A Framework for Task-Based Learning (Longman Handbooks for by Jane Willis

By Jane Willis

*Lesson outlines convey how the framework can be utilized to devise classes *Includes over two hundred rules for initiatives for school room use and photocopiable concentration Pages to be used in instructor education periods

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1 Trigonometric identities sin x = O H cos x = A H tan x = O A sin x tan x = cos x csc x = H O sec x = H A cot x = A O x tan x = cos sin x H is the Hypotenuse, always being opposite the right angle. Relative to angle x, O is the Opposite and A is the Adjacent. 39 40 CHAPTER 5. 3. 4 cos 2x = (cos2 x) - (sin2 x) cos2 x = 1 + cos 2x 2 2 Hyperbolic functions sinh x = ex - e-x 2 cosh x = ex + e-x 2 tanh x = sinh x cosh x Note: all angles (x) must be expressed in units of radians for these hyperbolic functions.

A capital letter ”A” represents the sum of an arithmetic sequence. For instance, in the same even-counting sequence starting at 2, A4 is equal to the sum of all elements from a1 through a4 , which of course would be 2 + 4 + 6 + 8, or 20. an = an-1 + d an = a1 + d(n-1) Where: d = The "common difference" Example of an arithmetic sequence: -7, -3, 1, 5, 9, 13, 17, 21, 25 . . An = a1 + a2 + . . an An = n (a1 + an) 2 28 CHAPTER 4. 2 Geometric sequences A geometric sequence, on the other hand, is a series of numbers obtained by multiplying (or dividing) by the same value with each step.

2! + 3! 2 1 n! Pi Pi (π) is defined as the ratio of a circle’s circumference to its diameter. 14159 26535 89793 23846 26433 83279 50288 41971 69399 37511 Note: For both Euler’s constant (e) and pi (π), the spaces shown between each set of five digits have no mathematical significance. They are placed there just to make it easier for your eyes to ”piece” the number into five-digit groups when manually copying. 1 Definition of a logarithm If: by = x Then: logb x = y Where, b = "Base" of the logarithm ”log” denotes a common logarithm (base = 10), while ”ln” denotes a natural logarithm (base = e).

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