Question

If we know the values of the sine and cosine of $\mathit{x}$and $\mathit{y}$,we can find the value of$\mathbf{sin}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}$ by using the ___________ Formula for Sine. State the formula:

$\mathbf{sin}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}\mathbf{=}$______________ .

Short answer

If we know the values of the sine and cosine of$x$and $y$, we can find the value of$\sin (x+y)$ by using the addition Formula for Sine. The formula:

$\sin (x+y)=\sin \left(x\right)\cos \left(y\right)+\cos \left(x\right)\sin \left(y\right)$

Step-by-step solution

Step 1. Given information.

sine and cosine are the trigonometric functions, where $x$and $y$are the angles

If we know the values of $\sin \left(x\right)$, $\sin \left(y\right)$, $\cos \left(x\right)$and $\cos \left(y\right)$then by using addition formula of sine we can easily calculate the value of$\sin (x+y)$

Formula$\sin (x+y)=\sin \left(x\right)\cos \left(y\right)+\cos \left(x\right)\sin \left(y\right)$

Step 2. Determining the formula.

Let us split the angle of sine into $x$and $y$. Since we know the exact values of sine and cosine at ${30}^{\circ }$, ${45}^{\circ }$,${60}^{\circ }$ and${90}^{\circ }$So we got the values of $x$and $y$.

Note down the values of $\sin \left(x\right)$, $\sin \left(y\right)$, $\cos \left(x\right)$, and $\cos \left(y\right)$.

Then substitute in the addition formula of sine

$\sin (x+y)=\sin \left(x\right)\cos \left(y\right)+\cos \left(x\right)\sin \left(y\right)$to get the value of $\sin (x+y)$.