Tools

http://www.cymath.com/

http://www.mathportal.org/calculators.php

https://www.desmos.com/calculator

Sketch graphs of trigonometric functions

Graphs of sin θ, cos θ and tan θ

Remember sin graph starts at 0

Amplitude and period transformations

Calculator to test answers: https://www.desmos.com/calculator

Solving triangles

Right Angled Triangles

Sine Rule 
Determine the sides




Determine the angles



Cosine Rule
Determine the sides



Determine the angles

Trig Ratios

30°, 45° and 60°

45°, 45°, 90° 

When you work with trig and radians, π means the same as 180° because 2π radians is a full circle 360°

Trig Ratios for positive multiples of

30°, 45° and 60°

Negative Multiples of 30°, 60° and 45° trig ratios 



Reduction Formulae 

Trigonometric identities

Identities 



Proving trigonometric identities



Solving trigonometric equations using identities

Trigonometric equations

Using CAST




Equations with multiple angles




Trigonometric equations that factorise




Solving a trigonometric equation by using reciprocal function




Make everything sin and cos

Trigonometric functions

Basic Trigonometry: Sin, Cos, Tan 



Relation between the trigonometric functions

Differential Calculus

Average gradient

Change in yChange in x

=△y△x

=y2−y1x2−x1

Limits




Limits from first principles



Differentiation
Differentiate from first principles



Rules of differentiation



Notation 




The gradient of a curve




Turning points

Intro to Differential Calculus

Functional notation, average gradient, limits, derivative from first principles and rules

Sketch graphs

Straight line




You can also let y=0 and calculate x to draw the line.

The Circle




The Ellipse




The Parabola




The Hyperbola


Solve simultaneous equations graphically

The Circle and the tangent

The equation of a circle

(x-a)² + (y-b)² = r²



The equation of a tangent

Equations of straight lines

Two point form





One point and a gradient (Point slope form)





The distance between two points

,



The midpoint of a line segment

,

Different forms of the Straight line

The Gradient(Slope)-intercept form

$y=mx+c$

Where m = gradient = difference in y divided by the difference in x





The Intercept form





http://www.vitutor.com/geometry/line/intercept_form.html

The General form

$ax+bx+c=0$
$\mathrm{<br>}$ $$

Angle between points is given

If you are given an angle (Θ) and you must determine the gradient: The tangent of the angle is the gradient of the line: tanΘ = m


The equation is given

$y=mx+c$
m is the gradient.

Two points are given

$m=\frac{y_2-<!--\; -\; -->y_1}{x_2-<!--\; -\; -->x_1}$



Parallel lines
If two lines are parallel, their gradients are the same.




Perpendicular lines
If two lines are perpendicular, their gradients are opposite and inverse of one another.

Manipulation of technical formulee

Rules for manipulating formulas 

1. What you do on the one side of the equal sign you must do on the other side.
2. If you must make an exponent the subject, you use logs.
3. The inverse operation of a 'root' is a 'power'.
   $\text{T}heinverseof\sqrt[3]{}is({)}^3$

Make u the subject of:

$v=u+atv-<!--\; -\; -->at=u\therefore <!--\; \therefore \; -->u=v-<!--\; -\; -->at$

Word problems

Also a good read: http://www.themathpage.com/alg/word-problems.htm

Linear equations

Linear equations

If the variable occurs more than once first simplify. The variables must all be taken to one side and then added or subtracted.

If the variable occurs in the denominator of a fraction, it must be removed. This is done by multiplying by the variable throughout the equation.

Simultaneous linear equations













Solving simultaneous equations (one linear and one quadratic)

Logarithms

A logarithm is just another form of an exponent.

$2^3=8lo{g}_{2}8=3$

The logarithm tells you what the exponent is:


How to do log calculations without calculator.

If you have an unknown value in log form, you first change the equation to exponential form and then solve the unknown value.

$x=lo{g}_{5}2525=5^x{5}^2=5^x\therefore <!--\; \therefore \; -->x=2$

Logs, antilog, log_e and antilog_e a calculator.

