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Using the equation cos(a+b) = cos(a)cos(b) - sin(a)sin(b) or otherwise, show that cos(2x) = 2cos^2(x) - 1.

First let a = b = x such that: cos(a + b) = cos(a)cos(b) - sin(a)sin(b) becomes: cos(x + x) = cos(x)cos(x) - sin(x)sin(x) Leading to: cos(2x) = cos 2 (x) - sin 2 (x) Using the fact that sin 2 (y) + cos 2 (y)...
BH
Answered by Benjamin H. • Maths tutor
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A general function f(x) has the property f(-x)=-f(x). State a trigonometric function with this property and explain using the Maclaurin series expansion for this function why this property holds. Write down the integral in the limits -q to q of f(x) wrt x

Note that this property is the definition of an odd function, or draw a sketch of what this looks like in general about the horizontal axis. E.g. f(x)=sin(x) which has the expansion f(x)=x-((x^3)/3!)+((x^5)/...
JH
Answered by James H. • Maths tutor
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Rearrange to make x the subject of the equation: 4(3x+y) = 12-2y

To tackle this question, start by expanding the brackets: 12x + 4y = 12 - 2y Making x the subject of the equation requires x to be on its own. So, in this example, we need to first get rid of the 4y. Given t...
MR
Answered by Mana R. • Maths tutor
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Find the derivative of f(x)=x^3 sin(x)

Find the derivative of f(x)=x 3 sin(x). To do this calculation we need to use the product rule of differentiation: if f(x)=u(x)v(x), then the derivative is f'(x)=u'(x)v(x)+u(x)v'(x). In our case, u(x)=x 3 an...
MM
Answered by Mairi M. • Maths tutor
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Given that y=x^3 +2x^2, find dy/dx . Hence find the x-coordinates of the two points on the curve where the gradient is 4.

First of all we are asked to differentiate the function. This can be easily done by multiplying the coefficient of x by its exponent, and then decreasing the exponent by one. Therefore, dy/dx=3x^2+4x We are ...
IH
Answered by Imran H. • Maths tutor
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