Solve 3^{2}30^{circ}+sin^245^circ-1/3tan^260^circ

$$3 ^ { 2 } 30 ^ { \circ } + \sin ^ { 2 } 45 ^ { \circ } - \frac { 1 } { 3 } \tan ^ { 2 } 60 ^ { \circ }$$

Evaluate

$$\frac{539}{2}=269.5$$

Solution Steps

Calculate 3 to the power of 2 and get 9.

$$9\times 30+\left(\sin(45)\right)^{2}-\frac{1}{3}\left(\tan(60)\right)^{2}$$

Multiply 9 and 30 to get 270.

$$270+\left(\sin(45)\right)^{2}-\frac{1}{3}\left(\tan(60)\right)^{2}$$

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

$$270+\left(\frac{\sqrt{2}}{2}\right)^{2}-\frac{1}{3}\left(\tan(60)\right)^{2}$$

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

$$270+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{3}\left(\tan(60)\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 270 times \frac{2^{2}}{2^{2}}.

$$\frac{270\times 2^{2}}{2^{2}}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{3}\left(\tan(60)\right)^{2}$$

Since \frac{270\times 2^{2}}{2^{2}} and \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{270\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{3}\left(\tan(60)\right)^{2}$$

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

$$\frac{270\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{3}\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\frac{270\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{3}\times 3$$

Multiply \frac{1}{3} and 3 to get 1.

$$\frac{270\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2^{2}}{2^{2}}.

$$\frac{270\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{2^{2}}{2^{2}}$$

Since \frac{270\times 2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}} and \frac{2^{2}}{2^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{270\times 2^{2}+\left(\sqrt{2}\right)^{2}-2^{2}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{270\times 4+\left(\sqrt{2}\right)^{2}}{2^{2}}-1$$

Multiply 270 and 4 to get 1080.

$$\frac{1080+\left(\sqrt{2}\right)^{2}}{2^{2}}-1$$

The square of \sqrt{2} is 2.

$$\frac{1080+2}{2^{2}}-1$$

Add 1080 and 2 to get 1082.

$$\frac{1082}{2^{2}}-1$$

Calculate 2 to the power of 2 and get 4.

$$\frac{1082}{4}-1$$

Reduce the fraction \frac{1082}{4} to lowest terms by extracting and canceling out 2.

$$\frac{541}{2}-1$$

Subtract 1 from \frac{541}{2} to get \frac{539}{2}.

$$\frac{539}{2}$$