find the exact value using a sum or difference identity. sin 390° hint: sin(a ± b) = sin a cos b ± cos a sin…

Explanation:

Step1: Rewrite the angle

We know that (390^{\circ}=360^{\circ} + 30^{\circ}), so (\sin390^{\circ}=\sin(360^{\circ}+ 30^{\circ})). According to the angle - sum identity (\sin(A + B)=\sin A\cos B+\cos A\sin B), when (A = 360^{\circ}) and (B=30^{\circ}), (\sin(360^{\circ}+30^{\circ})=\sin360^{\circ}\cos30^{\circ}+\cos360^{\circ}\sin30^{\circ}).

Step2: Evaluate trigonometric values

We know that (\sin360^{\circ}=0) and (\cos360^{\circ}=1), (\cos30^{\circ}=\frac{\sqrt{3}}{2}), (\sin30^{\circ}=\frac{1}{2}). Substitute these values into the above formula: (\sin(360^{\circ}+30^{\circ})=0\times\frac{\sqrt{3}}{2}+1\times\frac{1}{2}).

Step3: Calculate the result

(0\times\frac{\sqrt{3}}{2}+1\times\frac{1}{2}=\frac{1}{2}).

Answer:

(\frac{1}{2})