PHP atan()函数

atan(float $num): float

<?php
// 计算反正切值
echo "atan(0) = " . atan(0) . " 弧度\n";           // 0
echo "atan(1) = " . atan(1) . " 弧度\n";           // 约 0.78539816339745 (π/4)
echo "atan(-1) = " . atan(-1) . " 弧度\n";         // 约 -0.78539816339745 (-π/4)
echo "atan(0.57735026918963) = " . atan(0.57735026918963) . " 弧度\n";  // 约 0.5235987755983 (π/6)
echo "atan(-0.57735026918963) = " . atan(-0.57735026918963) . " 弧度\n"; // 约 -0.5235987755983 (-π/6)

// 处理大数
echo "atan(1000) = " . atan(1000) . " 弧度\n";     // 约 1.5697963271282
echo "atan(-1000) = " . atan(-1000) . " 弧度\n";   // 约 -1.5697963271282

// 转换弧度到角度
$radians = atan(1);
$degrees = rad2deg($radians);
echo "atan(1) = " . $radians . " 弧度 (" . $degrees . " 度)\n";

// 验证 tan(atan(x)) = x
$x = 2.5;
$atan_x = atan($x);
$tan_atan_x = tan($atan_x);
echo "验证: tan(atan(" . $x . ")) = tan(" . $atan_x . ") = " . $tan_atan_x . "\n";
echo "误差: " . abs($tan_atan_x - $x) . "\n";
?>

<?php
// 创建一个角度计算类
class AngleCalculator {
    const PI = M_PI;

    // 计算角度（度）的反正切
    public static function atanDeg($value) {
        return rad2deg(atan($value));
    }

    // 计算正切为指定值的角度（度）
    public static function angleFromTangent($tangent) {
        $radians = atan($tangent);
        return rad2deg($radians);
    }

    // 计算斜边和邻边的夹角
    public static function angleFromOppositeAdjacent($opposite, $adjacent) {
        if ($adjacent == 0) {
            // 处理除零情况
            return $opposite > 0 ? 90 : ($opposite < 0 ? -90 : 0);
        }
        return self::angleFromTangent($opposite / $adjacent);
    }
}

// 使用示例
echo "常见正切值对应的角度：\n";
$tangentValues = [0, 0.5773503, 1, 1.7320508, 1000, -0.5773503, -1, -1.7320508, -1000];

foreach ($tangentValues as $tan) {
    $angle = AngleCalculator::angleFromTangent($tan);
    echo "tan(θ) = " . round($tan, 4) . " → θ = " . round($angle, 2) . "°\n";
}

echo "\n从对边和邻边计算角度：\n";
$sides = [
    [3, 4],     // 对边=3, 邻边=4
    [4, 3],     // 对边=4, 邻边=3
    [1, 1],     // 对边=1, 邻边=1 (45°)
    [1, 0],     // 对边=1, 邻边=0 (90°)
    [0, 1],     // 对边=0, 邻边=1 (0°)
    [-3, 4],    // 对边=-3, 邻边=4
    [3, -4]     // 对边=3, 邻边=-4
];

foreach ($sides as $i => $side) {
    $opposite = $side[0];
    $adjacent = $side[1];
    $angle = AngleCalculator::angleFromOppositeAdjacent($opposite, $adjacent);
    echo "对边=$opposite, 邻边=$adjacent → 角度 = " . round($angle, 2) . "°\n";
}

// 计算atan和atan2的关系
echo "\natan和atan2关系：\n";
$testPairs = [[1, 1], [1, -1], [-1, 1], [-1, -1], [0, 1], [1, 0]];
foreach ($testPairs as $pair) {
    $y = $pair[0];
    $x = $pair[1];
    $atan2_result = rad2deg(atan2($y, $x));
    if ($x != 0) {
        $atan_result = rad2deg(atan($y / $x));
        echo "点($x, $y): atan(y/x) = " . round($atan_result, 2) . "°, atan2(y, x) = " . round($atan2_result, 2) . "°\n";
    } else {
        echo "点($x, $y): atan2(y, x) = " . round($atan2_result, 2) . "° (atan无法计算，因为x=0)\n";
    }
}
?>

<?php
// 计算直线与x轴的夹角
class GeometryCalculator {

    // 计算两点确定的直线与x轴的夹角（弧度）
    public static function lineAngle($x1, $y1, $x2, $y2) {
        $dx = $x2 - $x1;
        $dy = $y2 - $y1;

        if (abs($dx) < 0.000001) {
            // 垂直线
            return $dy > 0 ? M_PI_2 : -M_PI_2;
        }

        return atan($dy / $dx);
    }

    // 计算三角形中的角度
    public static function triangleAngle($side1, $side2, $side3) {
        // 使用余弦定理计算角度
        // 角A的对边是side1
        $cosA = ($side2*$side2 + $side3*$side3 - $side1*$side1) / (2 * $side2 * $side3);

