Find the Exact Value cos(105 degrees )

Problem

cos(105)

Solution

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

cos(105)=cos(60+45)

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

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

3. Substitute the known exact values for the trigonometric functions of 60 and 45

cos(60)=1/2

cos(45)=√(,2)/2

sin(60)=√(,3)/2

sin(45)=√(,2)/2

4. Multiply the fractions in each term.

1/2⋅√(,2)/2−√(,3)/2⋅√(,2)/2=√(,2)/4−√(,6)/4

5. Combine the terms over a common denominator.

(√(,2)−√(,6))/4

Final Answer

cos(105)=(√(,2)−√(,6))/4

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