Question

find the exact value of the expression

sin 285$°$sin 75$°$

Short answer

$\sin 285°\sin 75°=-0.9330127019$

Step-by-step solution

Step 1. Rewriting the expression 

sin(270$°$+x)=$-$cos(x )

sin285$°$=sin(270$°$+15$°$) =$-\cos (15°)$

Given expression becomes

sin 285$°$sin 75$°$= $-$cos15$°$sin75$°$

Step 2. Calculations

cos(90$°$$-$x)=sin(x)

so, sin75$°$=cos(90$°-$75$°$)=cos15$°$

So now the expression becomes

$-$cos15 $°$cos15$°$=$-{\cos }^2(15°)$

We know that

cos(2x)= $2{\cos }^{2}x-1$

role="math" localid="1646416466227" $$\begin{gathered} {\cos }^{2}x=\frac{1+\cos 2x}{2} \\ -{\cos }^{2}15°=\frac{1+\cos 30°}{2}=\frac{1+{\displaystyle \frac{\sqrt{3}}{2}}}{2}=-0.9330127019 \end{gathered}$$

Therefore,role="math" localid="1646416523469" $\sin 285°\sin 75°=-0.9330127019$