Find the Exact Value cos(16)

Problem

cos(16)

Solution

1. Identify the task as finding the exact trigonometric value of cos(16)

2. Recognize that 16 is not a standard angle (like 30 45 or 60, but it can be expressed using the half-angle formula or sum/difference identities.

3. Relate 16 to known angles. One method is to use the fact that 16=60−44 or 16=15+1 but these do not lead to simple exact radicals.

4. Use the relationship with 80 and 72 or more commonly, the construction of a regular pentagon (18 and subtracting 2 However, the standard "exact value" for such angles usually involves nested radicals derived from cos(16)=cos(2⋅8) or cos(16)=cos(36−20)

5. Apply the half-angle formula repeatedly starting from cos(32) or using the roots of unity for a 360/16 approach. For most curriculum purposes, the exact value of cos(16) is expressed using the following radical form derived from the properties of a 15-17-8 triangle or complex trigonometric identities.

6. State the exact value using the radical expression derived from the half-angle of 32 or the combination of 72 and 60 related identities.

Final Answer

cos(16)=1/2√(,2+√(,2+2*cos(64)))

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