Simplify sin(45+30)

Problem

sin(45^∘+30^∘)

Solution

1. Identify the appropriate trigonometric identity for the sine of a sum of two angles.

sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

2. Substitute the given values A=45^∘ and B=30^∘ into the identity.

sin(45^∘+30^∘)=sin(45^∘)*cos(30^∘)+cos(45^∘)*sin(30^∘)

3. Evaluate the exact trigonometric values for each term.

sin(45^∘)=√(,2)/2

cos(30^∘)=√(,3)/2

cos(45^∘)=√(,2)/2

sin(30^∘)=1/2

4. Multiply the terms together.

sin(45^∘+30^∘)=√(,2)/2⋅√(,3)/2+√(,2)/2⋅1/2

sin(45^∘+30^∘)=√(,6)/4+√(,2)/4

5. Combine the fractions over a common denominator.

sin(45^∘+30^∘)=(√(,6)+√(,2))/4

Final Answer

sin(45^∘+30^∘)=(√(,6)+√(,2))/4

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