# How to determine the value of tangent?How to determine the value of tangent of angle if i know the value of sine of double angle?

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### 2 Answers

Let us assume you know the sine of 2x.

Let sin 2x = y, we need to find tan x in terms of y

tan x = sin x / cos x

=> sin x* cos x/ (cos x)^2

=> [(2*sin x * cos x)/2] / [(cos 2x + 1)/2]

=> (y/2)/[ (sqrt (1 - (sin 2x)^2) + 1)/2

=> y/[sqrt (1 - y^2) + 1

=> y/[1 + sqrt (1 - y^2)]

**tan x = y/[1 + sqrt (1 - y^2)]**

Let's say that we know the value of the sine of double angle:

sin 2x = a

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 and 2sinx*cos x = a

2 tan x = a/cos x

tan x = a/2 cos x

We know that:

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

1 + a^2/4(cos x)^2 = 1/(cos x)^2

a^2/4(cos x)^2 - 1/(cos x)^2 = -1

(4-a^2)/4(cos x)^2 = 1

4-a^2= 4(cos x)^2

cos x = [sqrt(4-a^2)]/2

sin x = a/2

tan x = a/[sqrt(4-a^2)]

Of course, it is important to establish the quadrant where the angle x is located.