Resultant vector of an isosceles triangle?

[atypical]

Homework Statement

what is the resultant vector of an isosceles triangle?

Homework Equations

R^2=a^2+b^2-4abcos(theta)

The Attempt at a Solution

The books answer R=2acos(theta/2)
Using the formula above, and knowing that a=b in a isosceles triangle I am getting:
R=sqrt[2a^2+2a^2cos(theta)]

 

[Stonebridge]

It depends a little.

If the two vectors are as shown in the 1st diagram, where the sum is CA + AB in the directions shown, the sum is the vector CB.
By the cosine rule its length R would be
R² = a² + a² - 2a.a cos θ
R² = 2a² - 2a² cos θ

If, on the other hand, you mean the two vectors AB + AC as in the lower diagram with directions shown, the sum is vector AC in the diagram on the right.
It looks like this is what the book's answer refers to.

 

[atypical]

Thanks for the response. Now I have one more question about that. In the attachment, the book talks about (1-cos(theta))=2sin^2(theta/2) gives the magnitude of R as R=2Asin(theta/2). I don't see how they jump from the first equation to the second. Can anyone explain?

 

[Stonebridge]

Thanks for the response. Now I have one more question about that. In the attachment, the book talks about (1-cos(theta))=2sin^2(theta/2) gives the magnitude of R as R=2Asin(theta/2). I don't see how they jump from the first equation to the second. Can anyone explain?

Yes, using the cos rule on the triangle gives
D² = 2A² - 2A² cos θ
D² = 2A² (1- cos θ)
Using the identity for (1 - cos θ) gives
D² = 2A² (2 sin² (θ/2))
D² = 4A² sin² (θ/2)

D =

 

[rwisz]

The resultant vector of an isosceles triangle would be the vector that represents the combined effect of all the individual vectors acting on the triangle. In this case, the resultant vector would be the net force acting on the triangle, which can be calculated using the formula R^2=a^2+b^2-2abcos(theta), where a and b are the magnitudes of the two equal sides of the triangle and theta is the angle between them. This formula can also be written as R=2acos(theta/2), as you have correctly calculated. This resultant vector represents the direction and magnitude of the total force acting on the triangle, which is important in analyzing the overall motion and equilibrium of the triangle.