Two sides of a triangle have constant lengths a and b, and the angle between them is theta. What value of theta will maximize the area of the triangle?
so far i have the formula and the derivative.
A=.5absin(theta)
A'=.5abcos(theta)
then i set the derivative to zero in order to get the critical point, but i don't know how to solve it because if I divide zero by .5, it will just be 0.
0=.5abcos(theta)
can anyone help me get past this step? thanks.
you almost in there already, think about the cos(theta) = 0, it means the theta=90 degree, so, you got it
To solve the equation 0 = 0.5abcos(theta), we need to set cos(theta) = 0.
The cosine function is equal to zero at certain angles. The values of theta for which cos(theta) = 0 are 90 degrees and 270 degrees (or pi/2 and 3pi/2 radians).
So, we have two possible values for theta: theta = 90 degrees or theta = 270 degrees (or theta = pi/2 or theta = 3pi/2 radians).
To determine which of these values maximizes the area, we need to evaluate the area function A(theta) = 0.5absin(theta) at these values of theta.
When theta = 90 degrees (or theta = pi/2 radians), the sine function is equal to 1, so A(theta) = 0.5absin(theta) = 0.5ab.
When theta = 270 degrees (or theta = 3pi/2 radians), the sine function is equal to -1, so A(theta) = 0.5absin(theta) = -0.5ab.
Therefore, the value of theta that maximizes the area of the triangle is theta = 90 degrees (or theta = pi/2 radians).
To find the value of theta that maximizes the area of the triangle, you need to solve the equation:
0 = 0.5abcos(theta)
Dividing by 0.5 won't lead to a solution because dividing anything by zero is undefined. However, you can solve the equation by using the inverse function of the cosine, which is the arccosine (or cos^(-1)) function:
cos^(-1)(0) = cos^(-1)(0.5abcos(theta))
The cosine function returns values between -1 and 1, so the only value of theta that satisfies the equation is when cos(theta) = 0. This occurs when theta is equal to π/2 or 90 degrees.
Therefore, the value of theta that maximizes the area of the triangle is 90 degrees.