Verify the Identity sin(3pi-x)=sin(x)

Problem

sin(3*π−x)=sin(x)

Solution

1. Identify the sine subtraction identity, which states sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

2. Substitute the values A=3*π and B=x into the identity.

sin(3*π−x)=sin(3*π)*cos(x)−cos(3*π)*sin(x)

3. Evaluate the trigonometric functions at 3*π Since 3*π is coterminal with π we have sin(3*π)=0 and cos(3*π)=−1

sin(3*π−x)=(0)*cos(x)−(−1)*sin(x)

4. Simplify the expression by performing the multiplication and addition.

sin(3*π−x)=0+sin(x)

sin(3*π−x)=sin(x)

Final Answer

sin(3*π−x)=sin(x)

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