Find the Exact Value sin(21)
Problem
Solution
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) Rewrite the angle using standard angles
45^∘ and24^∘ or60^∘ and39^∘ However, a more common exact form involves using the sum of45^∘ and30^∘ to find75^∘ or other combinations. For21^∘ we use the relation21^∘=45^∘−24^∘ or21^∘=51^∘−30^∘ Apply the triple angle and pentagon-based constants. The exact value for
sin(21^∘) is typically expressed using radicals involving the golden ratioϕ=(1+√(,5))/2 or roots of polynomials.Substitute the known exact values for
sin(21^∘) derived from the identitysin(21^∘)=sin(60^∘−39^∘) or similar combinations. The simplest radical form is:
Simplify the expression into its standard exact radical form.
Final Answer
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