Topic 3: Circular Functions and Trigonometry
Introduction
"The aims of this topic are to explore the circular functions and to solve problems using trigonometry. On examination papers, radian measure should be assumed unless otherwise indicated."
- From IB Math SL Guide
Circle
Radian Measure
There are 2π radians in a complete circle, and π radians in a half circle. Therefore as there are 360 degrees in a complete circle, and 180 degrees in a half circle, we can derive this equation to convert
Degrees = Radians * 180/π
Radians = Degrees * π/180
Length of an arc
The length of an arc is equal to s=r(θ), where r= radius, (θ)=inscribed angle in radians, and s=the length of the arc.
This formula is synonymous with the formula for the circumference of a circle where (theta)=2(pi).
Area of a Sector
A = (1/2)(θ)(r^2) where r is the radius.
Cosine and Sine (relative to Unit Circle)
sinθ=y cosθ=x tanθ=y/x CAST Beginning from the IV section will let you know which are positive (Cosine, All, Sine, Tangent)
| Quadrant | SIN | COS | TAN |
|---|---|---|---|
| I | + | + | + |
| II | + | - | - |
| III | - | - | + |
| IV | - | + | - |
Double Angle Formulae
sin2(θ)= 2sin(θ)cos(θ)
cos2(θ)= cos^2(θ)-sin^2(θ)= 2cos^2(θ)-1=1-2sin^2(θ)
Triangles
Area
Area of a triangle = (1/2) ab sin C