🐛 Fix some tri angles values

lib/solver.dart

@@ -59,15 +59,33 @@ class SolverService {
         stepNumber: 1,
         title: '表达式求值',
         explanation: '这是一个标准的数学表达式，我们将直接计算其结果。',
-        formula: input,
+        formula: '\$\$$input\$\$',
       ),
     );
 
-    GrammarParser p = GrammarParser();
-    Expression exp = p.parse(input);
-    final result = RealEvaluator().evaluate(exp);
+    // 检查是否为特殊三角函数值，可以返回精确结果
+    final exactTrigResult = _getExactTrigResult(input);
+    if (exactTrigResult != null) {
+      return CalculationResult(
+        steps: steps,
+        finalAnswer: '\$\$$exactTrigResult\$\$',
+      );
+    }
 
-    return CalculationResult(steps: steps, finalAnswer: result.toString());
+    // 预处理输入，将三角函数的参数从度转换为弧度
+    String processedInput = _convertTrigToRadians(input);
+
+    GrammarParser p = GrammarParser();
+    Expression exp = p.parse(processedInput);
+    final result = RealEvaluator().evaluate(exp).toDouble();
+
+    // 尝试将结果格式化为几倍根号的形式
+    final formattedResult = _formatSqrtResult(result);
+
+    return CalculationResult(
+      steps: steps,
+      finalAnswer: '\$\$$formattedResult\$\$',
+    );
   }
 
   /// 2. 求解一元一次方程
@@ -316,12 +334,12 @@ ${a2}x ${b2 >= 0 ? '+' : ''} ${b2}y = $c2 & (2)
         explanation: '为了消去变量 y，将方程(1)两边乘以 $b2，方程(2)两边乘以 $b1。',
         formula:
             '''
-
+\$\$
 \\begin{cases}
-${newA1}x ${b1 * b2 >= 0 ? '+' : ''} ${b1 * b2}y = $newC1 & (3) \\
+${newA1}x ${b1 * b2 >= 0 ? '+' : ''} ${b1 * b2}y = $newC1 & (3) \\\\
 ${newA2}x ${b1 * b2 >= 0 ? '+' : ''} ${b1 * b2}y = $newC2 & (4)
 \\end{cases}
-
+\$\$
 ''',
       ),
     );
@@ -360,14 +378,14 @@ ${newA2}x ${b1 * b2 >= 0 ? '+' : ''} ${b1 * b2}y = $newC2 & (4)
           explanation: '将 x = $x 代入原方程(2)中。',
           formula:
               '''
-
+\$\$
 \\begin{aligned}
 $a2($x) + ${b2}y &= $c2 \\\\
 ${a2 * x} + ${b2}y &= $c2 \\\\
 ${b2}y &= $c2 - ${a2 * x} \\\\
 ${b2}y &= ${c2 - a2 * x}
 \\end{aligned}
-
+\$\$
 ''',
         ),
       );
@@ -394,14 +412,14 @@ ${b2}y &= ${c2 - a2 * x}
           explanation: '将 x = $x 代入原方程(1)中。',
           formula:
               '''
-
+\$\$
 \\begin{aligned}
 $a1($x) + ${b1}y &= $c1 \\\\
 ${a1 * x} + ${b1}y &= $c1 \\\\
 ${b1}y &= $c1 - ${a1 * x} \\\\
 ${b1}y &= ${c1 - a1 * x}
 \\end{aligned}
-
+\$\$
 ''',
         ),
       );
@@ -422,6 +440,198 @@ ${b1}y &= ${c1 - a1 * x}
 
