use one or more of the six sum and difference identities to find the exact value of the expression.\n\n\\(…

use one or more of the six sum and difference identities to find the exact value of the expression.\n\n\\( \\sin \\left( 75 ^ { \\circ } \\right) \\)\n\nfind the exact value of the expression.\n\n\\( \\sin \\left( 75 ^ { \\circ } \\right) = \\square \\)\n(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize all denominators.)

use one or more of the six sum and difference identities to find the exact value of the expression.\n\n\\( \\sin \\left( 75 ^ { \\circ } \\right) \\)\n\nfind the exact value of the expression.\n\n\\( \\sin \\left( 75 ^ { \\circ } \\right) = \\square \\)\n(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Answer

Explanation:

Step1: Express (75^{\circ}) as a sum of two angles

We know that (75^{\circ}=45^{\circ}+30^{\circ}).

Step2: Use the sine sum identity (\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}), (\cos30^{\circ}=\frac{\sqrt{3}}{2}). Substitute these values into the formula: [ \begin{align*} \sin(75^{\circ})&=\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})