Question

if A and B are are acute angles satisfying sin A=

Answer 1

Answer:

Option ( 2 ) is correct.

Step-by-step explanation:

Given,

sinA = sin^2 B ... ( 1 )

2 cos^2 A = 3 cos^2 B ... ( 2 )

= > sinA = sin^2 B

Multiply both sides by 3 :

= > 3 x sinA = 3 x sin^2 B

= > 3 sinA = 3 sin^2 B

= > 3 sinA = 3( 1 - cos^2 B ) { sin^2 ∅ = 1 - cos^2 ∅ }

= > 3 sinA = 3 - 3 cos^2 B

= > 3 cos^2 B = 3 - 3 sinA

= > 2 cos^2 A = 3 - 3 sinA

= > 2 ( 1 - sin^2 A ) = 3 - 3 sinA { cos^2 ∅ = 1 - sin^2 ∅ }

= > 2 - 2 sin^2 A = 3 - 3 sinA

= > 2 sin^2 A - 3 sinA + 3 - 2 = 0

= > 2 sin^2 A - 3 sinA + 1 = 0

= > 2 sin^2 A - ( 2 + 1 )sinA + 1 = 0

= > 2 sin^2 A - 2 sinA - sinA + 1 = 0

= > 2sinA( sinA - 1 ) - ( sinA - 1 ) = 0

= > ( sinA - 1 )( 2 sinA - 1 ) = 0

Using Zero Product Rule :

= > sinA - 1 = 0 Or 2 sinA - 1 = 0

= > sinA = 1 Or sinA = 1 / 2

= > sinA = sin90° Or sinA = sin30° { From properties, sin90° = 1 and sin30° = 1 / 2 }

= > A = 90° Or A = 30°

Since A is an acute angle,its measure can't be more than 90°. So, A = 30°.

Substituting the value of A in ( 1 ) :

= > sinA = sin^2 B

= > sin30° = sin^2 B

= > 1 / 2 = sin^2 B

= > √{ 1 / 2 } = sinB

= > 1 / √2 = sinB

= > sin45° = sinB { From properties, sin45° = 1 / √2 }

= > 45° = B

Hence,

= > A + B = 30° + 45°

= > A + B = 75°

Hence the required value of A + B is 75°.

Option ( 2 ) is correct.