Maths Theorems

## Other Theorems

The topics highlighted in bold are theorems while others are properties or proofs .

## Trigonometric Equations

1. General Solutions
	1. sinθ = 0 ; cosθ = 0 ; tanθ = 0 [Using Periodicity]
	2. sinθ = sinα ; cosθ = cosα ; tanθ = tanα [Using Factorization Formulae]
	3. sin2θ = sin2α ; cos2θ = cos2α ; tan2θ = tan2α [Convert to cos]
	4. acosθ + bsinθ = c , ... θ = 2nπ + α ± β [Divide throughout by **√**(a2+ b2) ]
2. **Sine Rule** [Find sine of angle being changed ]
3. **Cosine Rule** [Resolve & use Distance Formula]
4. **Projection Rule** [Use Cosine Rule]
5. **Half Angle Formulae**
	1. sin (A/2) [Use 1 - cosA]
	2. cos (A/2) [Use 1 + cosA]
	3. tan (A/2) [Use sin(A/2) / cos(A/2)]
6. **Area of Triangle** [similar to sine rule . Find sin C for changing C .]
7. **Hero's Formula** [Use Area of Triangle and expand sin A as 2sin(A/2)cos(A/2) & apply half angle Formulae .]
8. **Napier's Analogies** [Use sine Rule & factorization formulae]

## Pair of Straight Lines

1. **Combined Equation of Line through Origin** [Simpy multiply a1x + b1y = 0 and a2x + b2y = 0]
2. **Converse of Theorem 1** [Take b=0 and b=/= 0]
3. **Acute Angle between pair of straight lines** [Use formula of angle between two lines having slopes m1 , m2 ]

## Vectors

1. **Collinearity** . If ma + nb = 0 ; a & b are collinear
2. **Coplanarity** . if r = ax + by , then r,x,y are coplanar [using collinearity and parallelogram law]
3. **Converse of Theorem 2**
4. **If a,b,c are 3 non coplanar vectors** , then r can be defined as linear combination of a,b,c .
5. **Section Formula**
	1. Internal Division
	2. External Division
6. **Volume of Parallelopiped**
7. **The Volume of Tetrahedron**
8. **Geometrical Applications of Vectors**
	1. Medians of Triangle are concurrent
	2. Angle Bisectors of triangle are concurrent
	3. A Quadrilateral is a parallelogram if and only if diagonals bisect each other .
	4. Median of Trapezium is parallel to parallel sides and half of sum of lengths of parallel sides .
	5. Angle Subtended by a semicircle is a right angle
	6. Altitudes of triangle are concurrent

## 3D Geometry

1. **l2 + m2 + n2 = 1** [Resolve as x=rcosα y=rcosβ z=rcosγ]
2. Relation between direction cosines and direction ratios [Equate direction cosines to a constant]
3. Acute Angle between two lines [Using dot product formula]

## Line

1. **Equation of Line : r = a + λb** [Draw a line AP ; draw a vector b parallel to it ]
2. **Equation of line passing through A & B : r = (1-λ) a + λb** [Draw lines AP and AB]
3. Distance of a Point from line []
4. Skew Lines - Shortest Distance
5. Distance between Parallel Lines

## Plane

1. **Equation of plane : r.n = p** [Draw the normal vector and then dot product formula]
2. **r.n = a.n** [dot product condition]
3. Distance of a point from a Plane

## Differentiation

1. **If f(x) is differentiable at a point , then it is also continuous at that point .** [converse is not true]
2. **Composite Functions**
3. **Inverse Functions**
4. Derivatives of inverse trigonometric Ratios : sin-1x , cos-1x , tan-1x , cot-1x , sec-1x , cosec-1x
5. **Parametric Functions**

## Integration

1. **Method of Substitution**
2. **Integration by Parts u.v** [integra of v = w]
3. **ex[f(x) + f'(x)]** [begin with differentiation of question]
4. Integrals :
	1. tan x , cot x , sec x , cosec x [Method of Substitution]
	2. C.T.S. [Multiply divide by 2a & adjust ] OR [Substitute x = asin , a tan ,etc. ]
	3. **√(a2-x2) , √(x2+a2) , √(x2-a2) .** [u.v]

## Definite Integration

1. Reversing the limits
2. Substitution
3. Splitting of Limits
4. Limit 0 to a
5. Limit a to b
6. Dividing Limit 2a
7. Even-odd Functions

**NOTE :**

The topics highlighted in bold are theorems while others are properties or proofs .