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

Question 35670: If angles A, B, and C are the sides of a triangle such that sin(A+B)=1/cosC and cos(A+B)=cosC, then show that the triangle is a right triangle.
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
>
> If angles A, B, and C are the sides of a triangle such that sin(A+B)=1/cosC
> and cos(A+B)=cosC, then show that the triangle is a right triangle.
> SINCE IN A TRIANGLE A+B+C=180
SIN(A+B)=SIN(180-C)=SIN(C)=1/COS(C)
HENCE SIN(C)*COS(C)=1.......SIN(2C)/2=1......SIN2C=2....WRONG...IT MUST BE
SIN(A+B)=1-COSC...OR...=1
THEN SIN(C)=1-COS(C)
SIN(C)+COS(C)=1...DIVIDE THROUGHOUT BY SQRT(2)
SIN(C)/SQRT(2)+COS(C)/SQRT(2)=1/SQRT(2).......
PUT SIN(45)=COS(45)=1/SQRT(2)
SIN(C)COS(45)+COS(C)SIN(45)=SIN(45)
SIN(45+C)=SIN(45)=SIN(180-45)=SIN(135)
45+C=135
C=90...RIGHT ANGLED TRIANGLE.
COS(A+B)=COS(180-C)=-COS(C)=COS(C)...GIVEN
COS(C)=0...OR....C=90..RIGHT ANGLED TRIANGLE
>
>
RELATED QUESTIONS
If A, B, C are angles of a triangle, prove that cosA + cosB + cosC =... (answered by MathLover1)

If anglesA,B, and C are the sides of a triangle such that sin(A+B)=1/cos(C) and... (answered by venugopalramana)

in triangle ABC, a=2, b=3, and c=4. What is the value of... (answered by stanbon)

in triangle ABC, a = 7, b = 5, and c = 8. What is the value of... (answered by Alan3354)

In a triangle if CosA/a = CosB/b = CosC/c and if a = 2 then find the area of triangle... (answered by robertb)

In triangle ABC, a=8, b=9, and cosC= 2/3. Find... (answered by ikleyn)

In ΔABC, Given (a/cosA)=(b/cosB)=(c/cosC) . Prove that ΔABC is an equilateral... (answered by Edwin McCravy)

If a, b, and c are integers and are the sides of a triangle ABC such that b=a(a-c)... (answered by venugopalramana)

If a, b, and c are integers and are the sides of triangle ABC such that b=a(a-c), show... (answered by venugopalramana)