Trigonometry

1.  Law of Sines.

a)  Let ABC be an acute triangle.   Let   a=|BC|,   b=|CA| and c=|AB|.
Prove the law of sines:

sin(A)/a = sin(B)/b = sin(C)/c.

Hint:   You only have to prove that sin(A)/a = sin(B)/b,  since the rest of it is just a matter
of changing the letters.    Let h be the height of the perpendicular dropped from C to line AB.
Express h via trig in two ways,  using the two right triangles formed.

b)  Let  a be an obtuse angle and let b be its  supplement.   Should we take
sin(a)=sin(b) or sin(a)=-sin(b)?   Show that the choice is forced by insisting that the
law of sines remain true for obtuse triangles.
 

2.  Law of Cosines.

a)  Let ABC be an acute triangle.   Let   a=|BC|,   b=|CA| and c=|AB|.
Prove the law of cosines:

a^2=b^2+c^2-2bc cos(A).

Hint:   Drop a perpendicular from C as in problem 1.     Let D denote the point where
it hits side AB.   Let h=|CD|.
Let c1=|BD| and let c2=|DA|.    Use one of your right triangles to get  a formula for
a^2 in terms of c1 and h.
Eliminate h from this formula by using the other right triangle.

b)  Now let a be an obtuse angle and let b be its  supplement.
Should we take cos(a)=cos(b) or cos(a)=-cos(b)?   Show that the
choice is forced by insisting that the law of cosines remain true for obtuse triangles.

3. Area  Let ABC be a  triangle.   Let   a=|BC|,   b=|CA| and c=|AB|.     Show that

area(ABC)=1/2 ab sin(C).   (Check both the acute and obtuse case for angle C.)