Q)If sin x=12/13 then find the value of the following in fraction:

1)3 tan x - cos x
2)5 sin x - 4 tan x
3)2 sin x - cos x + tan x

I had solved some of these parts but these are a little tricky.

this is a 5-12-13 triangle

cos = 5/13

tan = 12/5

To find the values of the given expressions, we need to first substitute the value of sin x, which is 12/13, into the expressions. Then, we can simplify the expressions to obtain the final answer.

1) 3 tan x - cos x:

To find tan x, we can use the relation: tan x = sin x / cos x.
Substituting sin x = 12/13 into the equation: tan x = (12/13) / cos x.

Now, we need to find cos x. Using the Pythagorean identity, we have:
sin^2 x + cos^2 x = 1
(12/13)^2 + cos^2 x = 1
(144/169) + cos^2 x = 1
cos^2 x = 169/169 - 144/169
cos^2 x = 25/169
cos x = sqrt(25/169) = 5/13 (taking the positive value since sin x is positive)

Substituting sin x = 12/13 and cos x = 5/13 into the expression:
3 tan x - cos x = 3 * (12/13) / (5/13) - 5/13
= 36/5 - 5/13
To find the common denominator, multiply 13 and 5.
= (36/5) * (13/13) - (5/13) * (5/5)
= 468/65 - 25/65
= 443/65

Therefore, the value of 3 tan x - cos x is 443/65.

2) 5 sin x - 4 tan x:

Substituting sin x = 12/13 into the expression:
5 sin x - 4 tan x = 5 * (12/13) - 4 * (12/13) / (5/13)
= 60/13 - 48/13
= 12/13

Therefore, the value of 5 sin x - 4 tan x is 12/13.

3) 2 sin x - cos x + tan x:

Substituting sin x = 12/13 into the expression:
2 sin x - cos x + tan x = 2 * (12/13) - 5/13 + (12/13) / (5/13)
= 24/13 - 5/13 + 12/5
= 24/13 - 5/13 + 60/65
= 79/65

Therefore, the value of 2 sin x - cos x + tan x is 79/65.