A Question from 1968!

[ScholMaths]

This question is from a 1968 Scholarship maths exam from New Zealand:

Find A and B, trig expressions independent of x such that:

sinx/[sin(x-a).sin(x-b)] can be expressed in the form A/sin(x-a) + B/sin(x-b)

Have tried compound angle formulae and sums to product formulae but can't get it out. Any help appreciated.

 

[Deleted member 4993]

This question is from a 1968 Scholarship maths exam from New Zealand:

Find A and B, trig expressions independent of x such that:

sinx/[sin(x-a).sin(x-b)] can be expressed in the form A/sin(x-a) + B/sin(x-b)

Have tried compound angle formulae and sums to product formulae but can't get it out. Any help appreciated.

You are sure that there are four independent parameters involved in this problem - namely - a, A, b and B ?

 

[soroban]

Hello, ScholMaths!

This question is from a 1968 Scholarship maths exam from New Zealand:

\(\displaystyle \text{Find }A\text{ and }B\text{, trig expressions independent of }x\text{, such that:}\)

. . \(\displaystyle \dfrac{\sin x}{\sin(x-a)\sin(x-b)}\text{ can be expressed in the form }\:\,\dfrac{A}{\sin(x-a)} + \dfrac{B}{\sin(x-b)}\)

It appears to be a problem in Partial Fraction Decomposition.


\(\displaystyle \text{We have: }\:\dfrac{\sin x}{\sin(x-a)\sin(x-b)} \;=\; \dfrac{A}{\sin(x-a)} + \dfrac{B}{\sin(x-b)}\)


\(\displaystyle \text{Multiply through by }\sin(x-a)\sin(x-b):\)

. . \(\displaystyle \sin x \;=\;A\sin(x-b) + B\sin(x-a)\)


\(\displaystyle \text{Let }x = a:\;\;\sin a \:=\:A\sin(a-b) + 0 \quad\Rightarrow\quad A \:=\:\dfrac{\sin a}{\sin(a-b)} \)

\(\displaystyle \text{Let }x = b:\;\;\sin b \:=\:0 + B\sin(b-a) \quad\Rightarrow\quad B \:=\:\dfrac{\sin b}{\sin(b-a)} \:=\:-\dfrac{\sin b}{\sin(a-b)} \)


\(\displaystyle \text{Therefore: }\:\dfrac{\sin x}{\sin(x-a)\sin(x-b)} \;=\;\dfrac{\frac{\sin a}{\sin(a-b)}}{\sin(x-a)} - \dfrac{\frac{\sin b}{\sin(a-b)}}{\sin(x-b)} \)

 

[ScholMaths]

Thanks

Thanks Soroban. That's what happens when the trig chapter in the textbook I'm working through is before the chapter on partial fractions! Now I'm reading up on partial fractions.