Solve 2sin^{2}60^{circ}+3cos^260^circ+4tan^260^circ

$$2 \sin ^ { 2 } 60 ^ { \circ } + 3 \cos ^ { 2 } 60 ^ { \circ } + 4 \tan ^ { 2 } 60 ^ { \circ }$$

Evaluate

$$\frac{57}{4}=14.25$$

Solution Steps

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

$$2\times \left(\frac{\sqrt{3}}{2}\right)^{2}+3\left(\cos(60)\right)^{2}+4\left(\tan(60)\right)^{2}$$

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

$$2\times \frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+3\left(\cos(60)\right)^{2}+4\left(\tan(60)\right)^{2}$$

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

$$\frac{2\left(\sqrt{3}\right)^{2}}{2^{2}}+3\left(\cos(60)\right)^{2}+4\left(\tan(60)\right)^{2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(\sqrt{3}\right)^{2}}{2}+3\left(\cos(60)\right)^{2}+4\left(\tan(60)\right)^{2}$$

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

$$\frac{\left(\sqrt{3}\right)^{2}}{2}+3\times \left(\frac{1}{2}\right)^{2}+4\left(\tan(60)\right)^{2}$$

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

$$\frac{\left(\sqrt{3}\right)^{2}}{2}+3\times \frac{1}{4}+4\left(\tan(60)\right)^{2}$$

Multiply 3 and \frac{1}{4} to get \frac{3}{4}.

$$\frac{\left(\sqrt{3}\right)^{2}}{2}+\frac{3}{4}+4\left(\tan(60)\right)^{2}$$

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

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

$$\frac{\left(\sqrt{3}\right)^{2}}{2}+\frac{3}{4}+12$$

Add \frac{3}{4} and 12 to get \frac{51}{4}.

$$\frac{\left(\sqrt{3}\right)^{2}}{2}+\frac{51}{4}$$

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

$$\frac{2\left(\sqrt{3}\right)^{2}}{4}+\frac{51}{4}$$

Since \frac{2\left(\sqrt{3}\right)^{2}}{4} and \frac{51}{4} have the same denominator, add them by adding their numerators.

$$\frac{2\left(\sqrt{3}\right)^{2}+51}{4}$$

The square of \sqrt{3} is 3.

$$\frac{3}{2}+\frac{51}{4}$$

Add \frac{3}{2} and \frac{51}{4} to get \frac{57}{4}.

$$\frac{57}{4}$$