Expand Using Sum/Difference Formulas sin(20)

Problem

sin(20)

Solution

1. Identify two angles whose sum or difference equals 20 using common reference angles. We can use 60 and 40 or 45 and 25 but the standard expansion uses the identity sin(A−B) Let A=60 and B=40 or A=50 and B=30 Let's use A=50 and B=30

2. Apply the formula for the sine of a difference, which is sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

3. Substitute the values A=50 and B=30 into the identity.

sin(50−30)=sin(50)*cos(30)−cos(50)*sin(30)

4. Evaluate the known trigonometric values for 30 where sin(30)=1/2 and cos(30)=√(,3)/2

sin(20)=sin(50)⋅√(,3)/2−cos(50)⋅1/2

5. Simplify the expression by factoring out the common denominator.

sin(20)=(√(,3)*sin(50)−cos(50))/2

Final Answer

sin(20)=sin(50)*cos(30)−cos(50)*sin(30)

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