Find the Exact Value cos(9)

Problem

cos(9^∘)

Solution

1. Identify the angle as 9^∘ which can be expressed as the difference between two angles with known exact values, such as 45^∘ and 36^∘

2. Apply the cosine difference formula cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B) where A=45^∘ and B=36^∘

3. Recall the exact values for 45^∘

cos(45^∘)=√(,2)/2

sin(45^∘)=√(,2)/2

4. Recall the exact values for 36^∘

cos(36^∘)=(1+√(,5))/4

sin(36^∘)=√(,10−2√(,5))/4

5. Substitute these values into the difference formula:

cos(9^∘)=√(,2)/2⋅(1+√(,5))/4+√(,2)/2⋅√(,10−2√(,5))/4

6. Simplify the expression by combining the terms over a common denominator:

cos(9^∘)=(√(,2)+√(,10))/8+√(,20−4√(,5))/8

7. Further simplify the second term by factoring out √(,4) from the radical:

cos(9^∘)=(√(,2)+√(,10)+2√(,5−√(,5)))/8

Final Answer

cos(9^∘)=(√(,2)+√(,10)+2√(,5−√(,5)))/8

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