        // 使用acos计算角度
        $angleA = acos($cosA);

        // 也可以使用反正切计算
        // sinA = sqrt(1 - cosA*cosA)
        // tanA = sinA / cosA
        // $angleA = atan(sqrt(1 - $cosA*$cosA) / $cosA);

        return $angleA;
    }

    // 计算向量的方向角
    public static function vectorDirection($x, $y) {
        // 返回向量(x, y)与x轴的夹角
        if (abs($x) < 0.000001) {
            return $y > 0 ? M_PI_2 : -M_PI_2;
        }

        $angle = atan($y / $x);

        // 调整角度到正确的象限
        if ($x < 0) {
            $angle += M_PI;
        }

        // 确保角度在[-π, π]范围内
        if ($angle > M_PI) {
            $angle -= 2 * M_PI;
        }

        return $angle;
    }
}

// 使用示例
echo "计算直线与x轴的夹角：\n";
$points = [
    [0, 0, 1, 0],    // 水平线 (0°)
    [0, 0, 0, 1],    // 垂直线 (90°)
    [0, 0, 1, 1],    // 45°线
    [0, 0, 1, -1],   // -45°线
    [0, 0, -1, 1],   // 135°线
    [0, 0, -1, -1]   // -135°线
];

foreach ($points as $point) {
    $x1 = $point[0]; $y1 = $point[1];
    $x2 = $point[2]; $y2 = $point[3];
    $angle = GeometryCalculator::lineAngle($x1, $y1, $x2, $y2);
    $angleDeg = rad2deg($angle);
    echo "点($x1,$y1)到点($x2,$y2): 角度 = " . round($angleDeg, 2) . "°\n";
}

echo "\n计算三角形角度：\n";
$sideA = 3;
$sideB = 4;
$sideC = 5;

$angleA = GeometryCalculator::triangleAngle($sideA, $sideB, $sideC);
$angleB = GeometryCalculator::triangleAngle($sideB, $sideA, $sideC);
$angleC = GeometryCalculator::triangleAngle($sideC, $sideA, $sideB);

echo "三角形边长: a = $sideA, b = $sideB, c = $sideC\n";
echo "角A: " . round(rad2deg($angleA), 2) . "°\n";
echo "角B: " . round(rad2deg($angleB), 2) . "°\n";
echo "角C: " . round(rad2deg($angleC), 2) . "°\n";
echo "角度总和: " . round(rad2deg($angleA + $angleB + $angleC), 2) . "°\n";

echo "\n计算向量方向角：\n";
$vectors = [[1, 0], [0, 1], [1, 1], [-1, 1], [-1, -1], [1, -1]];
foreach ($vectors as $vector) {
    $x = $vector[0];
    $y = $vector[1];
    $direction = GeometryCalculator::vectorDirection($x, $y);
    $directionDeg = rad2deg($direction);
    echo "向量($x, $y): 方向角 = " . round($directionDeg, 2) . "°\n";
}
?>

<?php
// 抛体运动中的角度计算
class ProjectileMotion {

    // 计算抛体运动的发射角度以达到指定目标
    public static function launchAngle($v0, $x, $y, $g = 9.81) {
        // 公式: tan(θ) = (v0² ± √(v0⁴ - g(gx² + 2yv0²))) / (gx)
        // 可能存在两个解

        $v0_2 = $v0 * $v0;
        $v0_4 = $v0_2 * $v0_2;
        $discriminant = $v0_4 - $g * ($g * $x * $x + 2 * $y * $v0_2);

        if ($discriminant < 0) {
            throw new InvalidArgumentException("无法达到目标点");
        }

        $sqrt_discriminant = sqrt($discriminant);

        $tan1 = ($v0_2 + $sqrt_discriminant) / ($g * $x);
        $tan2 = ($v0_2 - $sqrt_discriminant) / ($g * $x);

        $angle1 = atan($tan1);
        $angle2 = atan($tan2);

        return [
            'angle1_rad' => $angle1,
            'angle1_deg' => rad2deg($angle1),
            'angle2_rad' => $angle2,
            'angle2_deg' => rad2deg($angle2)
        ];
    }

    // 计算抛体落点
    public static function landingPoint($v0, $angle_deg, $g = 9.81) {
        $angle_rad = deg2rad($angle_deg);
        $range = ($v0 * $v0 * sin(2 * $angle_rad)) / $g;
        $max_height = ($v0 * $v0 * sin($angle_rad) * sin($angle_rad)) / (2 * $g);

        return [
            'range' => $range,
            'max_height' => $max_height,
            'time_of_flight' => (2 * $v0 * sin($angle_rad)) / $g
        ];
    }
}

