Solve 4/3tan^{2}30+sin^260-3cos^260+3/4tan^260-2tan^245

$$\frac { 4 } { 3 } \tan ^ { 2 } 30 + \sin ^ { 2 } 60 - 3 \cos ^ { 2 } 60 + \frac { 3 } { 4 } \tan ^ { 2 } 60 - 2 \tan ^ { 2 } 45$$

Evaluate

$$\frac{25}{36}\approx 0.694444444$$

Solution Steps

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

$$\frac{4}{3}\times \left(\frac{\sqrt{3}}{3}\right)^{2}+\left(\sin(60)\right)^{2}-3\left(\cos(60)\right)^{2}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

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

$$\frac{4}{3}\times \frac{\left(\sqrt{3}\right)^{2}}{3^{2}}+\left(\sin(60)\right)^{2}-3\left(\cos(60)\right)^{2}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

Multiply \frac{4}{3} times \frac{\left(\sqrt{3}\right)^{2}}{3^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\left(\sqrt{3}\right)^{2}}{3\times 3^{2}}+\left(\sin(60)\right)^{2}-3\left(\cos(60)\right)^{2}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

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

$$\frac{4\left(\sqrt{3}\right)^{2}}{3\times 3^{2}}+\left(\frac{\sqrt{3}}{2}\right)^{2}-3\left(\cos(60)\right)^{2}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

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

$$\frac{4\left(\sqrt{3}\right)^{2}}{3\times 3^{2}}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-3\left(\cos(60)\right)^{2}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\times 3^{2} and 2^{2} is 108. Multiply \frac{4\left(\sqrt{3}\right)^{2}}{3\times 3^{2}} times \frac{4}{4}. Multiply \frac{\left(\sqrt{3}\right)^{2}}{2^{2}} times \frac{27}{27}.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}}{108}+\frac{27\left(\sqrt{3}\right)^{2}}{108}-3\left(\cos(60)\right)^{2}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

Since \frac{4\times 4\left(\sqrt{3}\right)^{2}}{108} and \frac{27\left(\sqrt{3}\right)^{2}}{108} have the same denominator, add them by adding their numerators.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}}{108}-3\left(\cos(60)\right)^{2}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

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

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}}{108}-3\times \left(\frac{1}{2}\right)^{2}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

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

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}}{108}-3\times \frac{1}{4}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

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

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}}{108}-\frac{3}{4}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 108 and 4 is 108. Multiply \frac{3}{4} times \frac{27}{27}.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}}{108}-\frac{3\times 27}{108}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

Since \frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}}{108} and \frac{3\times 27}{108} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27}{108}+\frac{3}{4}\left(\tan(60)\right)^{2}-2\left(\tan(45)\right)^{2}$$

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

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27}{108}+\frac{3}{4}\left(\sqrt{3}\right)^{2}-2\left(\tan(45)\right)^{2}$$

The square of \sqrt{3} is 3.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27}{108}+\frac{3}{4}\times 3-2\left(\tan(45)\right)^{2}$$

Multiply \frac{3}{4} and 3 to get \frac{9}{4}.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27}{108}+\frac{9}{4}-2\left(\tan(45)\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 108 and 4 is 108. Multiply \frac{9}{4} times \frac{27}{27}.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27}{108}+\frac{9\times 27}{108}-2\left(\tan(45)\right)^{2}$$

Since \frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27}{108} and \frac{9\times 27}{108} have the same denominator, add them by adding their numerators.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27+9\times 27}{108}-2\left(\tan(45)\right)^{2}$$

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

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27+9\times 27}{108}-2\times 1^{2}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27+9\times 27}{108}-2\times 1$$

Multiply 2 and 1 to get 2.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27+9\times 27}{108}-2$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{108}{108}.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27+9\times 27}{108}-\frac{2\times 108}{108}$$

Since \frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27+9\times 27}{108} and \frac{2\times 108}{108} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\times 4\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27+9\times 27-2\times 108}{108}$$

Do the multiplications.

$$\frac{16\left(\sqrt{3}\right)^{2}+27\left(\sqrt{3}\right)^{2}-3\times 27+243}{108}-2$$

The square of \sqrt{3} is 3.

$$\frac{16\times 3+27\left(\sqrt{3}\right)^{2}-3\times 27+243}{108}-2$$

Multiply 16 and 3 to get 48.

$$\frac{48+27\left(\sqrt{3}\right)^{2}-3\times 27+243}{108}-2$$

The square of \sqrt{3} is 3.

$$\frac{48+27\times 3-3\times 27+243}{108}-2$$

Multiply 27 and 3 to get 81.

$$\frac{48+81-3\times 27+243}{108}-2$$

Add 48 and 81 to get 129.

$$\frac{129-3\times 27+243}{108}-2$$

Multiply -3 and 27 to get -81.

$$\frac{129-81+243}{108}-2$$

Subtract 81 from 129 to get 48.

$$\frac{48+243}{108}-2$$

Add 48 and 243 to get 291.

$$\frac{291}{108}-2$$

Reduce the fraction \frac{291}{108} to lowest terms by extracting and canceling out 3.

$$\frac{97}{36}-2$$

Subtract 2 from \frac{97}{36} to get \frac{25}{36}.

$$\frac{25}{36}$$