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प्रश्न
If tan x = `1(1)/(3)`, find the value of : 4 sin2x - 3 cos2x + 2
उत्तर
Consider the diagram below :
tan x = `1(1)/(3)`
tan x = `(4)/(3)`
i.e.`"perpendicular"/"base" = (4)/(3)`
Therefore if length of base = 3x, length of perpendicular = 4x
Since
base2+ perpendicular2 = hypotenuse2 ..[ Using Pythagoras Theorem]
(3x)2 + (4x)2 = hypotenuse2
hypotenuse = 9x2+16x2+ 25x2
∴ hypotenuse = 5x
Now
sin x = `"perpendicular"/"hypotenuse" = (4x)/(5x) = (4)/(5)`
cos x = `"base"/"hypotenuse" = (3x)/(5x) = (3)/(5)`
Therefore
4 sin2 x – 3cos2 x +2
= `4(4/5)^2 – 3(3/5)^2+ 2`
= `(64)/(25) – (27)/(25)+ 2`
= `(87)/(25)`
=`3(12)/(25)`
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