Find the Exact Value cos(16)
Problem
Solution
Identify the task as finding the exact trigonometric value of
cos(16^∘) Recognize that
16^∘ is not a standard angle (like30^∘ 45^∘ or60^∘ , but it can be expressed using the half-angle formula or sum/difference identities.Relate
16^∘ to known angles. One method is to use the fact that16^∘=60^∘−44^∘ or16^∘=15^∘+1^∘ but these do not lead to simple exact radicals.Use the relationship with
80^∘ and72^∘ or more commonly, the construction of a regular pentagon (18^∘ and subtracting2^∘ However, the standard "exact value" for such angles usually involves nested radicals derived fromcos(16^∘)=cos(2⋅8^∘) orcos(16^∘)=cos(36^∘−20^∘) Apply the half-angle formula repeatedly starting from
cos(32^∘) or using the roots of unity for a360/16 approach. For most curriculum purposes, the exact value ofcos(16^∘) is expressed using the following radical form derived from the properties of a 15-17-8 triangle or complex trigonometric identities.State the exact value using the radical expression derived from the half-angle of
32^∘ or the combination of72^∘ and60^∘ related identities.
Final Answer
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