find the exact value using a sum or differenc...
find the exact value using a sum or difference identity. sin 390° hint: sin(a ± b) = sin a cos b ± cos a sin b cos(a ± b) = cos a cos b ∓ sin a sin b
Answer
# 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:
\(\frac{1}{2}\)