Solve sin^{2}30^{circ}+sin60^{circ}-sin^245^circ+cos^230^circ

$$\sin ^ { 2 } 30 ^ { \circ } + \sin 60 ^ { \circ } - \sin ^ { 2 } 45 ^ { \circ } + \cos ^ { 2 } 30 ^ { \circ }$$

Evaluate

$$\frac{\sqrt{3}+1}{2}\approx 1.366025404$$

Short Solution Steps

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

$$\left(\frac{1}{2}\right)^{2}+\sin(60)-\left(\sin(45)\right)^{2}+\left(\cos(30)\right)^{2}$$

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

$$\frac{1}{4}+\sin(60)-\left(\sin(45)\right)^{2}+\left(\cos(30)\right)^{2}$$

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

$$\frac{1}{4}+\frac{\sqrt{3}}{2}-\left(\sin(45)\right)^{2}+\left(\cos(30)\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{\sqrt{3}}{2} times \frac{2}{2}.

$$\frac{1}{4}+\frac{2\sqrt{3}}{4}-\left(\sin(45)\right)^{2}+\left(\cos(30)\right)^{2}$$

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

$$\frac{1+2\sqrt{3}}{4}-\left(\sin(45)\right)^{2}+\left(\cos(30)\right)^{2}$$

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

$$\frac{1+2\sqrt{3}}{4}-\left(\frac{\sqrt{2}}{2}\right)^{2}+\left(\cos(30)\right)^{2}$$

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

$$\frac{1+2\sqrt{3}}{4}-\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\left(\cos(30)\right)^{2}$$

The square of \sqrt{2} is 2.

$$\frac{1+2\sqrt{3}}{4}-\frac{2}{2^{2}}+\left(\cos(30)\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{1+2\sqrt{3}}{4}-\frac{2}{4}+\left(\cos(30)\right)^{2}$$

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

$$\frac{1+2\sqrt{3}-2}{4}+\left(\cos(30)\right)^{2}$$

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

$$\frac{-1+2\sqrt{3}}{4}+\left(\cos(30)\right)^{2}$$

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

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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