// 使用示例
echo "抛体运动计算：\n";
$v0 = 20; // 初速度 20 m/s
$x = 30;  // 目标x坐标 30 m
$y = 5;   // 目标y坐标 5 m

try {
    $angles = ProjectileMotion::launchAngle($v0, $x, $y);

    echo "初速度: {$v0} m/s, 目标点: ($x, $y) m\n";
    echo "可能的发射角度:\n";
    echo "角度1: " . round($angles['angle1_deg'], 2) . "° (" . round($angles['angle1_rad'], 4) . " 弧度)\n";
    echo "角度2: " . round($angles['angle2_deg'], 2) . "° (" . round($angles['angle2_rad'], 4) . " 弧度)\n";

    // 计算两个角度的轨迹
    foreach ([$angles['angle1_deg'], $angles['angle2_deg']] as $i => $angle) {
        $trajectory = ProjectileMotion::landingPoint($v0, $angle);
        echo "\n角度" . ($i+1) . " ($angle°):\n";
        echo "  射程: " . round($trajectory['range'], 2) . " m\n";
        echo "  最大高度: " . round($trajectory['max_height'], 2) . " m\n";
        echo "  飞行时间: " . round($trajectory['time_of_flight'], 2) . " s\n";
    }

} catch (Exception $e) {
    echo "错误: " . $e->getMessage() . "\n";
}

// 计算斜坡上的摩擦力角度
echo "\n斜坡上的物体：\n";
function calculateCriticalAngle($friction_coefficient) {
    // 当物体刚好开始滑动时，摩擦角 = atan(摩擦系数)
    $critical_angle_rad = atan($friction_coefficient);
    $critical_angle_deg = rad2deg($critical_angle_rad);

    return [
        'coefficient' => $friction_coefficient,
        'critical_angle_rad' => $critical_angle_rad,
        'critical_angle_deg' => $critical_angle_deg
    ];
}

$friction_coefficients = [0.1, 0.2, 0.3, 0.5, 0.7, 1.0];
echo "摩擦系数与临界角：\n";
foreach ($friction_coefficients as $mu) {
    $result = calculateCriticalAngle($mu);
    echo "摩擦系数 μ = $mu → 临界角 = " . round($result['critical_angle_deg'], 2) . "°\n";
}

// 计算力与水平面的夹角
echo "\n力的分解：\n";
function forceComponents($force, $angle_deg) {
    $angle_rad = deg2rad($angle_deg);
    $horizontal = $force * cos($angle_rad);
    $vertical = $force * sin($angle_rad);

    return [
        'force' => $force,
        'angle' => $angle_deg,
        'horizontal' => $horizontal,
        'vertical' => $vertical
    ];
}

$force = 100; // 100 N
$angles = [0, 30, 45, 60, 90];
foreach ($angles as $angle) {
    $components = forceComponents($force, $angle);
    echo "力 = {$force}N, 角度 = {$angle}°:\n";
    echo "  水平分量: " . round($components['horizontal'], 2) . " N\n";
    echo "  垂直分量: " . round($components['vertical'], 2) . " N\n";
}
?>

<?php
// 数值计算反正切的方法
class NumericalMethods {

    // 使用泰勒级数计算反正切
    public static function atanTaylor($x, $terms = 10) {
        // 泰勒展开：atan(x) = x - x³/3 + x⁵/5 - x⁷/7 + ...
        // 仅适用于 |x| < 1

        $result = 0;
        $sign = 1;

        for ($n = 0; $n < $terms; $n++) {
            $exponent = 2 * $n + 1;
            $term = pow($x, $exponent) / $exponent * $sign;
            $result += $term;
            $sign *= -1;
        }

        return $result;
    }

    // 使用反正切公式计算π
    public static function calculatePiMachin() {
        // Machin公式：π/4 = 4*atan(1/5) - atan(1/239)
        $term1 = 4 * atan(1/5);
        $term2 = atan(1/239);
        return 4 * ($term1 - $term2);
    }

    // 使用反正切公式计算π（高精度）
    public static function calculatePiHighPrecision($iterations = 10) {
        // 使用反正切公式：π = 16*atan(1/5) - 4*atan(1/239)
        $pi = 0;

        for ($n = 0; $n < $iterations; $n++) {
            // 计算atan(1/5)的级数项
            $x1 = 1/5;
            $term1 = pow($x1, 2*$n+1) / (2*$n+1);
            $term1 *= ($n % 2 == 0 ? 1 : -1);

