A triangle has sides A, B, and C. The angle between sides A and B is `(5pi)/12` and the angle between sides B and C is `pi/12`. If side B has a length of 3, what is the area of the triangle?

`color(blue)("Assumption: ")`

As `pi` is used in the angular measure it is assumed that the unit is radians. (Not stated)

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Tony B

`color(blue)("Method Plane")`

Determine `/_cba`

Using Sine Rule and `/_cba` determine length of side A
Determine h using `h=Asin((5pi)/12)`
Determine area `hxxB/2`

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`color(blue)("To determine" /_cba)`

Sum internal angles of a triangle is `180^0 = pi" radians"`

`=>/_cba= pi-(5pi)/12-pi/12`
`color(brown)(/_cba = pi/2 -> 90^o)`

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`color(blue)("Determine length of A")`

Using `B/(sin(b))=A/sin(a)`

`=> 3/(sin(pi/2)) =A/sin(pi/12)`

`=> A= (3xxsin(pi/12))/(sin(pi/2))`

But `sin(pi/2) = 1`

`color(blue)(=> A = 3xxsin(pi/12))`

`color(brown)("This is an exact value so keep it in this form for now to reduce error")` `color(brown)("on final calculation.")`
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`color(blue)("To determine h")`

`h=Asin((5pi)/12)`

`=>h=3xxsin(pi/12)xxsin((5pi)/12)`

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`color(blue)("To determine area")`

`"area "= B/2xxh`

`"area "= 3/2 xx3xx sin(pi/12)xxsin((5pi)/12)`

but `sin(pi/12)xxsin((5pi)/12)=1/4`

`"area "= 3/2 xx3xx1/4`

`color(green)("area " = 1.125 " units"^2 -> 1 1/8 " units"^2)`