,
Trigonometry: radians
 

[ radians ][ arc length ][ sector area][ small angles ]

 

 

 

 

What is a 'radian'?

radian definition

A radian is the angle subtended at the centre of a circle by an arc the same length as the radius of the circle.

Units

1C (meaning 1 radian)= 57.296 deg.

 

back to top

 

Arc length

radian definition #2

The arc length is proportional to its subtended angle.

Hence, if θ(theta) is in degrees and 'l' is the arc length:

equation for arc length

 

An angle can be expressed in radians by dividing the arc length by the radius.

Therefore θ in radians is given by:

arc length

Therefore for a circle(a 360 deg. angle), where the arc length is '2πr' and the radius is 'r' , the number of radians is 2πr/r , i.e. 2π .

 

back to top

 

Sector area

The area of a sector is proportional to the angle its arc subtends at the centre.

If a sector contains an angle of θo then its area is given by:

sector area

However, if θ is in radians, remembering there are 2π radians in a circle:

sector area #2

 

 

back to top

 

Small angles

small angles

For small angles(<10 deg.) there is a convergence between the value of the angle in radians with the value of its sine & tangent.

This approximate sine value may be expressed as:

sine theta approximates t o theta

The approximate cosine value is obtained thus:

small angle - cosine equation

 

 

 

 

back to top

 

your stop for the best in math, science & programming tutorials on the Net revision help to get a better result incremental success advanced physics for secondary/high school, including much in-depth content common to first year university courses your one stop for the best in math, science and programming tuition revision help for a better result incremental success advanced physics for high school/secondary and 1st year university fast-track learning for everyone

[ PURE MATHS ][ MECHANICS ][ STATISTICS ]

VIDEO

parametric differen.
equation of a tangent
equation of a normal
rate of change prob.1
rate of change prob.2
the Chain Rule
Chain Rule probs. #1
Chain Rule probs. #2
Chain Rule probs. #3
the Product Rule
parametric eqs.prob#1
parametric eqs.prob#2
intro. to integration
integration by parts 1
integration by parts 2
area under a curve
volumes of revolution
area between curves
Binomial Theorem
Bin. Theorem problems
Trig. Identities
Half Angle Formula
Double Angle Formula
Vectors,mod,resultant
Vector problems
Vector problems in 3D
MORE . . .
 

INTERACTIVE

 
derivative formula
tangents & normals
inverse functions
MORE . . .
 

EXAM PAPERS(.pdf)

 
Edxl C1 Pure specimen
Edxl C1 Pure answers
Edxl C2 Pure specimen
Edxl C2 Pure answers
Edxl C3 Pure specimen
Edxl C3 Pure answers
Edxl C4 Pure specimen
Edxl C4 Pure answers
MORE . . .

TOPIC NOTES(.pdf)

the derivative formula
tangents & normals
maxima & minima
chain rule
diffn.exponentials,logs
diffn.trigonometric fns.
product rule
quotient rule
parametric equations
implicit equations
differential equations
integration formula
int.'by substitution'
int.'by parts'
algebraic fractions
definite integrals
areas under curves
volumes of revolution
Trapezium Rule
integ. diff. equations
radians
sine, cosine, tangent
sec, cosec, cotan
Sine & Cosine Rules
Pythagorean ID's
compound angles
Sigma Notation
arithmetic progression
geometic progression
line between points
more straight lines
parametric equations
circles & ellipses
MORE . . .
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Google