Exact Answer:
[tex]\frac{\sqrt{15}+\sqrt{8}}{12}[/tex]
which can be written as (sqrt(15) + sqrt(8))/12
This approximates to about 0.55845087257947
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Work Shown:
Given info:
sin(A) = 1/3
sin(B) = 1/4
Based on the given info, we can isolate each variable using the arcsine function, so,
sin(A) = 1/3
arcsin(sin(A)) = arcsin(1/3)
A = arcsin(1/3)
and,
sin(B) = 1/4
arcsin(sin(B)) = arcsin(1/4)
B = arcsin(1/4)
The arcsine function cancels out the sine function. The rule is arcsin(sin(x)) = x where x is some angle between 0 and 90 degrees. Also, sin(arcsin(x)) = x will come in handy later.
The three angles of a triangle add to 180, which means,
A+B+C = 180
C = 180-(A+B)
sin(C) = sin(180-(A+B))
sin(C) = sin(x-y)
where, x = 180 and y = A+B
use this trig identity
sin(x-y) = sin(x)cos(y) - cos(x)sin(y)
to get the following
sin(C) = sin(x-y)
sin(C) = sin(x)cos(y) - cos(x)sin(y)
sin(C) = sin(180)cos(arcsin(1/3)+arcsin(1/4)) - cos(180)sin(arcsin(1/3)+arcsin(1/4))
sin(C) = 0cos(arcsin(1/3)+arcsin(1/4)) - (-1)sin(arcsin(1/3)+arcsin(1/4))
sin(C) = 0 + 1sin(arcsin(1/3)+arcsin(1/4))
sin(C) = sin(arcsin(1/3)+arcsin(1/4))
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Now let's reassign x and y.
Make x = arcsin(1/3) and y = arcsin(1/4)
note that
sin(x) = sin(arcsin(1/3)) = 1/3
sin(y) = sin(arcsin(1/4)) = 1/4
based on the rule sin(arcsin(x)) = x when x is some angle between 0 and 90 degrees.
furthermore,
cos(x) = cos(arcsin(1/3)) = sqrt(8)/3
cos(y) = cos(arcsin(1/4)) = sqrt(15)/4
based on the attached images below (figure 1 and figure 2). The values in red were solved for using the pythagorean theorem a^2+b^2 = c^2.
use this trig identity
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
to get
sin(C) = sin(arcsin(1/3)+arcsin(1/4))
sin(C) = sin(x+y)
sin(C) = sin(x)cos(y) + cos(x)sin(y)
sin(C) = sin(arcsin(1/3))cos(arcsin(1/4)) + cos(arcsin(1/3))sin(arcsin(1/4))
sin(C) = (1/3)cos(arcsin(1/4)) + cos(arcsin(1/3))(1/4)
sin(C) = (1/3)(sqrt(15)/4) + (sqrt(8)/3)(1/4)
sin(C) = (sqrt(15)/12) + (sqrt(8)/12)
sin(C) = (sqrt(15) + sqrt(8))/12
[tex]\sin(C) = \frac{\sqrt{15}+\sqrt{8}}{12}[/tex]
[tex]\sin(C) \approx 0.55845087257947[/tex]
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Extra info:
We can sqrt(8) simplify to get 2*sqrt(2), though this isnt much of a simplification (as it doesnt help cancel or reduce the fraction at all).
