A triangle has sides A, B, and C. Sides A and B are of lengths #7# and #4#, respectively, and the angle between A and B is #pi/6#. What is the length of side C?

Question

A triangle has sides A, B, and C.  Sides A and B are of lengths #7# and #4#, respectively, and the angle between A and B is #pi/6#. What is the length of side C?

Aaliyah Abdullah · Accepted Answer

#C=root2(65-14root2(3))# According to cosinus law #C=root2(A^2+B^2-2A*B*cos (pi/6))# #C=root2(49+16-28*root2(3)/2)#

Oliver Atkinson · Answer

undefined To find the length of side $C$ in the triangle given sides $A$ and $B$ and the angle between them, you can use the Law of Cosines, which states:

$$C^2 = A^2 + B^2 - 2AB\cos(\theta)$$

Where:
- $C$ is the length of the side opposite the given angle.
- $A$ and $B$ are the lengths of the other two sides.
- $\theta$ is the measure of the angle between sides $A$ and $B$.

Given that $A = 7$, $B = 4$, and $\theta = \frac{\pi}{6}$, we can plug these values into the formula:

$$C^2 = 7^2 + 4^2 - 2 \times 7 \times 4 \times \cos\left(\frac{\pi}{6}\right)$$

$$C^2 = 49 + 16 - 56 \times \frac{\sqrt{3}}{2}$$

$$C^2 = 65 - 28\sqrt{3}$$

Taking the square root of both sides:

$$C = \sqrt{65 - 28\sqrt{3}}$$

This is the exact length of side $C$. If you want a decimal approximation, you can calculate it further.