Solve 3cos^{2}60^{circ}+2cot^230^circ-5sin^245^circ

$$3 \cos ^ { 2 } 60 ^ { \circ } + 2 \cot ^ { 2 } 30 ^ { \circ } - 5 \sin ^ { 2 } 45 ^ { \circ }$$

Evaluate

$$\frac{17}{4}=4.25$$

Solution Steps

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

$$3\times \left(\frac{1}{2}\right)^{2}+2\left(\cot(30)\right)^{2}-5\left(\sin(45)\right)^{2}$$

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

$$3\times \frac{1}{4}+2\left(\cot(30)\right)^{2}-5\left(\sin(45)\right)^{2}$$

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

$$\frac{3}{4}+2\left(\cot(30)\right)^{2}-5\left(\sin(45)\right)^{2}$$

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

$$\frac{3}{4}+2\left(\sqrt{3}\right)^{2}-5\left(\sin(45)\right)^{2}$$

The square of \sqrt{3} is 3.

$$\frac{3}{4}+2\times 3-5\left(\sin(45)\right)^{2}$$

Multiply 2 and 3 to get 6.

$$\frac{3}{4}+6-5\left(\sin(45)\right)^{2}$$

Add \frac{3}{4} and 6 to get \frac{27}{4}.

$$\frac{27}{4}-5\left(\sin(45)\right)^{2}$$

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

$$\frac{27}{4}-5\times \left(\frac{\sqrt{2}}{2}\right)^{2}$$

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

$$\frac{27}{4}-5\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}$$

Express 5\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} as a single fraction.

$$\frac{27}{4}-\frac{5\left(\sqrt{2}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{27}{4}-\frac{5\times 2}{2^{2}}$$

Multiply 5 and 2 to get 10.

$$\frac{27}{4}-\frac{10}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{27}{4}-\frac{10}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$\frac{27}{4}-\frac{5}{2}$$

Subtract \frac{5}{2} from \frac{27}{4} to get \frac{17}{4}.

$$\frac{17}{4}$$