Find the Value Using the Unit Circle arctan(1)

Problem

arctan(1)

Solution

1. Identify the definition of the inverse tangent function, where y=arctan(x) means tan(y)=x for y in the interval (−π/2,π/2)

2. Set up the equation based on the given value.

tan(y)=1

3. Recall the definition of the tangent function on the unit circle.

tan(y)=sin(y)/cos(y)

4. Determine the angle in the interval (−π/2,π/2) where the sine and cosine values are equal, resulting in a quotient of 1

sin(y)=cos(y)

5. Locate the point on the unit circle in the first quadrant where x=y This occurs at an angle of 45 or π/4 radians, where both coordinates are √(,2)/2

tan(π/4)=√(,2)/2/√(,2)/2=1

Final Answer

arctan(1)=π/4

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