Find the Exact Value cos(66)

Problem

cos(66)

Solution

1. Identify the expression as the cosine of an angle in degrees. Since 66 is not a standard angle, use the sum formula cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

2. Decompose the angle into the sum of two standard angles: 66=60+6 However, 6 is not a standard angle. Instead, use 66=36+30

3. Recall the exact values for 30 and 36

cos(30)=√(,3)/2

sin(30)=1/2

cos(36)=(1+√(,5))/4

sin(36)=√(,10−2√(,5))/4

4. Apply the sum formula for cosine.

cos(66)=cos(36+30)

cos(66)=cos(36)*cos(30)−sin(36)*sin(30)

5. Substitute the known values into the formula.

cos(66)=((1+√(,5))/4)*(√(,3)/2)−(√(,10−2√(,5))/4)*(1/2)

6. Simplify the expression by multiplying the fractions.

cos(66)=(√(,3)+√(,15))/8−√(,10−2√(,5))/8

cos(66)=(√(,3)+√(,15)−√(,10−2√(,5)))/8

Final Answer

cos(66)=(√(,3)+√(,15)−√(,10−2√(,5)))/8

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