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Trapezium Rule

This guide covers key concepts in advanced mathematics, including trapezium rule integration, solving trig equations, and logarithmic identities. It discusses the application of sin and cos in finding angles, forming quadratic equations, and calculating the area under curves. Additionally, it explores arithmetic and geometric sequences, proofs, and important formulas related to circle geometry and triangle properties. With clear explanations and methods for solving equations, this resource serves as a valuable tool for mastering mathematical techniques and principles.

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Trapezium Rule

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Presentation Transcript

  1. Trapezium Rule • Formula given • 5 ordinates means n=4 strips

  2. Trig Equations 1 • Sin-1 0.5 • Bow ties to get 2 angles • As its 3x goes round the bow tie 2 more times • Take away 15, divide by 3 • Its only x between 0 and 180

  3. Trig Equations II • Subs sin 2 x = 1 – cos2 x or vice versa • Set up quadratic equation • Solve • Don’t forget bow ties Stays the same

  4. Integration • Rearrange to get powers of x • Cube root is 1/3 • Write powers on the bottom as negative powers

  5. Area under the curve • Solve curve = line to find A and B • Integrate between these x values • Integration is area under curve • Here shaded area = area under curve - trapezium

  6. APs • Use nth term = a + (n-1)d • Expect simultaneous equations to find a and d • Learn proof

  7. GPs • 3 formula to learn: • Nth term = a x r n-1 • Only if r is a between -1 and 1, • Set up equations • Expect quadratic equation, or to divide equations • Learn proof

  8. Radians • 3 formula to set up and solve equation • Arc length = rθ • Sector area = ½ r2θ • Triangle area = ½ r2 sin θ • Don’t forget area of circle = π r2 • Circumference of circle = πd

  9. Circles • C1 complete the square • Find the number in front of x • Halve and square • Repeat for y • Centre and radius can be written down • Radius and tangent meet at 90°, so grads x to -1 • If 2 circles touch, distance between centres is the 2 radius added

  10. Logs • Adopt a log eg log 10 100 = 2, as 10 2 = 100 • Log x + log y = log xy • Log x – log y = log (x/y) • Log x n = n log x • Use to write as a single log • Solve equation with x as a power

  11. Triangles • Also • Sin 30 = cos 60 = ½ • Sin 60 = cos 30 = √3 • 2

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