use one or more of the six sum and difference identities to find the exact value of the expression. \n\\(…
use one or more of the six sum and difference identities to find the exact value of the expression. \n\\( \\sin \\left(15^{\\circ}\\right) \\)\nfind the exact value of the expression. \n\\( \\sin \\left(15^{\\circ}\\right)= \\) \n(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expres
Answer
Explanation:
Step1: Express (15^{\circ}) as a difference of two known angles
We know that (15^{\circ}=45^{\circ} - 30^{\circ}). So, (\sin(15^{\circ})=\sin(45^{\circ}-30^{\circ})).
Step2: Apply the sine - difference identity
The sine - difference identity is (\sin(A - B)=\sin A\cos B-\cos A\sin B). Here (A = 45^{\circ}) and (B=30^{\circ}). We know that (\sin45^{\circ}=\frac{\sqrt{2}}{2}), (\cos45^{\circ}=\frac{\sqrt{2}}{2}), (\sin30^{\circ}=\frac{1}{2}), and (\cos30^{\circ}=\frac{\sqrt{3}}{2}). Substitute these values into the formula: [ \begin{align*} \sin(45^{\circ}-30^{\circ})&=\sin45^{\circ}\cos30^{\circ}-\cos45^{\circ}\sin30^{\circ}\ &=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}\ &=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\ &=\frac{\sqrt{6}-\sqrt{2}}{4} \end{align*} ]
Answer:
(\frac{\sqrt{6}-\sqrt{2}}{4})