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

Want more problems? Check here!