Level 2 Further Mathematics AQA

This subject is broken down into 74 topics in 7 modules:

  1. Number 3 topics
  2. Trigonometry 17 topics
  3. Algebra 23 topics
  4. Coordinate Geometry (2 Dimensions Only) 11 topics
  5. Calculus 8 topics
  6. Matrix Transformations 4 topics
  7. Geometry 8 topics
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  • 7
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  • 74
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  • 23,239
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  • 2+
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This page was last modified on 28 September 2024.

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Further Mathematics

Trigonometry

Understanding Sine, Cosine, and Tangent Ratios

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Understanding Sine, Cosine, and Tangent Ratios

Understanding Sine, Cosine, and Tangent Ratios

Introduction to Ratios in Trigonometry

  • Trigonometry is a branch of mathematics that studies triangles, specifically the relationships among their angles and sides.
  • The three primary ratios in trigonometry are sine (sin), cosine (cos), and tangent (tan).
  • These ratios are built upon using the right-angled triangle, where one angle is exactly 90 degrees.

Sine Ratio

  • The sine of an angle is described as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
  • Mathematical representation: sin(θ) = Opposite/Hypotenuse.
  • Remember, the hypotenuse is the longest side of the right-angled triangle, always opposite the right angle.

Cosine Ratio

  • The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
  • Mathematical representation: cos(θ) = Adjacent/Hypotenuse.
  • Relevant to remember: the adjacent side is the side that forms the angle θ along with the hypotenuse.

Tangent Ratio

  • The tangent of an angle is the ratio of the sine of the angle to the cosine of the angle. For a right-angled triangle, it's the ratio of the length of the opposite side to the length of the adjacent side.
  • Mathematical representation: tan(θ) = sin(θ)/cos(θ) or tan(θ) = Opposite/Adjacent.
  • Note this importance: the tangent ratio only makes sense as long as the cosine of the angle isn't zero to avoid division by zero.

Usage and Practical Applications

  • These ratios are essential in countless situations where angles and distances need to be determined, such as in navigation, architecture, physics, engineering, and computer graphics.
  • Calculators often have specific buttons for these trigonometric functions, enabling the computation of the ratios for any given angle.
  • Sine and cosine ratios are crucial in a periodic phenomenon like sound and light waves, while tangent ratios predominantly find their use in slope calculations.

Remember SOHCAHTOA

  • An easy way to remember the three basic functions sin, cos, and tan is by the mnemonic SOHCAHTOA.
    • 'S' stands for Sine: Opposite/Hypotenuse.
    • 'C' stands for Cosine: Adjacent/Hypotenuse.
    • 'T' stands for Tangent: Opposite/Adjacent.
  • This acronym is typically one of the most beneficial memorisation aids when first learning about trigonometric ratios.

Course material for Further Mathematics, module Trigonometry, topic Understanding Sine, Cosine, and Tangent Ratios

Further Mathematics

Algebra

Solution of Linear and Quadratic Inequalities

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Solution of Linear and Quadratic Inequalities

Introduction to Linear and Quadratic Inequalities

  • Inequalities extend the concept of equations by introducing 'greater than' (>) and 'less than' (<) conditions.
  • These inequalities can exist between two algebraic expressions.
  • Solutions to inequalities are ranges of values that satisfy the given conditions.

Solving Linear Inequalities

  • Linear inequalities are solved in a manner very similar to linear equations.
  • The aim is to isolate the variable on one side of the inequality.
  • Add, subtract, multiply, or divide both sides just as in equations, but be aware that multiplying or dividing by negative numbers reverses the inequality direction.
  • For example, to solve the inequality 2x + 3 < 7, subtract 3 from both sides to get 2x < 4, then divide by 2 to get x < 2.

Solving Quadratic Inequalities

  • When dealing with quadratic inequalities, start by rearranging the inequality to one side i.e., in the form of ax^2 + bx + c < 0 or ax^2 + bx + c > 0.
  • Factorise or complete the square to find 'critical values' where the expression equals zero.
  • Take ranges between critical values and test a value in each range by substituting it into the factorised inequality.
  • For example, to solve (x - 1)(x - 3) > 0, the critical values are x = 1 and x = 3. Testing gives the solutions x < 1 or x > 3.

Representation of Solutions

  • Solutions to the inequalities can be represented on a number line.
  • The solutions to quadratic inequalities are typically ranges of values and can be represented on a number line with hollow or filled circles indicating whether the endpoints are included in the solution.

Remembering these key formulas and rules will enable you to tackle linear and quadratic inequalities comfortably and effectively.

Course material for Further Mathematics, module Algebra, topic Solution of Linear and Quadratic Inequalities

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