Analytic geometry

The Standard Form of the Equation of a Circle

(h, k) is the center

r is the radius

(x, y) is any point on the circle

Example 1: Write the standard form of the equation of the circle with center (5, 7) and r = 4.

*Plug in 5 for h, 7 for k, and 4 for r

*4 squared is 16

*Standard form of circle with center (5, 7) and radius 4

Putting it into standard form we get:

During solving this question,steps we should do:

-is identifying the cordinate of circle and the radius of circle from the question given or from the figure given

-next note down all the information

- Sketch diagram of the circle with all information to give you clear image as shown above.

-finally put all informaton into the equation.

Trigonometry Identities

Pythagorean_trigonometric_identity

Double-angle formulae

Half-angle formulae

During study i found it is harder to solve questions involving double angles and half angles because it has many idenities and they all have to be memorized.But they can be memorized and get used by doing a lot of trigonometry exercises with varies conditions.

Example
Prove that cosx cos2x + sinx sin2x = cosx

when solving this question it is adviseable to choose for LHS because is easier and have more information rather than RHS

Starting from the left hand side:

=cosx cos2x + sinx sin2x

= cos(-x) = cosx

1.we start by identifying any double angles or half angles or phytogorean identites.

2.In this question,we can found two double angles.

3.Next step. we substitute the angles with suitable identities to prove the RHS.

Logarithms

important feature of logarithms

The major property of logarithms is that they map multiplication to addition.This ability stems from the following identity: which by taking logarithms becomes

A related property is reduction of exponentiation to multiplication. Using the identity;

Reduction is important in calculation because it simplyfy the calculation.Besides it is useful for students as it take a little time to solve problems.

Group theory

From my opinion the group shown above is sometime difficult to compare.But finally i found both can be compared through calculation.

Change of base

While the re are several useful identities, the most important for calculator use lets one find logarithms with bases other than those built into the calculator (usually loge and log10). To find a logarithm with base b, using any other base k

Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other. So to calculate the log with base 2 of the number 16 with a calculator

Logarithmic identity

Polynomial

In mathematics, a polynomial is an expression constructed from one or more variables (also known as indeterminates) and constants, using the operations of addition, subtraction, multiplication, and raising to constant non-negative integer powers. For example,x^2-4x+7is a polynomial, but x^2-4/x-7x^3/2is not, because its second term involves division by a variable, and also because its third term contains an exponent that is not an integer.
Polynomials are one of the most important concepts in algebra and throughout mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary story problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics, and are used in calculus and numerical analysis to approximate other functions. Polynomials are used to construct polynomial rings, one of the most powerful concepts in algebra and algebraic geometry.