Can someone please assist me with this questions? Thank you in advance for your assistance.

If angles A, B, and C are the angles of a triangle such that sin(A+B)=1/sin(C) and cos(A+B)=cos(C), then show that the triangle is a right triangle

- April 29th 2009, 06:47 PMbbanburyWorking with Trig identies to show that a triangle is a right triangle
Can someone please assist me with this questions? Thank you in advance for your assistance.

If angles A, B, and C are the angles of a triangle such that sin(A+B)=1/sin(C) and cos(A+B)=cos(C), then show that the triangle is a right triangle - April 29th 2009, 07:32 PMlisczz
Hmm, interesting! I'm sure there are many ways to prove that ABC is a right triangle, but here's one method:

So let's assume ABC is a right triangle.

is the hypotenuse of the triangle

and are both legs of the triangle

Leg is the opposite side to angle ; leg isn't adjacent to angle . So the sine of A would be .

can be expanded to

The cosine of is only if angle is or Since a triangle cannot have a angle, angle must be , and ABC is a right triangle. - April 30th 2009, 06:33 AMGrandadTrigonometry
Hello bbanburyThis is very straightforward.

We know that, for any angle . And in a triangle , so in*any*triangle.

If, in addition, we know that , then

(since is impossible if )

So the triangle is right-angled at C.

(Notice that we don't need the extra fact about . But you could equally well use this instead, and say that in*any*triangle. So in this case .)

Grandad - April 30th 2009, 08:41 AMbbanburyA big Thank you
Thank you for your assistance...it makes sense. Have a great day.