What is sin x if tan x=2/3 and x is in the set (0,pi)?

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The range (0,pi) covers the first and the second quadrant where the values of the sine function are positive.

We'll apply, for the beginning, the Pythagorean identity:

(sin x)^2 + (cos x)^2=1

We'll divide the formula with the value  (sin x)^2:

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

But the ratio sin x /cos x= tan x and cos x/sin x=1/tan x

The formula will become:

1 + (cotx)^2 = 1/(sin x)^2

sin x = 1/sqrt[1+(cot x)^2]

sin x = 1/sqrt[1+(3/2)^2]

sin x= 1/sqrt(1+9/4)

sin x = 2/sqrt13 => sin x = 2sqrt13/13

The requested value for sin x is : sin x = 2sqrt13/13

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neela | High School Teacher | (Level 3) Valedictorian

Posted on

tanx is > 0  for o<x<pi/2 in (0.pi).

tanx = 2/3.

we know that sinxx = tanx/sqrt(1+tan^2x). We put tanx = 2/3

sinxx = (2/3)/sqrt(1+(2/3)^2)

sinx = 2/sqrt(3^2+2^2)

Sinx= 2/sqrt13.

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