Question

If 90° < <180°.cos=-12/13.then sin 20 =​

Answer 1

Answer:

There are 2 answers: + 2/131/2 = +0.832 = 0.832 and negative 0.832

sin(theta) = sinx = 12/13 = 0.923

arcsin(.923) = 67.4

67.4 is not in quadrant 2

but 180-67.4=112.6 is in quadrant 2 and has the same sine value

sin(112.6)=12/13=0.923

divide x= 112.6 by 2 to get 56.3

sin(x/2)=sin(56.3)=0.832

sin(theta) or sinx = 12/13 means cosx=5/13 as this is a 5-12-13 right triangle

because 132-122 =(- 5)2 169-144=25 sinx=12/13 cosx=5/13 tanx=12/-5=2.4

the half angle formula with cosx=-5 becomes

sin(x/2) = square root of (1-cosx)/2 = square root of 18/13/2 = 9/(13)1/2 = 3/131/2 = +0.832

The 5-12-13 right triangle is a slightly larger angle than the 1-31/2-2 triangle which has 60 degrees

so the 5-12-13 triangle is slightly more than 60 degrees

If it were 60, then x would equal 120 degrees in the 2nd quadrant. Half 120 = 60 = x/2 and sin(x/2)=.866 which is slightly more than sin(56.3) = 0.832

But x=67.4 and 112.6 are not the only x values that makes sinx=12/13

Add any integer multiple of 360 to 112.6 and the sine is still 12/13

x = 112.6, 472.6, 832.6 ...

x/2 = 56.3, 236.3, 416.3 ...

sin(56.3)= 0.832

sin(236.3)=-0.832

sin(416.3)=0.832

...

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