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Trig equation of the week….

OK  that title may be a bit misleading  – I’m not proposing a Trig problem every week…  Its also a post more aimed at A-Level students..  I’ll post on more GCSE matters later in the week

I seem to have become a member of a Q&A social networking site called Quora. Now this is very loosely edited, if it is at all, and some of the questions come with strange assumptions (e.g. How many Swedish people really wish they were Americans). The questions on Maths also cover a broad range – One asked how the questioner could find the highest Odd number under 100.

Yesterday though someone posted a question that is right in the frame that my A-level students can have a go at, so I thought I’d reproduce it here…

Rewrite  Sinx + √3Cos x  in the form A sin( x + θ) where θ > 0

OK read no further if you don’t want the answer…

Start by using the expansion of Sin(x + θ) – which by co-incidence I was teaching to a student the day I saw the question.

A sin( x + θ) = A(SinxCosθ + SinθCosx).

Now I’m going to multiply the A into the expression, and re-arrange slightly

ACosθSinX + ASinθCosx.

This is beginning to look like what we need,  but the co-efficients of Sinx and Cosx need matching.  This gives us

ACosθ = 1 (The co-efficient of Sinx)
ASinθ = √3 (The co-efficient of Cosx)

If we divide the second by the first we can eliminate A, for now

Tanθ=  √3..  so  θ = 60 Degrees

Cos60 = 1/2 so we can see from  ACosθ = 1  that A = 2.

 

And so we have the full solution

Sinx + √3Cos x  in the form 2Sin( x + 60)

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