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

Want more problems? Check here!