Angle alpha is divided into two parts A and B such that A B x and tan ...........

Mathematics (NDA) Trigonometry

Angle α is divided into two parts A and B such that A - B = x and tan A : tan B = p : q. The value of sin x is equal to -

( p + q )sinα / p-q

p sinα / p-q

( p-q / p+q )sinα

Let tan A = p and tan B = q

∴ sec A = √1+ p² and cos A = 1 / √1+p² , cos B = 1 / √1+q²

Sin x = sin (A - B)

= p / √1+p² x 1 / √1+q² – q / √1+q² x 1 / √1+p²

= p-q / √(1+p²)(1+ q²) →(1)

Now α = A + B

Or sinα = sin(A+B)

= p+q / √(1+p²)(1+q²)

Or √(1 + p²)(1 + q²) = p+q / sinα

Putting in (1):

Sin x =( p-q / p+q )sinα

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