Find the Exact Value cos(-75)

Problem

cos(−75)

Solution

1. Apply the even-odd identity for the cosine function, which states cos(−θ)=cos(θ)

cos(−75)=cos(75)

2. Rewrite the angle as a sum of two special angles from the unit circle.

75=45+30

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

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

4. Substitute the exact values for the sine and cosine of 45 and 30

cos(45)=√(,2)/2

cos(30)=√(,3)/2

sin(45)=√(,2)/2

sin(30)=1/2

5. Multiply the fractions in each term.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

6. Subtract the terms to find the final exact value.

√(,6)/4−√(,2)/4=(√(,6)−√(,2))/4

Final Answer

cos(−75)=(√(,6)−√(,2))/4

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