Solve 2tan30^circ/1+tan^230=

$$\frac { 2 \tan 30 ^ { \circ } } { 1 + \tan ^ { 2 } 30 } =$$

Evaluate

$$\frac{\sqrt{3}}{2}\approx 0.866025404$$

Solution Steps

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

$$\frac{2\times \frac{\sqrt{3}}{3}}{1+\left(\tan(30)\right)^{2}}$$

Express 2\times \frac{\sqrt{3}}{3} as a single fraction.

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3^{2}}{3^{2}}.

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

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

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

Divide \frac{2\sqrt{3}}{3} by \frac{3^{2}+\left(\sqrt{3}\right)^{2}}{3^{2}} by multiplying \frac{2\sqrt{3}}{3} by the reciprocal of \frac{3^{2}+\left(\sqrt{3}\right)^{2}}{3^{2}}.

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

Cancel out 3 in both numerator and denominator.

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

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{3}}{\left(\sqrt{3}\right)^{2}+3^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{6\sqrt{3}}{3+3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{6\sqrt{3}}{3+9}$$

Add 3 and 9 to get 12.

$$\frac{6\sqrt{3}}{12}$$

Divide 6\sqrt{3} by 12 to get \frac{1}{2}\sqrt{3}.

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