Calculate sin 2x if tan x=5 and x is in (0 , 90)?

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Since x is in the first quadrant, then the values of the trigonometric functions are positive.

We'll write the formula for sin 2x:

sin 2x = sin (x+x)

sin (x+x) = sinx*cosx + sinx*cosx

sin (x+x) = 2 sinx*cos x

We know that the tangent function is a ratio of sine and cosine functions:

tan x = sin x/cos x

tan x = 5 => sin x/cos x = 5=> sin x = 5 cos x

sin 2x = 2*5cos x*cosx = 10 (cos x)^2

We know that:

1 + (tan x)^2 = 1/(cos x)^2

1 + 25 = 1/(cos x)^2

26 = 1/(cos x)^2

(cos x)^2 = 1/26

sin 2x = 10/26

**sin 2x = 5/13**

We have to find sin 2x given that tan x = 5 and x is in (0 , 90)

As x is in (0 , 90) neither of sin x nor cos x are negative.

tan x = sin x / cos x = 5

=> sqrt(1 - (cos x)^2) / cos x = 5

=> (1 - (cos x)^2) = 25*(cos x)^2

=> (cos x)^2 = 1/26

=> cos x = sqrt (1/26)

1 - (sin x)^2 = 1/26

=> (sin x)^2 = 25/26

=> sin x = sqrt (25/26)

sin 2x = 2*sin x * cos x

=> 2*sqrt(25*1/26*26)

=> 2*5/26

=> 5/13

**The required value of sin 2x = 5/13**

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