Simplify sin(x/5)-sin(x/6)

Problem

sin(x/5)−sin(x/6)

Solution

1. Identify the appropriate sum-to-product identity for the difference of two sine functions.

sin(A)−sin(B)=2*cos((A+B)/2)*sin((A−B)/2)

2. Assign the values for the variables A and B based on the given expression.

A=x/5

B=x/6

3. Calculate the sum term (A+B)/2 by finding a common denominator for the fractions.

(A+B)/2=(x/5+x/6)/2

(A+B)/2=(6*x+5*x)/30/2

(A+B)/2=(11*x)/60

4. Calculate the difference term (A−B)/2 using the same common denominator.

(A−B)/2=(x/5−x/6)/2

(A−B)/2=(6*x−5*x)/30/2

(A−B)/2=x/60

5. Substitute these results back into the sum-to-product formula to obtain the simplified form.

sin(x/5)−sin(x/6)=2*cos((11*x)/60)*sin(x/60)

Final Answer

sin(x/5)−sin(x/6)=2*cos((11*x)/60)*sin(x/60)

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