Find the Exact Value sin(7.5)

Problem

sin(7.5^∘)

Solution

1. Identify the angle as a half of a known special angle. Since 7.5^∘=15^∘/2 we can use the half-angle formula for sine.

2. Recall the half-angle formula for sine, which is sin(θ/2)=±√(,(1−cos(θ))/2) Since 7.5^∘ is in the first quadrant, the value is positive.

3. Determine the value of cos(15^∘) using the difference formula cos(45^∘−30^∘)=cos(45^∘)*cos(30^∘)+sin(45^∘)*sin(30^∘)

4. Substitute the known values: cos(15^∘)=(√(,2)/2)*(√(,3)/2)+(√(,2)/2)*(1/2)=(√(,6)+√(,2))/4

5. Apply the half-angle formula with θ=15^∘ sin(7.5^∘)=√(,(1−(√(,6)+√(,2))/4)/2)

6. Simplify the expression inside the radical by finding a common denominator: sin(7.5^∘)=√(,(4−√(,6)−√(,2))/8)

7. Rationalize the denominator by multiplying the numerator and denominator inside the square root by 2 sin(7.5^∘)=√(,(8−2√(,6)−2√(,2))/16)

8. Extract the square root of the denominator: sin(7.5^∘)=√(,8−2√(,6)−2√(,2))/4

Final Answer

sin(7.5^∘)=√(,8−2√(,6)−2√(,2))/4

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