Solve {left(sin(pi/4)right)}^2-sin(-pi)sin(13pi/12)+cos(pi/3)

$${ \left( \sin ( \frac{ \pi }{ 4 } ) \right) }^{ 2 } - \sin ( - \pi ) \sin ( \frac{ 13 \pi }{ 12 } ) + \cos ( \frac{ \pi }{ 3 } )$$

Evaluate

$$0.7411809548974792$$

Solution Steps

Evaluate trigonometric functions in the problem

$${(\sin^{2}(\frac{\pi}{4}))} - \sin(-\pi) -0.2588190451025208 + \cos(\frac{\pi}{3})$$

Get the value of \sin(\frac{\pi }{4}) from trigonometric values table.

$$\left(\frac{\sqrt{2}}{2}\right)^{2}-\sin(-\pi )-0.2588190451025208+\cos(\frac{\pi }{3})$$

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\sin(-\pi )-0.2588190451025208+\cos(\frac{\pi }{3})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sin(-\pi ) times \frac{2^{2}}{2^{2}}.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{\sin(-\pi )\times 2^{2}}{2^{2}}-0.2588190451025208+\cos(\frac{\pi }{3})$$

Since \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} and \frac{\sin(-\pi )\times 2^{2}}{2^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(\sqrt{2}\right)^{2}-\sin(-\pi )\times 2^{2}}{2^{2}}-0.2588190451025208+\cos(\frac{\pi }{3})$$

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

$$\frac{\left(\sqrt{2}\right)^{2}-\sin(-\pi )\times 2^{2}}{2^{2}}-0.2588190451025208+\frac{1}{2}$$

Add -0.2588190451025208 and \frac{1}{2} to get \frac{301476193621849}{1250000000000000}.

$$\frac{\left(\sqrt{2}\right)^{2}-\sin(-\pi )\times 2^{2}}{2^{2}}+\frac{301476193621849}{1250000000000000}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2} and 1250000000000000 is 1250000000000000. Multiply \frac{\left(\sqrt{2}\right)^{2}-\sin(-\pi )\times 2^{2}}{2^{2}} times \frac{312500000000000}{312500000000000}.

$$\frac{312500000000000\left(\left(\sqrt{2}\right)^{2}-\sin(-\pi )\times 2^{2}\right)}{1250000000000000}+\frac{301476193621849}{1250000000000000}$$

Since \frac{312500000000000\left(\left(\sqrt{2}\right)^{2}-\sin(-\pi )\times 2^{2}\right)}{1250000000000000} and \frac{301476193621849}{1250000000000000} have the same denominator, add them by adding their numerators.

$$\frac{312500000000000\left(\left(\sqrt{2}\right)^{2}-\sin(-\pi )\times 2^{2}\right)[phone]}{1250000000000000}$$

The square of \sqrt{2} is 2.

$$\frac{2-\sin(-\pi )\times 2^{2}}{2^{2}}+\frac{301476193621849}{1250000000000000}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2-\sin(-\pi )\times 4}{2^{2}}+\frac{301476193621849}{1250000000000000}$$

Multiply -1 and 4 to get -4.

$$\frac{2-4\sin(-\pi )}{2^{2}}+\frac{301476193621849}{1250000000000000}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2-4\sin(-\pi )}{4}+\frac{301476193621849}{1250000000000000}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 1250000000000000 is 1250000000000000. Multiply \frac{2-4\sin(-\pi )}{4} times \frac{312500000000000}{312500000000000}.

$$\frac{312500000000000\left(2-4\sin(-\pi )\right)}{1250000000000000}+\frac{301476193621849}{1250000000000000}$$

Since \frac{312500000000000\left(2-4\sin(-\pi )\right)}{1250000000000000} and \frac{301476193621849}{1250000000000000} have the same denominator, add them by adding their numerators.

$$\frac{312500000000000\left(2-4\sin(-\pi )\right)[phone]}{1250000000000000}$$

Use the property \sin(-x)=-\sin(x).

$$\frac{2-4\left(-1\right)\sin(\pi )}{4}+\frac{301476193621849}{1250000000000000}$$

Get the value of \sin(\pi ) from trigonometric values table.

$$\frac{2-4\times 0}{4}+\frac{301476193621849}{1250000000000000}$$

Multiply -4 and 0 to get 0.

$$\frac{2+0}{4}+\frac{301476193621849}{1250000000000000}$$

Add 2 and 0 to get 2.

$$\frac{2}{4}+\frac{301476193621849}{1250000000000000}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}+\frac{301476193621849}{1250000000000000}$$

Add \frac{1}{2} and \frac{301476193621849}{1250000000000000} to get \frac{926476193621849}{1250000000000000}.

$$\frac{926476193621849}{1250000000000000}$$

Factor

$$\frac{13 \cdot 103 \cdot 109 \cdot 6347857799}{2 ^ {13} \cdot 5 ^ {16}} = 0.7411809548974791$$