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Induction Proofs, Trigonometric Equations, Calculus, Vectors - Coggle…
Induction Proofs
Proving Results Involving Series
Steps
Prove Sn is true for n = 1
Assume Sn is true for n = k
Try to prove Sn is true for n = k + 1
Hence, Sn is true for all positive integers n
Proving Divisibility
To be divisible
integer needs to be a factor of that number
Proving Results Involving Inequalities
Move entire equation to one side and leave the other side as zero
Trigonometric Equations
The Auxiliary Angle Method
Steps
Equals to co-efficient in actual equation
same for Rsin
R = co-efficient squared and then square rooted
Results for Sum & Differences of Angles
Double Angle Results
The T Formula
Solving Trig Equations Using the T Method
Tan
Negative
Quadrants 2 & 4
Positive
Quadrants 1 & 3
Final Check
Sub x = pi into original equation
Number not easily inversed
Change calc to radians
pi
Solving Trigonometric Equations
Calculus
Log and Exponential Functions
Rules
Product Rule
Quotient Rule
Power Rule
Chain Rule
Trigonometry Differentiation
Integration by Substitution
Steps
Let u = equation
Find
make dx subject
Integrate like normal
Sub original equation back into u
Solve
Limits
Have to be changed
Original limits are subbed into x in u
Limits are changed with results
Further Calculus Skills
Vectors
Further Operations with Vectors
Dot Products and the Angle between Two Vectors
Dot product / Scalar product
Angle between Two Vectors
magnitude
Found by squaring i and j value
Adding
Square rooting
Geometrical Applications
Lie on the same line
Same vectors
Same Direction
Different magnitude
Properties of Vectors
Parallel Vectors
Unit Vectors
Inverse of magnitude
Rationalise result
Vector Projection
Scalar projection of a onto b
Vector Projection
a perpendicular to b
Solving Motion Problems Using Vectors
velocity
Speed + Direction
Geometric Results and Proofs Using Vectors
Introduction to Vectors
Vector Quantity
Magnitude
Represent with
Horizontal
Vertical
Direction
Positive = right and up
Negative = left and down
Vector Arithmetic
Adding and Subtracting Vectors
Add vertical and horizontal components
Multiplying by a Scalar
Multiply each component by separate non-vector integer
Vectors in the Cartesian Plane
Magnitude of Vectors
square root of
x-value squared + y-value squared
Unit Vector
Magnitude of 1
Form
