find the exact value of the expression by using appropriate identities. do not use a calculator.\n\n\\(…

find the exact value of the expression by using appropriate identities. do not use a calculator.\n\n\\( \\sin 83 ^ { \\circ } \\cos 23 ^ { \\circ } - \\cos 83 ^ { \\circ } \\sin 23 ^ { \\circ } = \\)\n\n(type a fraction.)

find the exact value of the expression by using appropriate identities. do not use a calculator.\n\n\\( \\sin 83 ^ { \\circ } \\cos 23 ^ { \\circ } - \\cos 83 ^ { \\circ } \\sin 23 ^ { \\circ } = \\)\n\n(type a fraction.)

Answer

Explanation:

Step1: Use the sine subtraction formula

The formula for (\sin(A - B)=\sin A\cos B-\cos A\sin B). Let (A = 83^{\circ}) and (B=23^{\circ}).

Step2: Substitute values into the formula

We have (\sin83^{\circ}\cos23^{\circ}-\cos83^{\circ}\sin23^{\circ}=\sin(83^{\circ}- 23^{\circ})).

Step3: Simplify the expression

(\sin(83^{\circ}-23^{\circ})=\sin60^{\circ}). Since (\sin60^{\circ}=\frac{\sqrt{3}}{2}).

Answer:

(\frac{\sqrt{3}}{2})