Expand Using Sum/Difference Formulas cos(105)

Problem

cos(105)

Solution

1. Identify two angles with known trigonometric values that sum or subtract to 105

105=60+45

2. Apply the formula for the cosine of a sum, which is 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 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

4. Multiply the terms in the expression.

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

5. Simplify the expression by combining the fractions over a common denominator.

(√(,2)−√(,6))/4

Final Answer

cos(105)=(√(,2)−√(,6))/4

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