Question

If tan A = 2/3 , find the value of sin 2A and cos 2A​

Answer 1

$\boxed{\textbf{\large{explanation:}}}$

As we know,

Sin2Φ = ( 2 tan²Φ ) / √( 1 + tan²Φ)

And,

Cos 2Φ = ( 1 - tan²Φ )/ ( 1 + tan²Φ)

therefor,

we have given tanA = 2/3

Sin2A = ( 2 tan²A ) / √( 1 + tan²A)

=)) ( 2 ( tan A) ²) / √(1 + (tanA)²)

=)) ( 2 (2/3)²) / √(1 + (2/3)² )

=)) ( 2 x ( 4 / 9 )) / √( 1 + (4/9))

=)) ( 8 / 9 ) / √ ( (9 + 4) / 9 )

=)) ( 8/ 9) / ( √13 / 3 )

=)) ( 8 / 9 ) x ( 3 / √13 )

=)) ( 8 / 3√13 )

Sin2A = ( 8 / 3√13 )

____________________________

Now,

Cos2A = ( 1 - tan²A )/ ( 1 + tan²A)

=)) ( 1 - (tan A)² ) / ( 1 +( tanA)²)

=)) ( 1 - (2/3)² ) / (1 + ( 2/3)²)

=)) ( 1 - (4 / 9 ) ) / ( 1 + (4 /9 ))

=)) (( 9 - 4) / 9 ) / (( 9 + 4 ) / 9 )

=)) ( 5 / 9 ) / ( 13 / 9 )

=)) ( 5 / 9 ) x ( 9 / 13 )

=)) 5 / 13

Cos 2A = 5/13