Find the Exact Value cos(21)

Problem

cos(21^∘)

Solution

1. Identify the task as finding the exact trigonometric value of cos(21^∘)

2. Recognize that 21^∘ is not a standard angle (like 30^∘ 45^∘ or 60^∘ and cannot be easily expressed as a simple sum or difference of standard angles using integers.

3. Express the angle using the sum of 45^∘ and 72^∘ or other combinations involving 3^∘ increments, as 3^∘=18^∘−15^∘ Note that 21^∘=36^∘−15^∘

4. Apply the cosine difference identity cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B) where A=36^∘ and B=15^∘

5. Substitute the known exact values:

cos(36^∘)=(1+√(,5))/4

sin(36^∘)=√(,10−2√(,5))/4

cos(15^∘)=(√(,6)+√(,2))/4

sin(15^∘)=(√(,6)−√(,2))/4

6. Calculate the product:

cos(21^∘)=((1+√(,5))/4)*((√(,6)+√(,2))/4)+(√(,10−2√(,5))/4)*((√(,6)−√(,2))/4)

7. Simplify the expression into a single fraction.

Final Answer

cos(21^∘)=((√(,6)+√(,2))*(1+√(,5))+(√(,6)−√(,2))√(,10−2√(,5)))/16

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