Question

Sina/sinb=√3/2, cosa/cosb=√5/2. Than find 5tan

Answer 1

The complete question is:

Sina/sinb=√3/2, cosa/cosb=√5/2. Than find tan a + tan b.

Given:

(i) Sin a /sin b=√3/2

(ii) cos a / cos b = √5/2

To find:

(i) tan a + tan b

Solution:

Given,

sin a/ sin b = √3/2

⇒ sin a = (√3sin b)/2 ...(1)

cos a / cos b = √5/2

⇒ cos a = (√5cos b)/2 ...(2)

Squaring (1) and (2) and then adding them, we get

sin² a + cos² a = 3 sin²/4  + 5 cos² b/4

We know the identity, sin² x + cos² x = 1, so,

3 sin² b/4  + 5 cos² b/4 = 1

⇒ 3 sin² b  + 5 cos² b = 4    ...(3)

As cos² b = 1 - sin² b

So, replacing it in (3), we get,

3 sin² b  + 5 (1 - sin² b) = 4

⇒ 5 - 2 sin² b  = 4

⇒ 2 sin² b = 1

⇒ sin² b = 1/2

⇒ sin b = 1/√2

cos² b = 1 - sin² b

= 1 - 1/2

= 1/2

So, cos b = 1/√2

tan b = sin b / cos b = (1/√2)/(1/√2) = 1

So, tan b = 1

Now, sin a = √3/2(sin b)

= √3/2 (1/√2)

= √3/(2√2)

= √6/4

Again, cos a = √5/2(cos b)

cos a = √5/2(1/√2)

= √10/4

tan a = sin a / cos a

= (√6/4)/(√10/4)

= √(6/10)

= √(3/5)

tan a = √(3/5)

So, tan a + tan b = 1 + √(3/5)

So, the answer is 1 + √(3/5)