A triangle has sides A, B, and C. Sides A and B have lengths of 5 and 7, respectively. The angle between A and C is #(pi)/12# and the angle between B and C is # (2pi)/3#. What is the area of the triangle?

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Answer 1 · 2016-02-10T16:28:39

Area is #12.3745#.

Formula for area of a triangle in terms of its two sides is half the product of two sides multiplied by sine of the angle between the two sides.

Here we have sides A and B, but we do not know the angle between them.

Fr this, as sum of internal angles of a triangle is #pi# and other two angles are #pi/12# and "2pi/3#,

third angle is #pi - pi/12 - (2pi)/3# i.e. #(3pi)/12# or #pi/4#

Hence, as #sin (pi/2)# is #(1/sqrt2)#

Area of triangle is #(1/2)*5*7*sin (pi/4)#

or #(1/2)*5*7*(1/sqrt2)#,

i.e. #12.3745#