Question

How do you use the angle sum identity to find the exact value of $\sin 255{}^\circ$?

Answer 1

In this problem we need to calculate the value of $\sin 255{}^\circ$ by using angle sum identity. For this we will write the given angle $255{}^\circ$ as the sum of the two angles which are $225{}^\circ$ and $30{}^\circ$. From these we will apply the trigonometric formula $\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$. Here we need to calculate the value of both $\sin 225{}^\circ$ and $\cos 225{}^\circ$. Again, we will write the angle $225{}^\circ$ as a sum of $180{}^\circ$ and $45{}^\circ$. Now we will apply the same formula $\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$ to calculate the value of $\sin 225{}^\circ$ and we will use the formula $\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B$ to calculate the value of $\cos 225{}^\circ$. After calculating the values of $\sin 225{}^\circ$ and $\cos 225{}^\circ$ we will simply calculate the value of $\sin 255{}^\circ$ which is our required value.

Complete step by step answer:
Given that, $\sin 255{}^\circ$.
Writing the given angle $255{}^\circ$ as the sum of the two angles which are $225{}^\circ$ and $30{}^\circ$. Mathematically we can write $255{}^\circ =225{}^\circ +30{}^\circ$. Now the value of $\sin 255{}^\circ$ will be
$\sin 255{}^\circ =\sin \left( 225{}^\circ +30{}^\circ \right)$
We have the trigonometric formula $\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$. Applying this formula in the above equation, then we will get
$\Rightarrow \sin 255{}^\circ =\sin 225{}^\circ \cos 30{}^\circ +\cos 225{}^\circ \sin 30{}^\circ ...\left( \text{i} \right)$
To find the value of $\sin 255{}^\circ$ we need to have the values of $\sin 225{}^\circ$ and $\cos 225{}^\circ$.
Considering the angle $225{}^\circ$. We can write the given angle $225{}^\circ$ as sum of $180{}^\circ$ and $45{}^\circ$. Mathematically we can write $225{}^\circ =180{}^\circ +45{}^\circ$.
Now the value of $\sin 225{}^\circ$ will be
$\Rightarrow \sin 225{}^\circ =\sin \left( 180{}^\circ +45{}^\circ \right)$
Applying the formula $\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$ in the above equation, then we will get
$\Rightarrow \sin 225{}^\circ =\sin 180{}^\circ \cos 45{}^\circ +\cos 180{}^\circ \sin 45{}^\circ$
We have the values $\sin 45{}^\circ =\cos 45{}^\circ =\dfrac{1}{\sqrt{2}}$, $\sin 180{}^\circ =0$, $\cos 180{}^\circ =-1$. Substituting these values in the above equation, then we will get
$\Rightarrow \sin 225{}^\circ =-\dfrac{1}{\sqrt{2}}$
Now the value of $\cos 225{}^\circ$ will be
$\Rightarrow \cos 225{}^\circ =\cos \left( 180{}^\circ +45{}^\circ \right)$
Applying the formula $\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B$ in the above equation, then we will get
$\Rightarrow \cos 225{}^\circ =\cos 180{}^\circ \cos 45{}^\circ -\sin 180{}^\circ \sin 45{}^\circ$
Applying the known values in the above equation, then we will have
$\Rightarrow \cos 225{}^\circ =-\dfrac{1}{\sqrt{2}}$
Now substituting the values of $\sin 225{}^\circ$ , $\cos 225{}^\circ$, $\cos 30{}^\circ =\dfrac{\sqrt{3}}{2}$ and $\sin 30{}^\circ =\dfrac{1}{2}$ in equation $\left( \text{i} \right)$. Then we will get
$\begin{aligned} & \Rightarrow \sin 255{}^\circ =\left( -\dfrac{1}{\sqrt{2}} \right)\left( \dfrac{\sqrt{3}}{2} \right)+\left( -\dfrac{1}{\sqrt{2}} \right)\left( \dfrac{1}{2} \right) \\\ & \Rightarrow \sin 255{}^\circ =-\dfrac{\sqrt{3}}{2\sqrt{2}}-\dfrac{1}{2\sqrt{2}} \\\ & \Rightarrow \sin 255{}^\circ =\dfrac{-\sqrt{3}-1}{2\sqrt{2}} \\\ \end{aligned}$
Hence the value of $\sin 255{}^\circ$ is $\dfrac{-\sqrt{3}-1}{2\sqrt{2}}$.

Note: In this problem we have calculated the values of $\sin 225{}^\circ$ , $\cos 225{}^\circ$ to find the value of $\sin 255{}^\circ$. We can use the trigonometric circle to calculate the values of $\sin 225{}^\circ$ , $\cos 225{}^\circ$ then we can easily find the value of $\sin 255{}^\circ$ in one or two steps from the equation $\left( \text{i} \right)$.