Solve 3tan^{2}30^circ+4tan45^circ+cos30^circoperatorname{ctg}30^circ

$$3 \tan ^ { 2 } 30 ^ { \circ } + 4 \tan 45 ^ { \circ } + \cos 30 ^ { \circ } \operatorname { ctg } 30 ^ { \circ }$$

Evaluate

$$\frac{13}{2}=6.5$$

Short Solution Steps

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

$$3\times \left(\frac{\sqrt{3}}{3}\right)^{2}+4\tan(45)+\cos(30)\cot(30)$$

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

$$3\times \frac{\left(\sqrt{3}\right)^{2}}{3^{2}}+4\tan(45)+\cos(30)\cot(30)$$

Express 3\times \frac{\left(\sqrt{3}\right)^{2}}{3^{2}} as a single fraction.

$$\frac{3\left(\sqrt{3}\right)^{2}}{3^{2}}+4\tan(45)+\cos(30)\cot(30)$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(\sqrt{3}\right)^{2}}{3}+4\tan(45)+\cos(30)\cot(30)$$

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

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

Multiply 4 and 1 to get 4.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{3}{3}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{\left(\sqrt{3}\right)^{2}}{3} times \frac{2}{2}. Multiply \frac{\sqrt{3}\sqrt{3}}{2} times \frac{3}{3}.

$$\frac{2\left(\sqrt{3}\right)^{2}}{6}+4+\frac{3\sqrt{3}\sqrt{3}}{6}$$

Since \frac{2\left(\sqrt{3}\right)^{2}}{6} and \frac{3\sqrt{3}\sqrt{3}}{6} have the same denominator, add them by adding their numerators.

$$\frac{2\left(\sqrt{3}\right)^{2}+3\sqrt{3}\sqrt{3}}{6}+4$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{2}{2}.

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

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

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

Do the multiplications in 4\times 2+\sqrt{3}\sqrt{3}.

$$\frac{\left(\sqrt{3}\right)^{2}}{3}+\frac{8+3}{2}$$

Do the calculations in 8+3.

$$\frac{\left(\sqrt{3}\right)^{2}}{3}+\frac{11}{2}$$

The square of \sqrt{3} is 3.

$$\frac{3}{3}+\frac{11}{2}$$

Divide 3 by 3 to get 1.

$$1+\frac{11}{2}$$

Add 1 and \frac{11}{2} to get \frac{13}{2}.

$$\frac{13}{2}$$