            // 计算atan(1/239)的级数项
            $x2 = 1/239;
            $term2 = pow($x2, 2*$n+1) / (2*$n+1);
            $term2 *= ($n % 2 == 0 ? 1 : -1);

            $pi += 16 * $term1 - 4 * $term2;
        }

        return $pi;
    }

    // 计算复数的辐角（argument）
    public static function complexArgument($real, $imag) {
        // 复数的辐角 = atan(imag/real)
        // 需要考虑象限

        if (abs($real) < 0.000001) {
            if ($imag > 0) return M_PI_2;
            if ($imag < 0) return -M_PI_2;
            return 0; // 原点
        }

        $angle = atan($imag / $real);

        // 调整到正确的象限
        if ($real < 0) {
            $angle += $imag >= 0 ? M_PI : -M_PI;
        }

        return $angle;
    }
}

// 使用示例
echo "泰勒级数计算反正切：\n";
$testValues = [0.1, 0.5, 0.9];
foreach ($testValues as $x) {
    $taylor = NumericalMethods::atanTaylor($x, 20);
    $builtin = atan($x);
    $error = abs($taylor - $builtin);
    echo "atan($x): 泰勒 = " . sprintf("%.10f", $taylor) .
         ", 内置 = " . sprintf("%.10f", $builtin) .
         ", 误差 = " . sprintf("%.2e", $error) . "\n";
}

echo "\n使用反正切公式计算π：\n";
$piMachin = NumericalMethods::calculatePiMachin();
$piHighPrecision = NumericalMethods::calculatePiHighPrecision(20);
echo "Machin公式计算: π = " . sprintf("%.15f", $piMachin) . "\n";
echo "级数展开计算: π = " . sprintf("%.15f", $piHighPrecision) . "\n";
echo "PHP常量M_PI: π = " . sprintf("%.15f", M_PI) . "\n";
echo "Machin公式误差: " . abs($piMachin - M_PI) . "\n";
echo "级数展开误差: " . abs($piHighPrecision - M_PI) . "\n";

echo "\n计算复数的辐角：\n";
$complexNumbers = [
    [1, 0],      // 正实轴
    [0, 1],      // 正虚轴
    [-1, 0],     // 负实轴
    [0, -1],     // 负虚轴
    [1, 1],      // 第一象限
    [-1, 1],     // 第二象限
    [-1, -1],    // 第三象限
    [1, -1]      // 第四象限
];

foreach ($complexNumbers as $i => $number) {
    $real = $number[0];
    $imag = $number[1];
    $arg = NumericalMethods::complexArgument($real, $imag);
    $argDeg = rad2deg($arg);
    echo "复数 {$real}+{$imag}i: 辐角 = " . round($argDeg, 2) . "° (" . round($arg, 4) . " 弧度)\n";
}

// 计算反正切的导数
echo "\n反正切函数的导数：\n";
function atanDerivative($x) {
    // d(atan(x))/dx = 1/(1+x²)
    return 1 / (1 + $x * $x);
}

function numericalDerivative($x, $h = 0.0001) {
    // 数值导数
    return (atan($x + $h) - atan($x - $h)) / (2 * $h);
}

$testPoints = [-2, -1, 0, 1, 2];
foreach ($testPoints as $x) {
    $exact = atanDerivative($x);
    $numeric = numericalDerivative($x);
    $error = abs($exact - $numeric);
    echo "x = $x: 精确导数 = " . round($exact, 6) .
         ", 数值导数 = " . round($numeric, 6) .
         ", 误差 = " . sprintf("%.2e", $error) . "\n";
}

// 计算反正切的积分
echo "\n反正切函数的积分：\n";
function atanIntegral($x) {
    // ∫atan(x)dx = x·atan(x) - 0.5·ln(1+x²)
    return $x * atan($x) - 0.5 * log(1 + $x * $x);
}

function numericalIntegral($a, $b, $n = 1000) {
    // 数值积分（辛普森法则）
    $h = ($b - $a) / $n;
    $sum = atan($a) + atan($b);

    for ($i = 1; $i < $n; $i++) {
        $x = $a + $i * $h;
        $sum += 4 * atan($x - $h/2) + 2 * atan($x);
    }

    $sum += 4 * atan($b - $h/2);
    return $sum * $h / 6;
}

$a = 0;
$b = 1;
$exactIntegral = atanIntegral($b) - atanIntegral($a);
$numericIntegral = numericalIntegral($a, $b);
$error = abs($exactIntegral - $numericIntegral);

echo "∫atan(x)dx 从 $a 到 $b:\n";
echo "精确积分: " . round($exactIntegral, 6) . "\n";
echo "数值积分: " . round($numericIntegral, 6) . "\n";
echo "误差: " . sprintf("%.2e", $error) . "\n";
?>