Solve sqrt{1+cos200^circ/2}

$$\sqrt { \frac { 1 + \cos 200 ^ { \circ } } { 2 } }$$

Evaluate

$$\frac{\sqrt{301536896070458}}{100000000}\approx 0.173648178$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\sqrt{\frac{1 + -0.9396926207859084}{2}}$$

Subtract 0.9396926207859084 from 1 to get 0.0603073792140916.

$$\sqrt{\frac{0.0603073792140916}{2}}$$

Expand \frac{0.0603073792140916}{2} by multiplying both numerator and the denominator by 10000000000000000.

$$\sqrt{\frac{603073792140916}{20000000000000000}}$$

Reduce the fraction \frac{603073792140916}{20000000000000000} to lowest terms by extracting and canceling out 4.

$$\sqrt{\frac{150768448035229}{5000000000000000}}$$

Rewrite the square root of the division \sqrt{\frac{150768448035229}{5000000000000000}} as the division of square roots \frac{\sqrt{150768448035229}}{\sqrt{5000000000000000}}.

$$\frac{\sqrt{150768448035229}}{\sqrt{5000000000000000}}$$

Factor 5000000000000000=50000000^{2}\times 2. Rewrite the square root of the product \sqrt{50000000^{2}\times 2} as the product of square roots \sqrt{50000000^{2}}\sqrt{2}. Take the square root of 50000000^{2}.

$$\frac{\sqrt{150768448035229}}{50000000\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{150768448035229}}{50000000\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{150768448035229}\sqrt{2}}{50000000\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{150768448035229}\sqrt{2}}{50000000\times 2}$$

To multiply \sqrt{150768448035229} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{301536896070458}}{50000000\times 2}$$

Multiply 50000000 and 2 to get 100000000.

$$\frac{\sqrt{301536896070458}}{100000000}$$