In the triangle the length of side a is 5 ft and angle m A=60 degrees. Find the exact lengths of sides b and c and show work
To find the lengths of sides b and c, we can use the Law of Sines. The Law of Sines states that in any triangle with sides a, b, and c, and opposite angles A, B, and C respectively, the ratio of the length of a side to the sine of the opposite angle is constant. In equation form, this is given by:
a/sin(A) = b/sin(B) = c/sin(C)
In this problem, we know that side a has a length of 5 ft and angle A is 60 degrees. Let's denote side b as x ft and side c as y ft.
Using the Law of Sines, we can set up the following ratios:
5/sin(60) = x/sin(B)
5/sin(60) = y/sin(C)
Applying the sin(60) = √3/2, we have:
5/(√3/2) = x/sin(B)
5/(√3/2) = y/sin(C)
To find the values of sin(B) and sin(C), we need to use the fact that the sum of the angles in a triangle is 180 degrees. Since the triangle has angles 60, B, and C, we know that:
60 + B + C = 180
Simplifying, we have:
B + C = 120
Now, we have two equations with two unknowns:
5/(√3/2) = x/sin(B) ...(1)
5/(√3/2) = y/sin(C) ...(2)
B + C = 120 ...(3)
Let's solve this system of equations:
From equation (3), we have C = 120 - B.
Substituting this into equation (2), we have:
5/(√3/2) = y/sin(120 - B)
Using the identity sin(120 - B) = sin(120)cos(B) - cos(120)sin(B), which gives sin(120 - B) = √3/2 * cos(B) - 1/2 * sin(B), we have:
5/(√3/2) = y/[(√3/2) * cos(B) - (1/2) * sin(B)]
Simplifying, we have:
5/(√3/2) = y/[√3cos(B)/2 - sin(B)/2]
Multiplying both sides by 2/√3, we have:
(10/√3) = y/[cos(B)/√3 - sin(B)/√3]
Simplifying further, we have:
(10/√3) = y/[cos(B) - sin(B)] ...(4)
Now, let's solve the equation (1):
5/(√3/2) = x/sin(B)
Multiplying both sides by 2/√3, we have:
(10/√3) = x/[√3sin(B)/2]
Simplifying, we have:
(10/√3) = x/[sin(B)/√3]
Again, simplifying, we have:
(10/√3) = x/[sin(B)] ...(5)
From equations (4) and (5), we can see that x = y.
Let's solve equation (4) for y:
(10/√3)[cos(B) - sin(B)] = y
Expanding, we have:
(10/√3)cos(B) - (10/√3)sin(B) = y ...(6)
And equation (5) becomes:
(10/√3) = x/[sin(B)] ...(7)
Since x = y, equation (7) can be rewritten as:
(10/√3) = y/[sin(B)] ...(8)
Now, let's consider equations (6) and (8):
(10/√3)cos(B) - (10/√3)sin(B) = y ...(6)
(10/√3) = y/[sin(B)] ...(8)
Equating the right-hand sides of equations (6) and (8), we have:
y/(sin(B)) = y/(sin(B))
From this equation, we can see that cos(B) = 1 and sin(B) = 1. We know that cos(B) = 1 when angle B = 0 degrees.
Now, let's substitute B = 0 into equation (5):
(10/√3) = x/[sin(0)]
sin(0) = 0, so we have:
(10/√3) = x/0
Since the right-hand side is undefined, this leads to a contradiction. Therefore, it is not possible to find the exact lengths of sides b and c given that side a has a length of 5 ft and angle A is 60 degrees.