A Level FSMQ Additional Maths OCR

This subject is broken down into 29 topics in 7 modules:

  1. Algebra 4 topics
  2. Enumeration 6 topics
  3. Coordinate Geometry 4 topics
  4. Pythagoras and Trigonometry 4 topics
  5. Calculus 3 topics
  6. Numerical Methods 4 topics
  7. Exponentials and Logarithms 4 topics
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  • 7
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  • 29
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  • 10,779
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This page was last modified on 28 September 2024.

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FSMQ Additional Maths

Algebra

Algebra: Algebraic Manipulation

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Algebra: Algebraic Manipulation

Basic Algebraic Manipulation

  • Understand the properties of real numbers, including: commutative, associative, distributive, and identity.

  • Familiarise yourself with the operations of addition, subtraction, multiplication, division and exponentiation of algebraic expressions.

  • Learn and apply the order of operations (BIDMAS: Brackets, Indices, Division and Multiplication, Addition and Subtraction).

  • Be confident in factorising quadratics, cubics and higher order polynomials using common factorisation, the difference of two squares and The Factor Theorem.

Equations and Inequalities

  • Develop skills in solving linear, quadratic, cubic and higher order equations, as well as simultaneous equations using analytical and graphical methods.

  • Understand and apply the concepts of absolute value equations and inequalities, and know how to solve them.

  • Familiarise yourself with the concepts of rational equations and inequalities.

Algebraic Fractions

  • Know how to simplify complex algebraic fractions, including those with polynomials in the numerator or denominator.

  • Be able to add, subtract, divide and multiply algebraic fractions, and to simplify to lowest terms.

  • Learn how to solve equations involving algebraic fractions.

Powers and Roots

  • Recall the laws of indices and apply them to manipulate expressions involving powers and roots.

  • Be vigilant on checking the conditions under which the rules of algebra holds.

  • Understand how to simplify and rationalise surds.

  • Be able to express complex numbers in a+bi form.

Further Techniques and Applications

  • Gain confidence in working with complex numbers particularly, the arithmetic of complex numbers, the modulus-argument form and the use of De Moivre's Theorem.

  • Have a good understanding of the binomial theorem for rational indices, and their use in series expansion and approximations.

  • Learn to solve equations with variable powers using the logarithm properties.

  • Use algebraic manipulation to simplify, rearrange and solve real-world problems.

Course material for FSMQ Additional Maths, module Algebra, topic Algebra: Algebraic Manipulation

FSMQ Additional Maths

Pythagoras and Trigonometry

Pythagoras and Trigonometry: Trigonometric Identities

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Pythagoras and Trigonometry: Trigonometric Identities

Section 1: Introduction to Trigonometric Identities

  • Trigonometric identities are mathematical formulas relating the ratios of different angles in a right-angled triangle.
  • They established the relationship between sine, cosine, and tangent of different angles.
  • It's essential to understand and remember key radian measures: (pi) radians = 180 degrees, (pi/2) radians = 90 degrees and (2pi) radians = 360 degrees.

Section 2: Fundamental Trigonometric Identities

  • The Pythagorean identities: the squares of the sine and cosine of an angle add up to one. Symbolically, (sin^2(x) + cos^2(x) = 1).
  • The quotient identities: relate tangent and cotangent to sine and cosine. Stated as (tan(x) = sin(x)/cos(x)) and (cot(x) = cos(x)/sin(x)).
  • The reciprocal identities: relate secant, cosecant, and cotangent to cosine, sine, and tangent respectively. Highlighted as (sec(x) = 1/cos(x)), (csc(x) = 1/sin(x)), (cot(x) = 1/tan(x)).

Section 3: Compound Angle Identities

  • The sum and difference identities: they allow the simplification of expressions involving the sine, cosine, or tangent of multiple angles. Given as (sin(a ± b) = sin(a) cos(b) ± cos(a) sin(b)), (cos(a ± b) = cos(a) cos(b) ∓ sin(a) sin(b)) and (tan(a ± b) = (tan(a) ± tan(b)) / (1 ∓ tan(a)tan(b))).

Section 4: Double Angle and Half Angle Identities

  • The double-angle identities: depict the cosine and sine of double angles. Formulated as (sin(2a) = 2 sin(a) cos(a)) and (cos(2a) = cos^2(a) - sin^2(a)) or (cos(2a) = 2cos^2(a) - 1) or (cos(2a) = 1 - 2sin^2(a)).
  • The half-angle identities: show the sine, cosine and tangent of half-angles. Noted as (sin(a/2) = ± sqrt{[1 - cos(a)]/2}), (cos(a/2) = ± sqrt{[1 + cos(a)]/2}) and (tan(a/2) = ± sqrt{[1 - cos(a)] / [1 + cos(a)]}) or (tan(a/2) = sin(a) / [1 + cos(a)]) or (tan(a/2) = [1 - cos(a)] / sin(a)).

Remember, the sign (plus or minus) depends on the quadrant where the angle is located.

Course material for FSMQ Additional Maths, module Pythagoras and Trigonometry, topic Pythagoras and Trigonometry: Trigonometric Identities

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