Question

Solve by 'CROSS MULTIPLICATION' method 

5. Find sin theta and cos theta in terms of a, b, m, n if 

a sin theta + b cos theta = m , 

b sin theta - a cos theta = n


Ans: 

$( \frac{am + bn}{a {}^{2} + b {}^{2} } , \: \frac{bm - an}{a { }^{2} + b {}^{2} } )$


Class 10

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Answer 1

$\bold{Answer :}$

Let us learn the rule first.

Two equations are -

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Then the solution be presented as

x/(b₁c₂ - b₂c₁) = y/(a₂c₁ - a₁c₂) = 1/(a₁b₂ - a₂b₁)

Given equations are

a sinθ + b cosθ - m = 0 ...(i)

b sinθ - a cosθ - n = 0 ...(ii)

Then by cross-multiplucation method, we get

(sinθ)/(- bn - am) = (cosθ)/(- bm + an) = 1/(- aa - bb)

or, (sinθ)/(- am- bn) = (cosθ)/(- bm + an) = 1/(- a² - b²)

Then,

sinθ = (- am - bn)/(- a² - b²)

= (am + bn)/(a² + b²)

and

cosθ = (- bm + an)/(- a² - b²)

= (bm - an)/(a² + b²)

Hence, the required solution be

( (am + bn)/(a² + b²), (bm - an)/(a² + b²) )

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