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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