Solve 4^2+{left(2sqrt{3}right)}^2-2*4*2sqrt{3}cos(frac{sqrt{3}}{2})

$${ 4 }^{ 2 } + { \left(2 \sqrt{ 3 } \right) }^{ 2 } -2 \times 4 \times 2 \sqrt{ 3 } \cos ( \frac{ \sqrt{ 3 } }{ 2 } )$$

Evaluate

$$-\frac{4999428853211853\sqrt{3}}{312500000000000}+28\approx 0.290352696$$

Solution Steps

Evaluate trigonometric functions in the problem

$${(4)} ^ {2} + {(2 \sqrt{3})} ^ {2} - 2 \cdot 4 \cdot 2 \sqrt{3} 0.9998857706423706$$

Calculate 4 to the power of 2 and get 16.

$$16+\left(2\sqrt{3}\right)^{2}-0.9998857706423706\times 2\times 2\times 4\sqrt{3}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$16+2^{2}\left(\sqrt{3}\right)^{2}-0.9998857706423706\times 2\times 2\times 4\sqrt{3}$$

Calculate 2 to the power of 2 and get 4.

$$16+4\left(\sqrt{3}\right)^{2}-0.9998857706423706\times 2\times 2\times 4\sqrt{3}$$

The square of \sqrt{3} is 3.

$$16+4\times 3-0.9998857706423706\times 2\times 2\times 4\sqrt{3}$$

Multiply 4 and 3 to get 12.

$$16+12-0.9998857706423706\times 2\times 2\times 4\sqrt{3}$$

Add 16 and 12 to get 28.

$$28-0.9998857706423706\times 2\times 2\times 4\sqrt{3}$$

Multiply 0.9998857706423706 and 2 to get 1.9997715412847412.

$$28-1.9997715412847412\times 2\times 4\sqrt{3}$$

Multiply 1.9997715412847412 and 2 to get 3.9995430825694824.

$$28-3.9995430825694824\times 4\sqrt{3}$$

Multiply 3.9995430825694824 and 4 to get 15.9981723302779296.

$$28-15.9981723302779296\sqrt{3}$$

Factor

$$-\frac{4999428853211853 \sqrt{3}}{312500000000000} + 28 = 0.290352696$$