11. x and Y are complementary angles and Tan x = 3√3. Find the value of (1)/(2 - tan \: y\\ ) Hence rationalise the surd​

Question

11. x and Y are complementary angles and Tan x = 3√3. Find the value of


`(1)/(2 - tan \: y\\ )`
Hence rationalise the surd​

Answer 1

`(1)/(2-tan(y)) = (3√(3))/(6√(3)-1)`

After Rationalising:

`(1)/(2-tan(y)) = (54 +3√(3))/(107)`

Explanation:

tan(x) = 3√3

x + y = 90 (complimentary angles)

⇒ x = 90 - y

tan(
`(\pi)/(2)` - θ) = cot(θ)

⇒ tan(90 - θ) = cot(θ)

⇒ tan(90 - y) = cot(y)

⇒ tan(x) = cot(y)

⇒ cot(y) = 3√3

⇒
`(1)/(tan(y))` = 3√3

⇒ tan(y) =
`(1)/(3√(3))`

`2-tan(y) = 2 - (1)/(3√(3)) \\\\= (2*3√(3) -1)/(3√(3) ) \\\\= (6√(3) -1)/(3√(3) )\\\\\\\implies (1)/(2-tan(y)) \\\\= (3√(3))/(6√(3)-1)`

Multiply and divide by : 6(√3) + 1

`(3√(3))/(6√(3)-1)(6√(3)+1)/(6√(3)+1)\\ \\= ((3*6*√(3)*√(3)) + 3√(3))/(6^2 * (√(3))^2 -1^2)\\\\= (18*3 +3√(3))/(36*3 -1)\\\\= (54 +3√(3))/(108-1)\\\\= (54 +3√(3))/(107)`