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How do I construct a proof by induction?

There are typically 4 steps: proving the base case, making an assumption, making the inductive step and finally concluding the proof. The base case consists of proving that a statement is true for n = 1, the...
AF
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Using the definitions of hyperbolic functions in terms of exponentials show that sech^2(x) = 1-tanh^2(x)

tanh(x) = ((e x -e -x )/2)/((e x +e -x )/2) 1 - tanh 2 (x) = 1-((e x -e -x )/(e x +e -x )) 2 = ((e 2x +e -2x +2)-(e 2x +e -2x -2))/(e x +e -x ) 2 = (2e x .2e -x )/(e x +e -x ) 2 = 4/(e x +e -x ) 2 = sech 2 x
CB
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Core 3: Find all the solutions of 2cos(2x) = 1-2sin(x) in the interval 0<x<360

2cos(2x) = 1-2sin(x) 2(1-2sin 2 (x)) = 1-2sin(x) -- by our trigonometric functions 2-4sin 2 (x) = 1-2sin(x) 4sin 2 (x)-2sin(x)-1=0 -- rearrange sin(x) = (2±(4-4(4)(-1)) 0.5 )/8 -- quadratic formula x = 54 &a...
CB
Answered by Chris B. • Maths tutor
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x^3 + 2x^2 - 9x - 18 = (x^2 - a^2)(x + b) where a,b are integers. Work out the three linear factors of x^3 + 2x^2 - 9x - 18. (Note: x^3 indicates x cubed and x^2 indicates x squared).

There are a few different ways to approach this problem. The most obvious is to attempt to factorise x 3 + 2x 2 - 9x - 18. However it is very difficult to approach the problem like this. fortunately the ques...
CB
5119 Views

Jo has the following 9 coins in her purse: 10p, 2p, 2p, 50p, 10p, 1p, 20p, 2p, 2p. Work out the median, mean, mode and range.

To work out the median of a set of numbers we first place them in order: 1p, 2p, 2p, 2p, 2p, 10p, 10p, 20p, 50p Then the median is the middle value in the list, in this case 2p. To work out the mean we must ...
CB
Answered by Chris B. • Maths tutor
6024 Views