   /// ---- 辅助函数 ----
 
+  /// 获取精确三角函数结果
+  String? _getExactTrigResult(String input) {
+    final cleanInput = input.replaceAll(' ', '').toLowerCase();
+
+    // 匹配 sin(角度) 模式
+    final sinMatch = RegExp(r'^sin\((\d+(?:\+\d+)*)\)$').firstMatch(cleanInput);
+    if (sinMatch != null) {
+      final angleExpr = sinMatch.group(1)!;
+      final angle = _evaluateAngleExpression(angleExpr);
+      if (angle != null) {
+        return _getSinExactValue(angle);
+      }
+    }
+
+    // 匹配 cos(角度) 模式
+    final cosMatch = RegExp(r'^cos\((\d+(?:\+\d+)*)\)$').firstMatch(cleanInput);
+    if (cosMatch != null) {
+      final angleExpr = cosMatch.group(1)!;
+      final angle = _evaluateAngleExpression(angleExpr);
+      if (angle != null) {
+        return _getCosExactValue(angle);
+      }
+    }
+
+    // 匹配 tan(角度) 模式
+    final tanMatch = RegExp(r'^tan\((\d+(?:\+\d+)*)\)$').firstMatch(cleanInput);
+    if (tanMatch != null) {
+      final angleExpr = tanMatch.group(1)!;
+      final angle = _evaluateAngleExpression(angleExpr);
+      if (angle != null) {
+        return _getTanExactValue(angle);
+      }
+    }
+
+    return null;
+  }
+
+  /// 计算角度表达式（如 30+45 = 75）
+  int? _evaluateAngleExpression(String expr) {
+    final parts = expr.split('+');
+    int sum = 0;
+    for (final part in parts) {
+      final num = int.tryParse(part.trim());
+      if (num == null) return null;
+      sum += num;
+    }
+    return sum;
+  }
+
+  /// 获取 sin 的精确值
+  String? _getSinExactValue(int angle) {
+    // 标准化角度到 0-360 度
+    final normalizedAngle = angle % 360;
+
+    switch (normalizedAngle) {
+      case 0:
+      case 360:
+        return '0';
+      case 30:
+        return '\\frac{1}{2}';
+      case 45:
+        return '\\frac{\\sqrt{2}}{2}';
+      case 60:
+        return '\\frac{\\sqrt{3}}{2}';
+      case 75:
+        return '1 + \\frac{\\sqrt{2}}{2}';
+      case 90:
+        return '1';
+      case 120:
+        return '\\frac{\\sqrt{3}}{2}';
+      case 135:
+        return '\\frac{\\sqrt{2}}{2}';
+      case 150:
+        return '\\frac{1}{2}';
+      case 180:
+        return '0';
+      case 210:
+        return '-\\frac{1}{2}';
+      case 225:
+        return '-\\frac{\\sqrt{2}}{2}';
+      case 240:
+        return '-\\frac{\\sqrt{3}}{2}';
+      case 270:
+        return '-1';
+      case 300:
+        return '-\\frac{\\sqrt{3}}{2}';
+      case 315:
+        return '-\\frac{\\sqrt{2}}{2}';
+      case 330:
+        return '-\\frac{1}{2}';
+      default:
+        return null;
+    }
+  }
+
+  /// 获取 cos 的精确值
+  String? _getCosExactValue(int angle) {
+    // cos(angle) = sin(90 - angle)
+    final complementaryAngle = 90 - angle;
+    return _getSinExactValue(complementaryAngle.abs());
+  }
+
+  /// 获取 tan 的精确值
+  String? _getTanExactValue(int angle) {
+    // tan(angle) = sin(angle) / cos(angle)
+    final sinValue = _getSinExactValue(angle);
+    final cosValue = _getCosExactValue(angle);
+
+    if (sinValue != null && cosValue != null) {
+      if (cosValue == '0') return null; // 未定义
+      return '\\frac{$sinValue}{$cosValue}';
+    }
+
+    return null;
+  }
+
+  /// 将三角函数的参数从度转换为弧度
+  String _convertTrigToRadians(String input) {
+    String result = input;
+
+    // 正则表达式匹配三角函数调用，如 sin(30), cos(45), tan(60)
+    final trigPattern = RegExp(
+      r'(sin|cos|tan|asin|acos|atan)\s*\(\s*([^)]+)\s*\)',
+      caseSensitive: false,
+    );
+
+    result = result.replaceAllMapped(trigPattern, (match) {
+      final func = match.group(1)!;
+      final arg = match.group(2)!;
+
+      // 如果参数已经是弧度相关的表达式（包含 pi 或 π），则不转换
+      if (arg.contains('pi') || arg.contains('π') || arg.contains('rad')) {
+        return '$func($arg)';
+      }
+
+      // 将度数转换为弧度：度 * π / 180
+      return '$func(($arg)*($pi/180))';
+    });
+
+    return result;
+  }
+
+  /// 将数值结果格式化为几倍根号的形式
+  String _formatSqrtResult(double result) {
+    // 处理负数
+    if (result < 0) {
+      return '-${_formatSqrtResult(-result)}';
+    }
+
+    // 处理零
+    if (result == 0) return '0';
+
+    // 检查是否接近整数
+    final rounded = result.round();
+    if ((result - rounded).abs() < 1e-10) {
+      return rounded.toString();
+    }
+
+    // 计算 result 的平方，看它是否接近整数
+    final squared = result * result;
+    final squaredRounded = squared.round();
+
+    // 如果 squared 接近整数，说明 result 是某个数的平方根
+    if ((squared - squaredRounded).abs() < 1e-6) {
+      // 寻找最大的完全平方数因子
+      int maxSquareFactor = 1;
+      for (int i = 2; i * i <= squaredRounded; i++) {
+        if (squaredRounded % (i * i) == 0) {
+          maxSquareFactor = i * i;
+        }
+      }
+
+      final coefficient = sqrt(maxSquareFactor).round();
+      final remaining = squaredRounded ~/ maxSquareFactor;
+
+      if (remaining == 1) {
+        // 完全平方数，直接返回系数
+        return coefficient.toString();
+      } else if (coefficient == 1) {
+        return '\\sqrt{$remaining}';
+      } else {
+        return '$coefficient\\sqrt{$remaining}';
+      }
+    }
+
+    // 如果不是平方根的结果，返回原始数值（保留几位小数）
+    return result
+        .toStringAsFixed(6)
+        .replaceAll(RegExp(r'\.0+$'), '')
+        .replaceAll(RegExp(r'\.$'), '');
+  }
+
   String _expandExpressions(String input) {
     String result = input;
     int maxIterations = 10; // Prevent infinite loops
@@ -544,29 +754,68 @@ ${b1}y &= ${c1 - a1 * x}
 
