Question

In triangle ABC, if sinA = 1/3 and sinB = 1/4, find sinC.

Answer 1

Exact Answer:

`(√(15)+√(8))/(12)`

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

`\sin(C) = (√(15)+√(8))/(12)`

`\sin(C) \approx 0.55845087257947`

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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).