Algorithm to find the principle argument:

Step 1: 1^st draw a trigonometric circle and mark the quadrant, in which the angle may lie.

Step 2: Select anticlockwise direction for 1^st and 2^nd quadrants and select the clockwise direction for 3^rd  and 4^th quadrants.

Step 3: Find angle in the 1^st rotation.

Step 4: Select numerically least angle among these 2 values. The angle hence found will be principal value.

Step 5: In case, 2 angles 1 with positive sign and the other with negative sign qualify for numerically least angle, then it is convention to select angle with positive sign as principal value.

Example: If tanθ = - 1, then θ will lie in 2^nd or 4^th quadrant.

           For 2^nd quadrant we will select anticlockwise and for 4^th quadrant. we will select clockwise direction.

           In 1^st circle 2 values -Π/4 and 3Π/4 are obtained.

           Among these 2, -Π/4 is numerically least angle. Thus principal value is -Π/4.

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