Find the Exact Value sin(68)
Problem
Solution
Identify the goal as finding the exact value of
sin(68^∘) Since68^∘ is not a standard angle, it can be expressed as the sum of two angles, such as60^∘+8^∘ or30^∘+38^∘ but these do not lead to a standard exact form easily.Apply the sum-to-product or double-angle identities if possible. However,
68^∘ is typically expressed in terms of radicals using the values for18^∘ and15^∘ or by using the half-angle and sum formulas.Recognize that
68^∘=60^∘+8^∘ or90^∘−22^∘ A common way to express this exact value involves nested radicals derived from the pentagon constants (sin(18^∘) and standard trigonometric identities.Use the identity
sin(68^∘)=cos(22^∘) The exact value forcos(22^∘) is derived from the half-angle formulacos(44/2) or by relating it to the roots of specific polynomials.State the exact value in its standard radical form. The value is:
Simplify the expression into a single combined form if required.
Final Answer
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