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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