Question

A triangle has sides A, B, and C. Sides A and B have lengths of 8 and 15, respectively. The angle between A and C is `(7pi)/24` and the angle between B and C is `(5pi)/8`. What is the area of the triangle?

Answer 1

If `alpha, beta` and `gamma` are tha angles opposite to `A,B` and `C`, the theorem of sines states that:

`fracA sinalpha = frac B sin beta = frac C sin gamma`

We know that:

`alpha = (5pi)/8`

`beta = (7pi)/24`

and as the sum of the internal angles of a triangle always equals `pi`

`gamma = pi-alpha-beta=pi/12`

We can use this to determine `C`:

`C = frac A sinalpha sin gamma`

Then we can use Eron's formula to calculate the area from the sides:

`S= sqrt(p(p-A)(p-B)(p-C))`

where `p=frac (A+B+C) 2`

As the angles are not such as the sine is immediately known we can use approximate values or do a bit of computation using trigonometric formulas.

`sin alpha = sin((5pi)/8) = sin (1/2*(5pi)/4) = sqrt((1-cos((5pi)/4))/2)=sqrt((1+cos((pi)/4))/2)=sqrt((1+sqrt(2)/2)/2)=sqrt(2+sqrt(2))/2`

`sin gamma = sin(pi/12) = ((sqrt(3)-1)/(2sqrt(2)))`

(see: https://socratic.org/questions/find-the-exact-value-of-sin-pi-12-cos-11pi-12-and-tan-7pi-12-2)

Then:

`A=8`
`B=15`
`C=8* frac cancel 2 (sqrt(2+sqrt(2))) * ((sqrt(3)-1)/(cancel 2sqrt(2)))`