Solve sin(30)cos(45)+left(sin(60)right)^2+left(cos(60)right)^2

$$\sin ( 30 ) \cos ( 45 ) + { \left( \sin ( 60 ) \right) }^{ 2 } + { \left( \cos ( 60 ) \right) }^{ 2 }$$

Evaluate

$$\frac{\sqrt{2}}{4}+1\approx 1.353553391$$

Short Solution Steps

Get the value of \sin(30) from trigonometric values table.

$$\frac{1}{2}\cos(45)+\left(\sin(60)\right)^{2}+\left(\cos(60)\right)^{2}$$

Get the value of \cos(45) from trigonometric values table.

$$\frac{1}{2}\times \frac{\sqrt{2}}{2}+\left(\sin(60)\right)^{2}+\left(\cos(60)\right)^{2}$$

Multiply \frac{1}{2} times \frac{\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{2}}{2\times 2}+\left(\sin(60)\right)^{2}+\left(\cos(60)\right)^{2}$$

Get the value of \sin(60) from trigonometric values table.

$$\frac{\sqrt{2}}{2\times 2}+\left(\frac{\sqrt{3}}{2}\right)^{2}+\left(\cos(60)\right)^{2}$$

To raise \frac{\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\sqrt{2}}{2\times 2}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+\left(\cos(60)\right)^{2}$$

Get the value of \cos(60) from trigonometric values table.

$$\frac{\sqrt{2}}{2\times 2}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+\left(\frac{1}{2}\right)^{2}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{\sqrt{2}}{2\times 2}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+\frac{1}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2\times 2.

$$\frac{\sqrt{2}}{4}+\frac{\left(\sqrt{3}\right)^{2}}{4}+\frac{1}{4}$$

Since \frac{\sqrt{2}}{4} and \frac{\left(\sqrt{3}\right)^{2}}{4} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{2}+\left(\sqrt{3}\right)^{2}}{4}+\frac{1}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2\times 2.

$$\frac{\sqrt{2}}{4}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+\frac{1}{4}$$

Since \frac{\sqrt{2}}{4} and \frac{1}{4} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{2}+1}{4}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$\frac{\sqrt{2}+1}{4}+\frac{\left(\sqrt{3}\right)^{2}}{4}$$

Since \frac{\sqrt{2}+1}{4} and \frac{\left(\sqrt{3}\right)^{2}}{4} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{2}+1+\left(\sqrt{3}\right)^{2}}{4}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{2}+1}{4}+\frac{3}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\sqrt{2}+1}{4}+\frac{3}{4}$$

Since \frac{\sqrt{2}+1}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{2}+1+3}{4}$$

Do the calculations in \sqrt{2}+1+3.

$$\frac{\sqrt{2}+4}{4}$$