Term 1: Pure Math 3

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Term 1: Pure Math 3 by Mind Map: Term 1: Pure Math 3

1. Trigonometry

1.1. The Cosecant, Secant and Cotangent Ratios

1.1.1. cosec θ =1/sinθ

1.1.2. sec θ = 1/cosθ

1.1.3. cot θ = cos θ/sinθ or 1/tan θ

1.2. Compound Angle Formulae

1.2.1. sin(A+B) = sin A cos B + cos A sin B

1.2.2. sin(A-B) = sin A cos B - cos A sin B

1.2.3. cos(A+B) = cos A cos B + sin A sin B

1.2.4. cos(A-B) = cos A cos B - sin A sin B

1.2.5. tan(A+B) = tan A + tan B/1- tan A tan B

1.2.6. tan(A-B) = tan A - tan B/1+ tan A tan B

1.3. Double Angle Formulae

1.3.1. sin 2A = 2 sin A cos A

1.3.2. cos 2 A = cos^2 A - sin^2 A = 2cos^2 A -1 = 1 - 2sin^2 A

1.3.3. tan^2 A = 2 tan A/ 1- tan^2 A

1.4. Further Trigonometric Identities

1.4.1. 1+ tan^2x = sec^2x

1.4.2. 1 + cot^2x = cosec^2x

1.5. Expressing asinθ + bcosθ in the form R sin(θ+α) or R cos(θ+α)

1.5.1. R = √a^2 + b^2, tan α = b/α

1.5.2. max. value of asinθ + bcosθ is √a^2 + b^2 when sin(θ+α) = 1

1.5.3. min. value of asinθ + bcosθ is -√a^2 + b^2 when sin(θ+α) = -1

2. Logarithm and Exponential Functions

2.1. a^b = y can be annotated as loga y = b

2.2. loge x = ln x log10 x = lg x

2.3. Laws of Logarithm

2.3.1. loga x + loga y = loga (xy)

2.3.2. loga x – loga y = loga (x/y)

2.3.3. loga x^n = n loga x

2.4. Natural logarithm (ln x)

2.4.1. derivative of ln x is 1/x

2.4.2. integral of 1/x is ln x + c

3. Differentiation

3.1. Dy/Dx

3.2. Chain Rule d/dx[f(g(x))]=f'(g(x))g'(x)

3.3. Quotient Rule d/dx[f(x)/g(x)]= g(x)f'(x)-f(x)g'9(x)/[g(x)]^2

3.4. Constant Rule d/dx[c]=0

4. Modulus Inequalities

4.1. properties

4.1.1. |a| ≤ b => -b ≤ a ≤ b

4.1.2. |a| ≥ b => a ≤ -b OR a ≥ b

4.1.3. |a| ≥ |b| => a² ≥ b²

4.1.4. |a| ≤ |b| => a² ≤ b²

5. Partial Fractions

5.1. type1

5.1.1. rational fraction: (px + q)/(ax + b)

5.1.2. partial fraction: A/(ax + b)

5.2. type 2

5.2.1. rational fraction: (px + q)/(ax + b)n

5.2.2. partial fraction: A1/(ax + b) + .......... An/(ax + b)n

5.3. type 3

5.3.1. rational fraction: (px2 + qx + r)/(ax2 + bx + c)

5.3.2. partial fraction: (Ax + B)/(ax2 + bx + c)

5.4. type 4

5.4.1. rational fraction: (px2 + qx + r)/(ax2 + bx + c)n

5.4.2. partial fraction: (A1x + B1)/(ax2 + bx + c) + ...(Anx + Bn)/(ax2 + bx + c)n

5.5. uses

5.5.1. useful for finding binomial approximations

5.5.2. useful for integrating a rational function

6. Polynomials

6.1. synthetic division

6.2. long division