Sine Rule for Non Right-Angled Triangles
Sin A ÷ a = Sin B ÷ b = Sin C ÷ c
Cos Rule for Non Right-Angled Triangles
a² = b² + c² – 2bc cos A, or for finding angles
Sin = opposite ÷ hypotenuse, Cos = Adjacent ÷ hypotenuse, Tan = Opposite ÷ Adjacent
For triangles when you need to find out the area but there is no way of finding out the height the formula that you use is:
Golden Rule of Algebra
Change the side; change the sign!
Quadratic Equation Formula
-b ± √(b² - 4ac)
Two numbers that multiply to make the number at the end and add to make the number in the middle with the "x" after it.
Area and Volume
Area of a Square or Rectangle = Two of the sides multiplied
Area of a Parallelogram = base x height
Area of a Trapezium = ½base (b¹ + b²)
Area of a Circle = π x radius²
Area of a Triangle = ½base x height
Volume of a Cube or a Rectangular Prism = height x width x depth
Volume of any Irregular Prism or a Cylinder = Area of the base x height
Volume of a Pyramid or Cone = ⅓ area of the base x height
Volume of a Sphere = 4/3 π x radius³
Perimeter of a Circle = Pi x diameter
Direct and Inverse Variation and Proportionality
For questions involving direct variation such as: d (is directly proportional) to v; you would divide d by v to find a constant which is known a "k", which can then be used to solve the problem.
Example: Cost of Apples (d) = price per apple (k) x number of apples bought
For questions involving inverse proportion you it would be k = d x v ; because d is inversely proportional to 1 over v. Which is the same as d = k over v.
A matrix is a group of numbers encased in a large set of brackets. We always read matrices in the order: Rows then Columns.
e.g. (7 3 5 9) = a 1 x 4 matrix.
To add and subtract matrices you just add or subtract the corresponding numbers.
e.g. ( 7 3 5 9) + (3 6 0 1) = (10 9 5 10), to add or subtract matrices they must have the same order. They must be in the same format.
When multiplying a matrix by a scalar - the scalar is the # outside the bracket. So you just multiply whatever is inside the bracket by whatever is outside the bracket.
When multiplying 2 matrices you have to multiply the row of the first matrix by the column of the second matrix. But you cannot multiply certain matrices: e.g.
To find the inverse of a matrix you must first find the determinant of the matrix. You can do this by using the formula:
Determinant = 1 over (ad - bc), a and d being the first and last numbers in the matrix and b and c being the other two. Once you have found the determinant you write it outside the matrix and then alter the numbers inside the matrix. You make a and d swap places and make b and c negative (unless one or both are negative in which case they become positive).
You can leave it like this (which is acceptable in the exam, and easier :P) or you can multiply the determinant by all of the numbers inside the matrix, basically using the determinant as a scalar. However there is more room for error is you do this and you risk losing marks so leaving the determinant outside the bracket is less risky just remember to alter the numbers inside the matrix.
Rules of Indices
When multiplying indices you add the powers
When dividing indices you subtract the powers
When there is an index outside both inside and outside a bracket you multiply the powers
When the index is negative it means there is a reciprocal (e.g. get rid of the minus sign and put the number at the bottom of a fraction, like one over 3 to the power of 7 is the same as 3 to the power of -7)
When the index is a fraction the top number (numerator) is a power and the bottom number (denominator) is a root. (e.g. five to the power of 2 thirds is the same as the cube root of five squared)
This may seem obvious but anything to the power of one is just itself, we just don't bother putting the one there (wastes time ;P)
- Functions are always written in the form: a(x) = an equation
- So solve a function you simply substitute the number that you are given in the bracket for the "x" in the equation.
- When trying to solve a composite function you always solve the second function of outside the bracket first and then apply it to the first function outside the bracket.
- To find the inverse of a function you form an equation with the function in the form y = and then the equation with x.
- You then make x the subject of the formula and when writing your answer swap the places of x and y.
- E.g. y = x + 2
- So x = y - 2
- But you would write the inverse of the function (?) as x - 2
-1, 0, 0, 1 = Reflection in the line x = 0
1, 0, 0, -1 = Reflection in the line y = 0
0, 1, 1, 0 = Reflection in the line y = x
0, -1, -1, 0 = Reflection in the line y = -x
0, 1, -1, 0 = Rotation Clockwise about (0,0) 90°
0, -1, 1, 0 = Rotation Anticlockwise about (0,0) 90°
-1, 0, 0, -1 = Rotation 180° about (0,0)
MUST REMEMBER THESE FOR EXAM!!!!!!