   Map<int, double> _parsePolynomial(String side) {
     final coeffs = <int, double>{};
+    // 扩展模式以支持 sqrt 函数
     final pattern = RegExp(
-      r'([+-]?(?:\d*\.?\d*)?)x(?:\^(\d+))?|([+-]?\d*\.?\d+)',
+      r'([+-]?(?:\d*\.?\d*|sqrt\(\d+\)))x(?:\^(\d+))?|([+-]?(?:\d*\.?\d*|sqrt\(\d+\)))',
     );
     var s = side.startsWith('+') || side.startsWith('-') ? side : '+$side';
 
     for (final match in pattern.allMatches(s)) {
       if (match.group(3) != null) {
-        coeffs[0] = (coeffs[0] ?? 0) + double.parse(match.group(3)!);
+        // 常数项
+        final constStr = match.group(3)!;
+        final constValue = _parseCoefficientWithSqrt(constStr);
+        coeffs[0] = (coeffs[0] ?? 0) + constValue;
       } else {
+        // x 的幂次项
         int power = match.group(2) != null ? int.parse(match.group(2)!) : 1;
         String coeffStr = match.group(1) ?? '+';
-        double coeff = 1.0;
-        if (coeffStr.isNotEmpty && coeffStr != '+') {
-          coeff = coeffStr == '-' ? -1.0 : double.parse(coeffStr);
-        } else if (coeffStr == '-') {
-          coeff = -1.0;
-        }
+        final coeff = _parseCoefficientWithSqrt(coeffStr);
         coeffs[power] = (coeffs[power] ?? 0) + coeff;
       }
     }
     return coeffs;
   }
 
+  /// 解析包含 sqrt 函数的系数
+  double _parseCoefficientWithSqrt(String coeffStr) {
+    if (coeffStr.isEmpty || coeffStr == '+') return 1.0;
+    if (coeffStr == '-') return -1.0;
+
+    // 检查是否包含 sqrt 函数
+    final sqrtMatch = RegExp(r'sqrt\((\d+)\)').firstMatch(coeffStr);
+    if (sqrtMatch != null) {
+      final innerValue = int.parse(sqrtMatch.group(1)!);
+
+      // 对于完全平方数，直接返回整数结果
+      final sqrtValue = sqrt(innerValue.toDouble());
+      final rounded = sqrtValue.round();
+      if ((sqrtValue - rounded).abs() < 1e-10) {
+        // 检查是否有系数
+        final coeffPart = coeffStr.replaceFirst(sqrtMatch.group(0)!, '');
+        if (coeffPart.isEmpty) return rounded.toDouble();
+        if (coeffPart == '-') return -rounded.toDouble();
+
+        final coeff = double.parse(coeffPart);
+        return coeff * rounded;
+      }
+
+      // 对于非完全平方数，计算数值但保持高精度
+      final nonPerfectSqrtValue = sqrt(innerValue.toDouble());
+
+      // 检查是否有系数
+      final coeffPart = coeffStr.replaceFirst(sqrtMatch.group(0)!, '');
+      if (coeffPart.isEmpty) return nonPerfectSqrtValue;
+      if (coeffPart == '-') return -nonPerfectSqrtValue;
+
+      final coeff = double.parse(coeffPart);
+      return coeff * nonPerfectSqrtValue;
+    }
+
+    // 普通数值
+    return double.parse(coeffStr);
+  }
+
   List<double> _parseTwoVariableLinear(String equation) {
     final parts = equation.split('=');
     if (parts.length != 2) throw Exception("方程 $equation 格式错误");