Find the Exact Value sin(pi/6-pi/4)

Problem

sin(π/6−π/4)

Solution

1. Identify the appropriate trigonometric identity. Use the sine difference formula:

sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

2. Substitute the values A=π/6 and B=π/4 into the formula:

sin(π/6−π/4)=sin(π/6)*cos(π/4)−cos(π/6)*sin(π/4)

3. Evaluate the trigonometric functions for the known special angles:

sin(π/6)=1/2

cos(π/6)=√(,3)/2

sin(π/4)=√(,2)/2

cos(π/4)=√(,2)/2

4. Multiply the terms:

sin(π/6−π/4)=(1/2)*(√(,2)/2)−(√(,3)/2)*(√(,2)/2)

sin(π/6−π/4)=√(,2)/4−√(,6)/4

5. Combine the fractions over a common denominator:

sin(π/6−π/4)=(√(,2)−√(,6))/4

Final Answer

sin(π/6−π/4)=(√(,2)−√(,6))/4

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