Solve frac{5sin^{2}30^{circ}+cos^{2}45^{circ}+4tan^{2}60^{circ}}{2sin^230^circcos60^circ+tan45^circ}

$$\frac { 5 \sin ^ { 2 } 30 ^ { \circ } + \cos ^ { 2 } 45 ^ { \circ } + 4 \tan ^ { 2 } 60 ^ { \circ } } { 2 \sin ^ { 2 } 30 ^ { \circ } \cos 60 ^ { \circ } + \tan 45 ^ { \circ } }$$

Evaluate

$$11$$

Solution Steps

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

$$\frac{5\times \left(\frac{1}{2}\right)^{2}+\left(\cos(45)\right)^{2}+4\left(\tan(60)\right)^{2}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

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

$$\frac{5\times \frac{1}{4}+\left(\cos(45)\right)^{2}+4\left(\tan(60)\right)^{2}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

Multiply 5 and \frac{1}{4} to get \frac{5}{4}.

$$\frac{\frac{5}{4}+\left(\cos(45)\right)^{2}+4\left(\tan(60)\right)^{2}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

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

$$\frac{\frac{5}{4}+\left(\frac{\sqrt{2}}{2}\right)^{2}+4\left(\tan(60)\right)^{2}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

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

$$\frac{\frac{5}{4}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+4\left(\tan(60)\right)^{2}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

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

$$\frac{\frac{5}{4}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+4\left(\sqrt{3}\right)^{2}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{5}{4}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+4\times 3}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

Multiply 4 and 3 to get 12.

$$\frac{\frac{5}{4}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+12}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

Add \frac{5}{4} and 12 to get \frac{53}{4}.

$$\frac{\frac{53}{4}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{53}{4}+\frac{2}{2^{2}}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{53}{4}+\frac{2}{4}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

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

$$\frac{\frac{53}{4}+\frac{1}{2}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

Add \frac{53}{4} and \frac{1}{2} to get \frac{55}{4}.

$$\frac{\frac{55}{4}}{2\left(\sin(30)\right)^{2}\cos(60)+\tan(45)}$$

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

$$\frac{\frac{55}{4}}{2\times \left(\frac{1}{2}\right)^{2}\cos(60)+\tan(45)}$$

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

$$\frac{\frac{55}{4}}{2\times \frac{1}{4}\cos(60)+\tan(45)}$$

Multiply 2 and \frac{1}{4} to get \frac{1}{2}.

$$\frac{\frac{55}{4}}{\frac{1}{2}\cos(60)+\tan(45)}$$

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

$$\frac{\frac{55}{4}}{\frac{1}{2}\times \frac{1}{2}+\tan(45)}$$

Multiply \frac{1}{2} and \frac{1}{2} to get \frac{1}{4}.

$$\frac{\frac{55}{4}}{\frac{1}{4}+\tan(45)}$$

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

$$\frac{\frac{55}{4}}{\frac{1}{4}+1}$$

Add \frac{1}{4} and 1 to get \frac{5}{4}.

$$\frac{\frac{55}{4}}{\frac{5}{4}}$$

Divide \frac{55}{4} by \frac{5}{4} by multiplying \frac{55}{4} by the reciprocal of \frac{5}{4}.

$$\frac{55}{4}\times \frac{4}{5}$$

Multiply \frac{55}{4} and \frac{4}{5} to get 11.

$$11$$