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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