Question

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value.

$\sin {18}^{\circ }\cos {27}^{\circ }+\cos {18}^{\circ }\sin {27}^{\circ }$

Short answer

Thus, the exact value of this equation is $\frac{\sqrt{2}}{2}$.

Step-by-step solution

Step 1. Given information.

The given function $\sin {18}^{\circ }\cos {27}^{\circ }+\cos {18}^{\circ }\sin {27}^{\circ }$.

Use an addition or subtraction formula to write the expression.

Then appraise the expression to obtain the final answer.

Step 2. Determining the value.

The expression can be expressed as the sine of the sum of the angles${18}^{\circ }$ and${27}^{\circ }$ because according to the addition formula for Sines,

$\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$

Where $\alpha ={18}^{\circ }$and $\beta ={27}^{\circ }$

Now, $\sin {45}^{\circ }=\frac{\sqrt{2}}{2}$

$\sin {18}^{\circ }\cos {27}^{\circ }+\cos {18}^{\circ }\sin {27}^{\circ }=\sin ({18}^{\circ }+{27}^{\circ })$

$$\begin{gathered} =\sin {45}^{\circ } \\ =\frac{\sqrt{2}}{2} \end{gathered}$$