Simplify sin(3x)+sin(5x)

Problem

sin(3*x)+sin(5*x)

Solution

1. Identify the sum-to-product formula for sine, which states sin(A)+sin(B)=2*sin((A+B)/2)*cos((A−B)/2)

2. Assign the variables A=5*x and B=3*x to make the subtraction easier, noting that sin(3*x)+sin(5*x)=sin(5*x)+sin(3*x)

3. Substitute the values into the formula.

sin(5*x)+sin(3*x)=2*sin((5*x+3*x)/2)*cos((5*x−3*x)/2)

4. Simplify the arguments inside the trigonometric functions.

(5*x+3*x)/2=(8*x)/2=4*x

(5*x−3*x)/2=(2*x)/2=x

5. Combine the results to reach the final simplified form.

2*sin(4*x)*cos(x)

Final Answer

sin(3*x)+sin(5*x)=2*sin(4*x)*cos(x)

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