indicated.
Link to Trigonometric Functions
Radians are intrinsically linked to trigonometric functions, as explored in AHL 2.9. Using radians often simplifies trigonometric equations and makes certain properties more apparent.
Periodicity
The period of sine and cosine functions in radians is $2\pi$, which aligns naturally with the full rotation of a circle.
Small Angle Approximations
For small angles (close to 0 radians), we have the following approximations:
- $\sin \theta ≈ \theta$
- $\tan \theta ≈ \theta$
- $\cos \theta ≈ 1 - \frac{\theta^2}{2}$
These approximations are most accurate when $\theta$ is expressed in radians.
ExampleCompare $\sin(0.1)$ with 0.1:
$\sin(0.1) ≈ 0.0998$ (calculator value) 0.1 = 0.1000
The difference is only about 0.0002, demonstrating the accuracy of the small angle approximation.
Derivatives of Trigonometric Functions
The derivatives of trigonometric functions are more elegantly expressed when using radians:
- $\frac{d}{dx}(\sin x) = \cos x$
- $\frac{d}{dx}(\cos x) = -\sin x$
- $\frac{d}{dx}(\tan x) = \sec^2 x$
These simple forms only hold when x is in radians.
TipWhen solving problems involving calculus and trigonometric functions, always ensure you're working in radians!
This diagram would illustrate how a radian is defined on the unit circle and how it relates to the sine and cosine functions.
In conclusion, radians provide a natural and powerful way to measure angles, simplifying many calculations in advanced mathematics and physics. Their connection to the fundamental constant π and their utility in trigonometry and calculus make them an essential tool for higher-level mathematical study.
