Solve {left(sin(10^{circ})right)}^{2}+{left(sin(20^{circ})right)}^{2}+{left(sin(30^{circ})right)}^{2}+{left(sin(40^{circ})right)}^{2}+{left(sin(50^{circ})right)}^{2}+{left(sin(60^{circ})right)}^2+{left(sin(70^circ)right)}^2+sin(80^circ)

$${ \left( \sin ( 10 ^ { \circ } ) \right) }^{ 2 } + { \left( \sin ( 20 ^ { \circ } ) \right) }^{ 2 } + { \left( \sin ( 30 ^ { \circ } ) \right) }^{ 2 } + { \left( \sin ( 40 ^ { \circ } ) \right) }^{ 2 } + { \left( \sin ( 50 ^ { \circ } ) \right) }^{ 2 } + { \left( \sin ( 60 ^ { \circ } ) \right) }^{ 2 } + { \left( \sin ( 70 ^ { \circ } ) \right) }^{ 2 } + \sin ( 80 ^ { \circ } )$$

Evaluate

$$4.0149614426192535330793324181709789$$

Solution Steps

Evaluate trigonometric functions in the problem

$$0.17364817766693033 ^ {2} + 0.3420201433256687 ^ {2} + {(\sin^{2}(30))} + 0.6427876096865393 ^ {2} + 0.766044443118978 ^ {2} + {(\sin^{2}(60))} + 0.9396926207859083 ^ {2} + 0.984807753012208$$

Calculate 0.17364817766693033 to the power of 2 and get 0.0301536896070458014258128850739089.

$$0.0301536896070458014258128850739089+0.3420201433256687^{2}+\left(\sin(30)\right)^{2}+0.6427876096865393^{2}+0.766044443118978^{2}+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

Calculate 0.3420201433256687 to the power of 2 and get 0.11697777844051095979530830215969.

$$0.0301536896070458014258128850739089+0.11697777844051095979530830215969+\left(\sin(30)\right)^{2}+0.6427876096865393^{2}+0.766044443118978^{2}+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

Add 0.0301536896070458014258128850739089 and 0.11697777844051095979530830215969 to get 0.1471314680475567612211211872335989.

$$0.1471314680475567612211211872335989+\left(\sin(30)\right)^{2}+0.6427876096865393^{2}+0.766044443118978^{2}+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

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

$$0.1471314680475567612211211872335989+\left(\frac{1}{2}\right)^{2}+0.6427876096865393^{2}+0.766044443118978^{2}+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

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

$$0.1471314680475567612211211872335989+\frac{1}{4}+0.6427876096865393^{2}+0.766044443118978^{2}+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

Add 0.1471314680475567612211211872335989 and \frac{1}{4} to get \frac{3971314680475567612211211872335989}{10000000000000000000000000000000000}.

$$\frac{3971314680475567612211211872335989}{10000000000000000000000000000000000}+0.6427876096865393^{2}+0.766044443118978^{2}+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

Calculate 0.6427876096865393 to the power of 2 and get 0.41317591116653479173440361044449.

$$\frac{3971314680475567612211211872335989}{10000000000000000000000000000000000}+0.41317591116653479173440361044449+0.766044443118978^{2}+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

Add \frac{3971314680475567612211211872335989}{10000000000000000000000000000000000} and 0.41317591116653479173440361044449 to get \frac{8103073792140915529555247976780889}{10000000000000000000000000000000000}.

$$\frac{8103073792140915529555247976780889}{10000000000000000000000000000000000}+0.766044443118978^{2}+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

Calculate 0.766044443118978 to the power of 2 and get 0.586824088833465120492663764484.

$$\frac{8103073792140915529555247976780889}{10000000000000000000000000000000000}+0.586824088833465120492663764484+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

Add \frac{8103073792140915529555247976780889}{10000000000000000000000000000000000} and 0.586824088833465120492663764484 to get \frac{13971314680475566734481885621620889}{10000000000000000000000000000000000}.

$$\frac{13971314680475566734481885621620889}{10000000000000000000000000000000000}+\left(\sin(60)\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

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

$$\frac{13971314680475566734481885621620889}{10000000000000000000000000000000000}+\left(\frac{\sqrt{3}}{2}\right)^{2}+0.9396926207859083^{2}+0.984807753012208$$

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

$$\frac{13971314680475566734481885621620889}{10000000000000000000000000000000000}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+0.9396926207859083^{2}+0.984807753012208$$

Calculate 0.9396926207859083 to the power of 2 and get 0.88302222155948885963114385600889.

$$\frac{13971314680475566734481885621620889}{10000000000000000000000000000000000}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+0.88302222155948885963114385600889+0.984807753012208$$

Add \frac{13971314680475566734481885621620889}{10000000000000000000000000000000000} and 0.88302222155948885963114385600889 to get \frac{22801536896070455330793324181709789}{10000000000000000000000000000000000}.

$$\frac{22801536896070455330793324181709789}{10000000000000000000000000000000000}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+0.984807753012208$$

The square of \sqrt{3} is 3.

$$\frac{22801536896070455330793324181709789}{10000000000000000000000000000000000}+\frac{3}{2^{2}}+0.984807753012208$$

Calculate 2 to the power of 2 and get 4.

$$\frac{22801536896070455330793324181709789}{10000000000000000000000000000000000}+\frac{3}{4}+0.984807753012208$$

Add \frac{22801536896070455330793324181709789}{10000000000000000000000000000000000} and \frac{3}{4} to get \frac{30301536896070455330793324181709789}{10000000000000000000000000000000000}.

$$\frac{30301536896070455330793324181709789}{10000000000000000000000000000000000}+0.984807753012208$$

Add \frac{30301536896070455330793324181709789}{10000000000000000000000000000000000} and 0.984807753012208 to get \frac{40149614426192535330793324181709789}{10000000000000000000000000000000000}.

$$\frac{40149614426192535330793324181709789}{10000000000000000000000000000000000}$$