Question

$°$find the exact value of the expression

sin255$°$-sin15$°$

Short answer

sin255-sin15$°=-1.2247448714$

Step-by-step solution

Step 1. Rewriting the expression   

$$\begin{gathered} \sin (270°-x)=-\cos x \\ \sin 255°=\sin (270°-15°)=-\cos 15° \\ -\cos 15°=-\sin 75° \\ ourexpressionbecomes \\ \sin 255°-\sin 15°=-(\sin 75°+\sin 15°) \end{gathered}$$

Step 2. calculations

Now our expression is$$\begin{gathered} \sin 255°-\sin 15°=-(\sin 75°+\sin 15°) \\ \sin x+\sin y=2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right) \\ -(\sin 75°+\sin 15°)=-2\sin 45°\cos 30° \\ =-2\frac{1}{\sqrt{2}}\frac{\sqrt{3}}{2}=-1.2247448714 \end{gathered}$$