Angle Fit Range

Angle Fit Range is a targeting puzzle that teaches inverse trigonometry by making you use it to aim a cannon. A rotating cannon sits at the centre of a dark targeting range. Targets glow at specific positions — each one at a known x and y coordinate from the cannon. The trig panel below the canvas updates sixty times per second as the cannon rotates: the current angle in degrees, sin, cos, tan, and cot, all live. The player rotates the cannon toward each target by dragging or using fine adjustment buttons, then holds the aim steady within three degrees for four tenths of a second to lock. Score is based on precision at the moment of locking. The mathematics the game builds: the required aim angle for a target at position x, y is arctan of y over x. This formula appears in the target information panel for every target. In the first level, all targets are in the first quadrant — x and y are both positive — and arctan works correctly every time. The player builds intuition for arctan as a function that converts a ratio into an angle: a steep target with large y and small x requires an angle near ninety degrees; a shallow target requires an angle near zero. The second level introduces targets in all four quadrants. For a target at negative x and positive y — in the second quadrant — the formula arctan of y over x returns a negative angle, placing the beam in the fourth quadrant, pointing the wrong direction. The panel shows this failure explicitly: arctan gives negative forty-five degrees with a red cross, while atan2 gives one hundred thirty-five degrees with a green check. The player must use the atan2 result. After ten targets across four quadrants, the reason is clear: arctan can only return values between negative ninety and positive ninety degrees because tangent repeats every one hundred eighty degrees and a valid inverse function requires a restricted domain. atan2 uses the signs of both coordinates to select the correct quadrant, returning values across the full circle. The trig panel makes the asymptotes of tangent and cotangent visible without any explanation. Rotating past eighty-nine degrees, the tan display reads a very large number, then the symbol for infinity, then a very large negative number. This is not an error — it is the tangent function crossing its vertical asymptote at ninety degrees. Rotating past zero degrees in the other direction, cot does the same. Watching both happen while physically controlling the angle produces more intuitive understanding of asymptotic behaviour than any graph in a textbook. The third level removes the coordinate display and shows only the tangent or cotangent value required. Two angles always share the same tangent — the player must look at the target position to determine which quadrant applies. Some targets show a cotangent requirement instead of tangent, requiring the player to compute the reciprocal mentally before finding the angle. After completing the third level, the relationships between the six trig functions are no longer abstract — they are things the player has navigated under time pressure.