∠CAB = (π - π/4 - (π/2 - x)) rad

= (π/4 + x) rad


Using the sine rule,


AC / ∠ABC = BC / ∠CAB

AC / sin π/4 = BC / sin(π/4 + x)


AC / BC = sin π/4 / sin(π/4 + x)


= (√2/2) / (sin π/4 cos x + cos π/4 sin x)

= (√2/2) / (√2/2 cos x + √2/2 sin x)

= 1/(cos x + sin x)


(Shown)

using the small angle approximation,

sin x ≈ x
cos x ≈ 1 + ½x²


So,

AC/BC ≈ 1 / (1 + ½x² + x)

AC/BC ≈ 1 / (1 + x + ½x²)

What we realise is that :

As x gets sufficiently small, ∠CAB is getting closer to π/4 or 45° (π/4 + x ≈ π/4) and ∠BCA is getting nearer to π/2 or 90° (π/2 - x ≈ π/2).


This means triangle ABC is getting closer to a right angled isosceles triangle where AC and BC are the identical legs of the triangle .


i.e AC ≈ BC and so AC/BC ≈ 1


Since AC/BC = 1/(cos x + sin x) which we found earlier,


It means that 1/(cosx + sinx) ≈ 1 which leads to this result :

cosx + sin x ≈ 1

Which leads to :

1/(cos x + sin x) ≈ (cosx + sin x)/ 1
1/(cos x + sin x) ≈ cos x + sin x


And since we used the small angle approximation that cos x + sin x ≈ 1 + x + ½x²,

then AC/BC ≈ 1 + x + ½ x²


(Basically, 1/(1 + x + ½ x²) ≈ (1 + x + ½ x²) / 1)

So a = 1, b = 1, c = ½

hi really appreciate the help once again! however the answer is a=1, b=-1, c=1.5

Oh. Perhaps I interpreted the question differently.

Since they mentioned x is sufficiently small for powers of x³ and above to be ignored, this means that x² and below still has to be included.

Actually the small angle approximation for cos x should be 1 - ½x² instead of 1 + ½x²

Come to think about it , they might actually want you to do this :


AC/BC

= 1 / (cos x + sin x)

= (cos x + sin x)-¹

≈ (1 - ½x² + x)-¹

(Using the small angle approximation)

= (1 + (x - ½x²))-¹

= 1 + (-1)(x - ½x²) + (-1)(-1 - 1)/2! (x - ½x²)² + ...

(Using the Maclaurin expansion for (1 + x)ⁿ in MF26. Here we replace x with (x - ½x²) and expand accordingly)


= 1 - x + ½x² + (x² - 2(x)(½x²) + (½x²)² + ...

= 1 - x + ½x² + x² + ...

(Ignoring terms x³ and higher)


≈ 1 - x + 3/2 x²


a = 1, b = -1 and c = 3/2 (or 1.5)

yup! i think that should be it:) thank you so much ☺️

Welcome