Question

If we know the values of the sine and cosine of $x$and $y$,we can find the value of$\mathbf{cos}\mathbf{(}\mathit{x}\mathbf{-}\mathit{y}\mathbf{)}$ by usingthe ___________ Formula for Cosine. State the formula:

$\mathbf{cos}\mathbf{(}\mathit{x}\mathbf{-}\mathit{y}\mathbf{)}$____________.

Short answer

If we know the values of the sine and cosine of $x$and $y$, we can find the value of $\cos (x-y)$by using the subtraction formula for Cosine.

The formula:$\cos (x-y)=\cos \left(x\right)\cos \left(y\right)+\sin \left(x\right)\sin \left(y\right)$

Step-by-step solution

Step 1. Given information.

We know that 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 then by using subtraction formula of cosine we can easily calculate the value of$\cos (x-y)$

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

Step 2. Determining the formula.

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 $\sin \left(x\right)$, $\sin \left(y\right)$, $\cos \left(x\right)$,$\cos \left(y\right)$ Then substitute in the subtraction formula of cosine$\cos (x-y)=\cos \left(x\right)\cos \left(y\right)+\sin \left(x\right)\sin \left(y\right)$ to get the value of $\cos (x-y)$.