1. Algebra
Common Formulas:
1. $ (a+b)^2 = a^2+2ab+b^2 $
2. $ (a+b)^3 = a^3+b^3+3a^2b+3ab^2 $
3. $ (a^2-b^2)=(a+b)(a-b) $
4. $ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca $
5. $ (a+b)^2 + (a-b)^2 = 2(a^2+b^2) $
6. $ (a+b)^2 - (a-b)^2 = 4ab $
7. $ (a-b)^3 = a^3-b^3-3a^2b+3ab^2 $
Solving Quadratic Equation: Let any quadratic equation be $ ax^2+bx+c = 0 $. Roots of the equation are given by, $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $
If $ {b^2-4ac} = 0 $ then roots are real and equal.
If $ {b^2-4ac} > 0 $ then roots are distinct and real.
If $ {b^2-4ac} < 0 $ then roots are imaginary.
If $ \alpha $ and $ \beta $ are two roots of the equations, then
Sum of roots: $ \alpha + \beta = \frac{-b}{a} $
Product of roots: $ \alpha \times \beta = \frac{c}{a} $
Difference of roots: $ \alpha - \beta = \frac{\sqrt{b^2 - 4ac}}{a} $
For example, Let $ x^2+x+1=0 $ is a quadratic equation and we need to find the roots of the equation.
For the given equation, $ b^2-4ac = 1 - (4 \times 1 \times 1) = 1-4 = -3 < 0 $
This means that the roots of the equation are imaginary.
Roots will be, $ x = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 - \sqrt{-3}}{2}, \frac{-1 + \sqrt{-3}}{2} $
For $ \sqrt{-1} = i $, then roots will be, \[x = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 - \sqrt{3}i}{2}, \frac{-1 + \sqrt{3}i}{2}\]
Binomial Expansion:
If we need to expend $ (1+x)^n $ in powers of x where n is positive integer, we expand it binomially.
Expansion will be
\[(1+x)^n = 1 + _{1}^{n}{x} + _{2}^{n}{x^2} + ....... + _{i}^{n}{x^i} + ..... + x^n \]
It can be written as \[(1+x)^n = \Sigma _{j=0}^{n} (_{j}^{n} C)(x^j)\]
where \[_{j}^{n} C = \frac{n!}{(n-j)! j!}, n! = n(n-1)(n-2).......3.2.1\] and $ 0! = 1 $.
The number of terms in the expansion of $ (1+x)^n $ are (n+1)
Binomial expansion for any index, i.e. if n is not a positive integer.
\[(1+x)^n = 1 + \frac{n}{1!}x + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + .......... \infty terms\]
If |x| << 1 then $ (1+x)^n = 1 + nx $ i.e. we can ignore highest power of the expansion.
For example: Expand $ (1 + x ) ^ {-2} $.
$ (1+x)^{-2} = 1 + \frac{-2}{1!}x + \frac{-2(-2-1)}{2!}x^2 + \frac{-2(-2-1)(-2-2)}{3!}x^3 + .......... \infty = 1 -2x -3x^2 - 4x^3 + ...... $.
Try Yourself:
Q1. Find the roots of the equations:
(a) $ x^2 + 2x + 3 = 0 $
(b) $ x^2 - 2x - 3 = 0 $
(c) $ x^2 + 30x + 1 = 0 $
(d) $ 2x^2 + x + 1 = 0 $
Q2. Expand following:
(a) $ (1+x)^7 $
(b) $ (1+x)^{-7} $
(c) $ (1+y)^{-1} $
(d) $ (1+z)^{-10} $
***Solutions of the above problems will be uploaded soon.....
2. Trigonometry
Relation between arc length, l, radius of the circle, r, and angle $ \theta $ subtended by the arc at the center, \[l = r \theta\]
Usefull Trigonometric Formulas for Right Angle Triangle
Let any right angle triangle with Right angle at B, as shown in the figure
click here to see large image.
1. $ sin A = \frac{a}{b} $
2. $ cosA = \frac{c}{b} $
3. $ tanA = \frac{sin A}{cosA}=\frac{a}{c} $
4. $ cosecA = \frac{1}{sinA}=\frac{b}{a} $
5. $ secA = \frac{1}{cosA}= \frac{b}{c} $
6. $ cot A = \frac{1}{tanA}= \frac{cosA}{sinA}= \frac{c}{a} $
7. $ sin^2 A + cos^2 A = 1 $
8. $ 1 + tan^2 A = sec^2 A $
9. $ 1+ cot^2 A = cosec^2 A $
Above formulas are valid only for Right angle triangle.
In general, for any triangle ABC where A, B and C are the angles, and a, b, and c are the sides opposite to angle A, B and C respectively.
1. $ \frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c} $
2. $ cos A = \frac{b^2+c^2-a^2}{2bc} $
Value of some trigonometric functions:
|
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0
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30
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45
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60
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90
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sin
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0
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$
\frac{1}{2} $
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$
\frac{1}{\sqrt{2}}$
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$
\frac{\sqrt{3}}{2}$
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1
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cos
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1
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$ \frac{\sqrt{3}}{2}$
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$ \frac{1}{\sqrt{2}}$
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$
\frac{1}{2} $
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0
|
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tan
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0
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$\frac{1}{\sqrt{3}}$
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1
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$
\sqrt{3} $
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$
\infty $
|
Compound Formula:
1. $ sin(A\pm B) = sinAcosB \pm sinBcosA $
2. $ cos(A\pm B) = cosAcosB \mp sinAsinB $
3. $ tan(A\pm B) = \frac{tanA\pm tanB}{1 \mp tanAtanB} $
4. $sin2A = 2sinAcosA $
5. $ cos2A = cos^2A - sin^2 A = 1 - 2sin^2 A = 2cos^2 A - 1 $
6. $tan2A = \frac{2tanA}{1-tan^2A} $
In the first quadrant, all trigonometric functions have positive values. In second quadrant, sine and cosec are positive and all others are negative. In third quadrant, tan and cot are positive and all others are negative. In fourth quadrant, cos and sec are positive and all other are negative.
1. $ sin(-\theta) = - sin \theta $
2. $ cos(-\theta) = cos \theta $
3. $ tan(-\theta) = - tan \theta $
