Skills Checklist | IB Math AAHL

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Skill Checklist

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

Track your progress:

Don't know

Working on it

Confident

📖 = 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

Skill Checklist

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

38 Skills Available

Track your progress:

Don't know

Working on it

Confident

📖 = 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

7 skills

Circumference & Area of a circle

SL 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.

Arc length (degrees)

SL 3.4

An arc is part of the circumference of a circle. It 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=360θ2πr=180θπr

Sector (degrees)

SL 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

SL 3.4

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

SL 3.4

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)

SL 3.4

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)

SL 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 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

8 skills

Sine and Cosine on the Unit Circle

SL 3.5

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

SL 3.5

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

SL 3.5

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 3.5

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=sinθk∈Z🚫

Symmetry About the X-axis

SL 3.5

sin(−θ)cos(−θ)=−sinθ=cosθ🚫

Symmetry About the Y-axis

SL 3.5

sin(π−θ)cos(π−θ)=sinθ=−cosθ🚫

Symmetry About the Origin

SL 3.5

sin(θ+π)cos(θ+π)=−sinθ=−cosθ🚫

Relating Angles Between Quadrants

SL 3.5

Once the values of sin and cos are memorized for critical angles in the first quadrant, you can use the rules of symmetry and periodicity to determine the values of critical angles in arbitrary quadrants.

Trigonometric Functions

3 skills

Sine and Cosine functions

SL 3.7

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

Sinusoidal Functions

SL 3.7

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

SL 3.7

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)

Reciprocal trig functions

4 skills

Reciprocal Trig Ratios

AHL 3.9

The reciprocal trig functions are functions of the form f(x)1, where f is sin,cos or tan. The three reciprocal trig functions are:

secθ cosecθ cotθ=cosθ1📖 =sinθ1📖 =tanθ1📖

Graph & domain & range of sec(x)

AHL 3.9

secx

Domain: x=2(2k+1)π=±2π,±23π…
Range: secx≤−1 or secx≥1.

Graph & domain & range of cosec(x)

AHL 3.9

cosecx

Domain: x=kπ=0,±π,±2π…
Range: cosecx≤−1 or cosecx≥1.

Graph & domain & range of cot(x)

AHL 3.9

cotx

Domain: x=kπ=0,±π,±2π…
Range: cotx∈R.

Trig Equations

5 skills

Solving sinθ=a

SL 3.8

Since sinθ represents the y-coordinate of a point on the unit circle, solving the equation

sinθ=a

is equivalent to drawing the line y=a, finding the intersections with the unit circle, and determining the angle of these intersections relative to the positive x-axis.

This helps visualize all the possible solutions. For

sinθ=a,0≤θ<2π

the solutions are

θ=sin−1(a),π−sin−1(a)

Solving cosθ=a

SL 3.8

Since cosθ represents the x-coordinate of a point on the unit circle, solving the equation

cosθ=a

is equivalent to drawing the line x=a, finding the intersections with the unit circle, and determining the angle of these intersections relative to the positive x-axis.

This helps visualize all the possible solutions. For

cosθ=a,0≤θ<2π

the solutions are

θ=cos−1(a),2π−cos−1(a)

Solving tan(x)=a

SL 3.8

Since tanθ represents the angle between the line y=xtanθ and the x-axis, solving

tanθ=a

is equivalent to drawing the line y=a, and measuring the minor and major angles it forms with the x-axis:

Solving trig equations algebraically in specific domain

SL 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 Quadratics

SL 3.8

Trigonometric functions can also show up in pseudo-quadratics - a quadratic where the variable being squared is not x but a trig function.

On exams, these equations often require using the Pythagorean identity sin2θ+cos2θ=1.

Inverse trig functions

3 skills

Inverse Sine (arcsin) Function

AHL 3.9

The inverse of sin is arcsin, also written sin−1. Its domain is the range of sin: (−1,1), and its range is the domain of sinx, restricted so that the inverse function passes the vertical line test: [−2π,2π].

The graph of y=arcsinx is the mirror image of sinx (restricted to −2π<x<2π) in the line y=x, giving an increasing function:

Inverse Cosine (arccos) Function

AHL 3.9

The inverse of cos is arccos, also written cos−1. Its domain is the range of cos: (−1,1), and its range is the domain of cosx, restricted so that the inverse function passes the vertical line test: [0,π].

The graph of y=arccosx is the mirror image of cosx (restricted to 0<x<π) in the line y=x, giving a decreasing function:

Inverse Tan (arctan) Function

AHL 3.9

The inverse of tan is arctan, also written tan−1. Its domain is the range of tan: all real numbers, and its range is the domain of tanx, restricted so that the inverse function passes the vertical line test: (−2π,2π).

The graph of y=arctanx is the mirror image of sinx (restricted to −2π<x<2π) in the line y=x, giving an increasing function:

Trigonometric Identities

3 skills

sin²θ+cos²θ=1

SL 3.6

For any value of θ:

sin2θ+cos2θ=1📖

Sine Double Angle Identity

SL 3.6

The double angle identity for sin states that

sin2θ=2sinθcosθ📖

Cosine Double Angle Identity

SL 3.6

The double angle identity for cosine states that

cos2θ =cos2θ−sin2θ=2cos2θ−1=1−2sin2θ📖

These three different forms come from leveraging sin2θ+cos2θ=1.

Further trigonometric identities

5 skills

Sine and Cosine Compound Angle Identities

AHL 3.10

sin(A±B) cos(A±B)=sinAcosB±cosAsinB📖 =cosAcosB∓sinAsinB📖

1+tan²θ=sec²θ

AHL 3.9

If we take the identity sin2θ+cos2θ=1 and divide through by cos2θ we find

1+tan2θ=sec2θ📖

1+cot²θ=cosec²θ

AHL 3.9

If we take the identity sin2θ+cos2θ=1 and divide through by sin2θ we find

1+cot2θ=cosec2θ📖

Tan Double Angle Identity

AHL 3.10

The double angle identity for tan states that

tan2θ=1−tan2θ2tanθ📖

Tan Compound Angle Identity

AHL 3.10

The compound angle identity for tan states that

tan(A±B)=1∓tanAtanBtanA±tanB📖