Solve sqrt{1+cos3^circ/2}-frac{sqrt{6}-sqrt{2}}{4}

$$\sqrt { \frac { 1 + \cos 3 ^ { \circ } } { 2 } } - \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$$

Evaluate

$$\frac{\sqrt{2}}{4}+\frac{\sqrt{9993147673772869}}{100000000}-\frac{\sqrt{6}}{4}\approx 0.74083828$$

Short Solution Steps

Evaluate trigonometric functions in the problem

$$\sqrt{\frac{1 + 0.9986295347545738}{2}} - \frac{\sqrt{6} - \sqrt{2}}{4}$$

Add 1 and 0.9986295347545738 to get 1.9986295347545738.

$$\sqrt{\frac{1.9986295347545738}{2}}-\frac{\sqrt{6}-\sqrt{2}}{4}$$

Expand \frac{1.9986295347545738}{2} by multiplying both numerator and the denominator by 10000000000000000.

$$\sqrt{\frac{19986295347545738}{20000000000000000}}-\frac{\sqrt{6}-\sqrt{2}}{4}$$

Reduce the fraction \frac{19986295347545738}{20000000000000000} to lowest terms by extracting and canceling out 2.

$$\sqrt{\frac{9993147673772869}{10000000000000000}}-\frac{\sqrt{6}-\sqrt{2}}{4}$$

Rewrite the square root of the division \sqrt{\frac{9993147673772869}{10000000000000000}} as the division of square roots \frac{\sqrt{9993147673772869}}{\sqrt{10000000000000000}}.

$$\frac{\sqrt{9993147673772869}}{\sqrt{10000000000000000}}-\frac{\sqrt{6}-\sqrt{2}}{4}$$

Calculate the square root of 10000000000000000 and get 100000000.

$$\frac{\sqrt{9993147673772869}}{100000000}-\frac{\sqrt{6}-\sqrt{2}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 100000000 and 4 is 100000000. Multiply \frac{\sqrt{6}-\sqrt{2}}{4} times \frac{25000000}{25000000}.

$$\frac{\sqrt{9993147673772869}}{100000000}-\frac{25000000\left(\sqrt{6}-\sqrt{2}\right)}{100000000}$$

Since \frac{\sqrt{9993147673772869}}{100000000} and \frac{25000000\left(\sqrt{6}-\sqrt{2}\right)}{100000000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{9993147673772869}-25000000\left(\sqrt{6}-\sqrt{2}\right)}{100000000}$$

Do the multiplications in \sqrt{9993147673772869}-25000000\left(\sqrt{6}-\sqrt{2}\right).

$$\frac{\sqrt{9993147673772869}-25000000\sqrt{6}[phone]\sqrt{2}}{100000000}$$

Factor

$$\frac{\sqrt{9993147673772869} + 25000000 \sqrt{2} - 25000000 \sqrt{6}}{100000000} = 0.7408382798730364$$