Find the Exact Value cos(-105)

Problem

cos(−105^∘)

Solution

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

cos(−105^∘)=cos(105^∘)

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

105^∘=60^∘+45^∘

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

cos(60^∘+45^∘)=cos(60^∘)*cos(45^∘)−sin(60^∘)*sin(45^∘)

4. Substitute the exact values for the sine and cosine of 60^∘ and 45^∘

cos(60^∘)=1/2

cos(45^∘)=√(,2)/2

sin(60^∘)=√(,3)/2

sin(45^∘)=√(,2)/2

5. Multiply the fractions in each term.

1/2⋅√(,2)/2=√(,2)/4

√(,3)/2⋅√(,2)/2=√(,6)/4

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

√(,2)/4−√(,6)/4=(√(,2)−√(,6))/4

Final Answer

cos(−105^∘)=(√(,2)−√(,6))/4

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