Find the Exact Value cos(67)

Problem

cos(67^∘)

Solution

1. Identify the task as finding the exact trigonometric value of cos(67^∘)

2. Determine if the angle is a standard reference angle. The angle 67^∘ is not a standard angle (like 30^∘ 45^∘ or 60^∘, nor is it a simple sum or difference of standard angles that results in a common radical form (like 15^∘ or 75^∘.

3. Recognize that without further context or specific identities (such as half-angle formulas for 134^∘, the exact value cannot be simplified into basic radicals.

4. Express the value using the complementary angle identity cos(θ)=sin(90^∘−θ)

cos(67^∘)=sin(90^∘−67^∘)

cos(67^∘)=sin(23^∘)

5. Conclude that since 67^∘ is not a multiple of 3^∘ its exact value in terms of square roots is extremely complex and typically left in its trigonometric form.

Final Answer

cos(67^∘)=cos(67^∘)

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