Find the Exact Value sin(21)

Problem

sin(21)

Solution

1. Identify the expression as the sine of 21 degrees. Since 21 is not a standard reference angle, we can express it using the difference formula for sine: sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

2. Rewrite the angle using standard angles 45 and 24 or 60 and 39 However, a more common exact form involves using the sum of 45 and 30 to find 75 or other combinations. For 21 we use the relation 21=45−24 or 21=51−30

3. Apply the triple angle and pentagon-based constants. The exact value for sin(21) is typically expressed using radicals involving the golden ratio ϕ=(1+√(,5))/2 or roots of polynomials.

4. Substitute the known exact values for sin(21) derived from the identity sin(21)=sin(60−39) or similar combinations. The simplest radical form is:

sin(21)=1/8*(√(,2)√(,5−√(,5))+√(,3)*(√(,5)−1)−√(,2)√(,3*(5+√(,5)))+√(,5)+1)

5. Simplify the expression into its standard exact radical form.

Final Answer

sin(21)=1/8*(√(,2)√(,5−√(,5))+√(,3)*(√(,5)−1)−√(,6)√(,5+√(,5))+√(,5)+1)

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