tanx = 0.5 find sinx and cosx
3 Answers
hala718 | High School Teacher | (Level 1) Educator Emeritus
tanx = 1/2
But we know that :
tanx = sinx/cosx = 1/2
==> 2sinx = cosx..........(1)
Also , we know that:
sin^2 x + cos^2 x = 1
==> cos x = sqrt(1-sin^2 x)..........(2)
Now substitute in (1)
==> 2sin x = sqrt(1-sin^2 x)
Square both sides:
==> 4sin^2 x = 1-sin^2 x
==> 5sin^2 x = 1
Divide by 5
==> sin^2 x = 1/5
==> sin x= 1/sqrt5= sqrt5/5
==> sin x = sqrt5/5
Now substitute in (1):
cosx = 2sinx
= 2(sqrt5/5)
= 2sqrt4/5
==> cos x = 2sqrt5/5
giorgiana1976 | College Teacher | (Level 3) Valedictorian
We know from the fundamental formula of trigonometry that:
(sin x)^2 + (cos x)^2 = 1
If we divide by (cos x)^2, both sides, we'll get:
(sin x/cos x)^2 + 1 = 1/(cos x)^2
But sin x / cos x = tan x
(tan x)^2 + 1 = 1/(cos x)^2
But tan x = 0.5 = 50/100 = 1/2
(1/2)^2 + 1 = 1/(cos x)^2
5/4 = 1/(cos x)^2
We'll cross multiply and we'll get:
4 = 5(cos x)^2
We'll divide by 5:
4/5 = (cos x)^2
cos x = +/- 2/sqrt5
cos x = +/- 2sqrt5/5
sin x = +/-sqrt(1 - 4/5)
sin x = +/-sqrt5/5
Conclusion: The values of sine and cosine have to be both positive, or both negative, in order to obtain the positive value of tan x = 0.5
neela | High School Teacher | (Level 3) Valedictorian
tanx = 0.5
Therefore sin x = tanx*cosx = tanx/ secx = tanx/sqrt(1+tan^2x).
So sinx = 0.5/sqrt(1+0.5^2) = 0.5/+(1.25)^(1/2) = 0.447213595 in 1st quadrant.
sinx = 0.5/-sqrt(1.0.5^2) = -0.447213595 in 3rd quadrant.
Similarly cosx = 1/secx = 1/sqrt(1+tan^2x) or 1/-srt(1+tan^2x)
cosx = 1/sqrt(1+0.5^2) = 0.894427191 or -0.894427191