Find the Exact Value sin(74)
Problem
Solution
Identify the angle as a sum of two angles with known trigonometric values. We can write
74^∘ as30^∘+44^∘ or44^∘+30^∘ but it is more useful to use the half-angle formula or sum-to-product identities. A common approach for74^∘ is to recognize it as2×37^∘ but37^∘ is an approximation. Instead, use the sum formulasin(A+B)=sin(A)*cos(B)+cos(A)*sin(B) with44^∘ and30^∘ or recognize74^∘=90^∘−16^∘ Apply the double angle formula for
cos(2*θ) to findsin(74^∘) using the identitysin(74^∘)=cos(16^∘) We can findcos(16^∘) by using the half-angle formula twice starting from64^∘ or using the values for18^∘ and2^∘ but the standard exact form forsin(74^∘) is derived from the values ofsin(18^∘) andcos(18^∘) or via the sum44^∘+30^∘ Use the sum formula with
44^∘ and30^∘ is difficult because44^∘ is not standard. Instead, use74^∘=45^∘+29^∘ or60^∘+14^∘ The most direct exact value forsin(74^∘) is often expressed using the radicals derived from the pentagon (sin(18^∘) and other identities.Substitute the known exact values. Using the identity
sin(74^∘)=cos(16^∘) and the half-angle formulacos(16^∘)=√(,(1+cos(32^∘))/2) This path is complex. A more common exact form uses the values forsin(75^∘−1^∘) However, the most simplified exact radical form forsin(74^∘) is:
Simplify the expression. Given the complexity of
74^∘ it is often represented ascos(16^∘) Using the values forsin(18^∘)=(√(,5)−1)/4 andcos(18^∘)=√(,10+2√(,5))/4 we can findsin(74^∘) throughsin(74^∘)=sin(90^∘−16^∘)
Final Answer
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