Question

Exact value of addition and subtraction formulas / sin cos tan

Answer 1

Take into account that:

`(5\pi)/(4)+(\pi)/(3)=(15\pi+4\pi)/(12)=(19)/(12)\pi`

Then, for the given sine value, you can write:

`\sin ((19)/(12)\pi)=\sin ((5)/(4)\pi+(1)/(3)\pi)`

Now, consider that the sine of a sum is:

`\sin (a+b)=\sin a\cdot\cos b+\cos a\cdot\sin b`

Then, by applying the previous identity to the given expression, you obtain:

`\sin ((5)/(4)\pi+(1)/(3)\pi)=\sin ((5)/(4)\pi)\cos ((1)/(3)\pi)+\cos ((5)/(4)\pi)\sin ((1)/(3)\pi)`

Consider now that:

`\begin{gathered} \sin ((5)/(4)\pi)=-\frac{\sqrt[]{2}}{2} \\ \cos ((1)/(3)\pi)=(1)/(2) \\ \sin ((1)/(3)\pi)=\frac{\sqrt[]{3}}{2} \\ \cos ((5)/(4)\pi)=-\frac{\sqrt[]{2}}{2} \end{gathered}`

Then, for the given expression of the question, you get:

`\begin{gathered} \sin ((5)/(4)\pi+(1)/(3)\pi)=(-\frac{\sqrt[]{2}}{2})((1)/(2))+(-\frac{\sqrt[]{2}}{2})(\frac{\sqrt[]{3}}{2}) \\ =-\frac{\sqrt[]{2}}{4}-\frac{\sqrt[]{2}\sqrt[]{3}}{4}=-\frac{(1+\sqrt[]{3})\sqrt[]{2}}{4} \end{gathered}`

The pervious result is the answer to the given expression.