what’s the area of the triangle

Question

Dyani attempts to find the area of ABC. What is her error? Find the correct area. Round to the nearest square centimeter.side b= 12cm, side a= 14cm, angle C= 65°

area= 1/2(AB)(BC)tan C

area= 1/2(14)((12) tan 65°

≈180cm^2

Answer 1

θ = 65°

Dyani’s Error

Dyani’s area formula has tan(65°) where she should have sin(65°). It would appear that she has the definitions of sine (opposite/hypotenuse) and tangent (opposite/adjacent) mixed up. [She also appears to be a bit confused about depicting the line segments meeting at angle C, but that’s a whole other issue.]

The Correct Area Formula

area = ½(base)(height)

where base = a = 14 cm

and height = h = bsin θ cm

So area = ½absin θ = ½(14)(12)sin(65°) = 76 cm²

Answer 2

It looks like she made an error using the formula.

Il she's using the Law of Sines, which that looks like, that should be sin C, not tan C in the formula.

Using sin C, the solution is

area = 1/2(AB)(BC) sin C

area= 1/2(14)((12) sin 65°

= 1/2(14)((12) (0.906)

= 76 cm^2