Find the Exact Value cos(105)

Problem

cos(105)

Solution

1. Identify the angle 105 as a sum of two special angles from the unit circle, such as 60 and 45

2. Apply the formula for the cosine of a sum, which is cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

3. Substitute A=60 and B=45 into the identity.

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

4. Evaluate the trigonometric functions for the special angles.

cos(60)=1/2

cos(45)=√(,2)/2

sin(60)=√(,3)/2

sin(45)=√(,2)/2

5. Multiply the terms together.

1/2⋅√(,2)/2=√(,2)/4

√(,3)/2⋅√(,2)/2=√(,6)/4

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

√(,2)/4−√(,6)/4=(√(,2)−√(,6))/4

Final Answer

cos(105)=(√(,2)−√(,6))/4

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