Find the Exact Value cos(37)

Problem

cos(37^∘)

Solution

1. Identify the nature of the angle. The angle 37^∘ is not one of the standard reference angles (such as 0^∘ 30^∘ 45^∘ 60^∘ or 90^∘ found on the unit circle.

2. Approximate using Pythagorean triples. In trigonometry, 37^∘ is often used as a common approximation for the smaller angle in a 3−4−5 right triangle.

3. Apply the definition of cosine. In a right triangle where the sides are 3 4 and 5 the angle opposite the side of length 3 is approximately 36.87^∘

4. Determine the ratio. For an angle θ≈36.87^∘ the cosine is the ratio of the adjacent side to the hypotenuse.

cos(37^∘)≈4/5

5. State the exact form. Since 37^∘ is not a multiple of 3^∘ its exact value cannot be expressed using only square roots. It remains in its trigonometric form unless represented by a specific infinite series or a decimal approximation.

Final Answer

cos(37^∘)=cos(37^∘)

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