Skills Checklist | IB Math AIHL

Skill Checklist

Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.

18 Skills Available

Track your progress:

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📖 = included in formula booklet • 🚫 = not in formula booklet

Track your progress:

Don't know

Working on it

Confident

📖 = included in formula booklet • 🚫 = not in formula booklet

Circles: Radians, arcs and sectors

6 skills

Circumference & Area of a circle

SL AI 3.4

The ratio of a circle's perimeter to its diameter is constant in all circles. This constant is called π (pi). Since the diameter is twice the radius, the circumference of a circle is

C=2πr📖

The area of a circle is

A=πr2📖

where r is the radius of the circle.

Sector (degrees)

SL AI 3.4

A sector is an area enclosed between a circular arc and the two radii of the circle touching each end of the arc:

The area of a circle is πr2, and there are 360 degrees of rotation in a circle. Therefore, a sector with central angle θ is 360°θ of a full circle, and has area

A=360°θπr2

Radian measure

AHL AI 3.7

One radian is the interior angle of an arc which has a length equivalent to the radius r of the circle. Since the circumference of a circle is given by 2πr, then, there are 2π total radians in a circle (the equivalent of 360°).

Converting Between Radians & Degrees

AHL AI 3.7

Since the perimeter of a full circle is 2πr, the angle θ corresponding to a full circle (360°) is

r2πrrad=360°

πrad=180°🚫

Some key angles in radians and degrees:

Degrees	Radians
0°	0
30°	6π
45°	4π
60°	3π
90°	2π

Arc length (radians)

AHL AI 3.7

An arc is defined by the radius r of the circle and the angle θ that the arc "sweeps out" over the circle's perimeter.

Since the arc length is a fraction of the overall circumference determined by the value of the angle θ, the arc length is calculated as

l=rθ

Sector area (radians)

AHL AI 3.7

A sector is an area enclosed between a circular arc and the two radii of the circle touching each end of the arc:

The area of a circle is πr2, and there are 2π radians in a circle. Therefore, a sector with central angle θ is 2πθ of a full circle, and has area

A=2πθ⋅πr2

A=21θr2📖

The Unit Circle

7 skills

Sine and Cosine on the Unit Circle

AHL AI 3.8

The key idea with triangles who's hypotenuse lie on the unit circle (that form an angle of θ with the x-axis) is that cosθ represents length of the base, and sinθ represents the height.

Take a look at the graph below and notice the following relationships always hold:

cosθsinθ=x-coordinate=y-coordinate

Key values of Sin, Cos & Tan

AHL AI 3.8

The following table shows the values of sinθ and cosθ for the so called critical angles θ. These are angles that give "nice" values for sin and cos:

θ (rad)	sinθ	cosθ
0	0	1
6π	21	2√3
4π	2√2	2√2
3π	2√3	21
2π	1	0

Quadrants

AHL AI 3.8

The unit circle can be divided into quadrants based on the sign of cosθ and sinθ. These correspond to the 4 quadrants produced by the intersection of the x and y axes. The quadrants are denoted Q1, Q2, Q3 and Q4.

Quadrant	sin	cos
Q1	+	+
Q2	+	-
Q3	-	-
Q4	-	+

Periodicity

SL AI 2.5

Since a full circle is 2π radians, adding 2π to any angle θ gives the same point on the unit circle. In fact, adding any integer multiple of 2π gives the same point:

cos(θ+2kπ)sin(θ+2kπ)=cosθ=sinθ

Symmetry About the X-axis

AHL AI 3.8

sin(−θ)cos(−θ)=−sinθ=cosθ

Symmetry About the Y-axis

AHL AI 3.8

sin(π−θ)cos(π−θ)=sinθ=−cosθ

Symmetry About the Origin

AHL AI 3.8

sin(θ+π)cos(θ+π)=−sinθ=−cosθ

Trigonometric Functions

3 skills

Sine and Cosine functions

SL AI 2.5

Notice that both sinx and cosx have a domain of x∈R and a range of (−1,1).

Sinusoidal Functions

SL AI 2.5

A sinusoidal function is a generalization of sin and cos to the form

asin(b(x+c))+d

acos(b(x+c))+d

Tan function

AHL AI 3.8

The tan function is defined by tanx=cosxsinx.

The domain is thus x=22k+1π (there are vertical asymptotes at those x′s), and the range is all real numbers R.

The function has roots at x=0,±π,±2π… (ie x=kπ where k∈Z)

Trig Equations

1 skill

Solving trig equations algebraically in specific domain

AHL AI 3.8

When we have a trig equation where the argument to the trig function is of the form ax+b, we need to find the domain of ax+b using the domain of x. For example, if 0≤x<2π and we have sin(2x+2π)=1, then

2⋅0+2π≤2x+2π<2⋅2π+2π

therefore

2π≤2x+2π<29π

Trigonometric Identities

1 skill

sin²θ+cos²θ=1

AHL AI 3.8

For any value of θ:

sin2θ+cos2θ=1📖