How to Solve a Trigonometry Triangle Question Using Cosine and Sine Functions

[mercury]

this is a question that was part of an mcq test.
and i did'nt have any clue as to how to begin!
finally i just tried plugging in values for A,B,C to see if they worked.
for ex. i put A=B=C=60 (degrees)- equilateral and so on..but i'd like to know the logical way to do it...

if A,B,C are the vertices of a triangle,
and if cosBcosC + sinAsinBsinC = 1

is the triangle
a) equilateral
b) isosceles
c) right angled isosceles
d) right angled but not isosceles
e) none of the above.

 

[Mulder]

Right, I'm assuming you know sin^2 +cos^2 = 1.

Well you can see this is nearly similar, and almost exactly the same if B = C. So there's two angles the same.

So now cos^2 + sinA sin^2 = 1

Which means sinA must be 1.

Therefore A can only take value of 90 degrees - right angle (450...etc useless here)

180 - 90..other two angles combine as total 90 degrees, so they must be 45 each since you've already found they're the same. Which leaves you a right angled isosceles.

 

[M98Ranger]

Solving trigonometry triangle questions using cosine and sine functions can seem daunting at first, but with practice and understanding of the concepts, it becomes easier. Here are the steps to solve this particular question:

1. Recall the trigonometric identities for cosine and sine functions. In this case, we have cosBcosC + sinAsinBsinC = 1. This is a variation of the cosine double angle identity, which states that cos(A+B) = cosAcosB - sinAsinB. By rearranging the terms, we can see that cosBcosC + sinAsinBsinC = cos(B+C) = 1.

2. Use the given information to determine the value of the angle B+C. In this case, we know that cos(B+C) = 1, which means that the angle (B+C) is either 0 or 360 degrees. However, since we are dealing with a triangle, the sum of all angles must be 180 degrees. Therefore, we can conclude that angle (B+C) = 180 degrees.

3. Use the triangle angle sum theorem to find the value of the remaining angle, A. The triangle angle sum theorem states that the sum of all angles in a triangle is 180 degrees. Therefore, we can calculate A by subtracting (B+C) from 180 degrees. In this case, A = 180 - (B+C) = 180 - 180 = 0 degrees.

4. Use the values of A, B, and C to determine the type of triangle. Since A = 0 degrees, we can conclude that the triangle is a right-angled triangle. Furthermore, since (B+C) = 180 degrees, we can also conclude that the triangle is not isosceles, as the angles B and C must be different. Therefore, the correct answer is d) right-angled but not isosceles.

In conclusion, when solving trigonometry triangle questions using cosine and sine functions, it is important to remember the trigonometric identities, use the given information to determine the values of the angles, and apply the relevant theorems to determine the type of triangle. With practice, you will become more comfortable and confident in solving these types of questions.