Expand Using Sum/Difference Formulas sin(30)

Problem

sin(30^∘)

Solution

1. Identify the goal to represent 30^∘ as a sum or difference of two common angles from the unit circle, such as 90^∘−60^∘ or 45^∘−15^∘ Using 90^∘−60^∘ is a standard choice.

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 A=90^∘ and B=60^∘ into the identity.

sin(90^∘−60^∘)=sin(90^∘)*cos(60^∘)−cos(90^∘)*sin(60^∘)

4. Evaluate the trigonometric values for the specific angles: sin(90^∘)=1 cos(60^∘)=1/2 cos(90^∘)=0 and sin(60^∘)=√(,3)/2

sin(30^∘)=(1)*(1/2)−(0)*(√(,3)/2)

5. Simplify the expression to find the final numerical value.

sin(30^∘)=1/2−0

sin(30^∘)=1/2

Final Answer

sin(30^∘)=1/2

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