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
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
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.
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