Use your calculator '2ndF log' to get antilog and '2ndF IN' to obtain log_e


Log Laws


Law 1

If the base and number are the same the log cancels out and the answer is 1.

log_aa 

= 1

Law 
If you have two logs and their bases are the same, you multiply the numbers.

log_ca + log_cb

= log_cab

Law 3
If you have two logs with the same base and you are subtracting, you divide the numbers.

log_ca - log_cb

= log_c^a/_b

Law 4
If you have a base, number and exponent you write the exponent in front of the log.

log_ab^m

= m log_ab

Law 5
If you have a log with a different base and number you can change the base.

$lo{g}_{a}b=\frac{lo{g}_{c}b}{lo{g}_{c}a}$

Simplify logs without a calculator




Changing the base of a log

Surds

A surd is a value in a root that cannot be determined unless you use a calculator.
Example: √2 (square root of 2) can't be simplified further so it is a surd

How to multiply surds

If you have two surds that are multiplied and their roots are the same, you put them under one root and multiply the values.

$\sqrt{3a}\times <!--\; \times \; -->\sqrt{3a}=\sqrt{3a\times <!--\; \times \; -->3a}=\sqrt{9a^2}=3a$


How to add and subtract surds

If you add and subtract surds, you always have to find a common factor and then simplify the expression.

$2\sqrt{2}+6\sqrt{8}=2\sqrt{2}+6\sqrt{2\times <!--\; \times \; -->4}\text{(write 8 as 2 times 4)}=2\sqrt{2}+6\times <!--\; \times \; -->2\sqrt{2}\text{(the root of 4 is 2)}=2\sqrt{2}+12\sqrt{2}=\sqrt{2}(2+12)\text{(take }\sqrt{2}\text{out as common factor)}=14\sqrt{2}$

Rationalizing denominators 

When fractions have a surd in the denominator we can rationalize the denominator so that it is not in surd form. We do this by multiplication.

Factors 

To simplify an expression you must always try to take out common factors.

$\sqrt{4-<!--\; -\; -->4x}=\sqrt{4(1-<!--\; -\; -->x)}=\sqrt{4}\times <!--\; \times \; -->\sqrt{1-<!--\; -\; -->x}\text{(}\sqrt{4}=2)=2\sqrt{1-<!--\; -\; -->x}$


Equations with surds 

Given an equation that contains a surd you must always first change the surd into a form that has no root sign and then solve the unknown values.

How to Simplify Radicals/Surds Summery

Exponents

Exponent Laws

Law 1
If you multiply, and the bases are the same you add the exponents.

a ⁿ × a ^m = a ^n+m

Law 2
If you divide, and the bases are the same you subtract the exponents.

a ⁿ ÷ a ^m = a ^n-m

Law 3
If you have an exponent inside a bracket and an exponent outside the bracket, you multiply the exponents.

(a ⁿ )^m = a ^nm

Law 4
Anything except zero to the power of zero is one.

a ⁰ = 1

Law 5
If you have a negative exponent and you want to make it positive, you change the value inside the bracket around. Only the sign of the exponent changes.

(a)^{-m = (1/a)m}

Law 6
If you have a fraction as an exponent you change it to root form. The denominator will then be the root.
$a^{\frac{m}{n}}=\sqrt[n]{a^m}$


Exponential equations

Algebraic Fractions

Simplification of fractions

Multiplication of fractions

Note 1: When you divide two fractions, you turn the fraction after the division sign around, and then you change the sign to a multiplication.

Note 2: Remember that you can only cancel out if there is a multiplication or division sign between terms.

How to add and subtract fractions

Cube Fractions

Function notation

Functional notation is the notation for expressing functions as  


Function values

You use function values when a value for x is given and you are asked to solve an equation. To solve the equation, you put x's value into each x in the equation.

The remainder theorem - using long division

The factor theorem

You use the factor theorem as an easy way to find out whether a term is a factor of an equation, without doing long division: