Find the Exact Value tan(100)

Problem

tan(100^∘)

Solution

1. Identify the angle and the required operation. The task is to find the exact value of tan(100^∘)

2. Apply the sum of angles formula for tangent, tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B)) by splitting 100^∘ into 60^∘+40^∘ or 45^∘+55^∘ However, 100^∘ is not a standard reference angle or a simple combination of standard angles (30^∘,45^∘,60^∘ that results in a simplified radical expression.

3. Recognize that 100^∘ is in the second quadrant. In the second quadrant, the tangent function is negative.

4. Relate to the reference angle. The reference angle is 180^∘−100^∘=80^∘ Therefore, tan(100^∘)=−tan(80^∘)

5. Determine if a simpler radical form exists. Since 100^∘ is not a multiple of 3^∘ or 15^∘ its exact value involves complex nested radicals or roots of high-degree polynomials that do not simplify into a standard form.

6. Conclude that the exact value is expressed in terms of the function itself or its relation to the reference angle.

Final Answer

tan(100^∘)=−tan(80^∘)

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