Question

Prove that : sin pi/10 + sin 13pi/10=-1/2

Answer 1

sin π 10 + sin 13 π 10
= - sin π 10 + sin π + 3 π 10
= - sin π 10 - sin 3 π 10
= - sin π 10 - sin π 2 - 2 π 10 =
= - sin π 10 - sin 2 π
= -1/2

Answer 2

Answer:

Hence proved that the given expression $\sin \left( \frac { \pi } { 10 } \right) + \sin \left( \frac { 13 \pi } { 10 } \right)$is equal to -1/2  

To find:

Prove that :  

$\sin \left( \frac { \pi } { 10 } \right) + \sin \left( \frac { 13 \pi } { 10 } \right) = - \frac { 1 } { 2 }$

Solution:

Given,

$\sin \left( \frac { \pi } { 10 } \right) + \sin \left( \frac { 13 \pi } { 10 } \right)$

Splitting out the terms, we get,

$\sin \left( \frac { 13 \pi } { 10 } \right) = \sin \left( \pi + \frac { 3 \pi } { 10 } \right)$

Substituting this in the above equation,

$= \sin \left( \frac { \pi } { 10 } \right) + \sin \left( \pi + \frac { 3 \pi } { 10 } \right)$

We know that, $\sin ( \pi + \theta ) = - \sin \theta$

Substituting this in the above equation, we get,

$\begin{array} { c } { = \sin \left( \frac { \pi } { 10 } \right) - \sin \left( \frac { 3 \pi } { 10 } \right) } \\\\ { = \sin \left( \frac { \pi } { 10 } \right) - \sin \left( \frac { \pi } { 2 } - \frac { 2 \pi } { 10 } \right) } \\\\ { = \sin \left( \frac { \pi } { 10 } \right) - \sin \frac { \pi } { 2 } } \\\\ { = - \frac { 1 } { 2 } } \end{array}$

Thus the value of

$\sin \left( \frac { \pi } { 10 } \right) + \sin \left( \frac { 13 \pi } { 10 } \right) = - \frac { 1 } { 2 }$

Hence proved that the given expression $\sin \left( \frac { \pi } { 10 } \right) + \sin \left( \frac { 13 \pi } { 10 } \right)$is equal to -1/2