Find the Exact Value cos(75 degrees )

Problem

cos(75)

Solution

1. Identify the angle as a sum of two special angles from the unit circle.

cos(75)=cos(45+30)

2. Apply the cosine sum formula, which states cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

cos(45+30)=cos(45)*cos(30)−sin(45)*sin(30)

3. Substitute the known exact values for the trigonometric functions of 45 and 30

cos(45)=√(,2)/2

cos(30)=√(,3)/2

sin(45)=√(,2)/2

sin(30)=1/2

4. Multiply the fractions in each term.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

5. Subtract the results to find the final exact value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

cos(75)=(√(,6)−√(,2))/4

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