在不使用外部库的情况下使用 C++ 中的伯努利级数展开计算角度的 Cosine/Sine/Tangent

Question

我似乎找不到错误。我需要一双新鲜的眼睛。我试图在不使用内置函数或库的情况下计算 C++ 中角度的 Cos/Sin/Tan 的值。这些是唯一的要求。

这是我目前得到的：我写了一个函数来计算指数值、阶乘和所需的值。

我不断收到错误，但我不知道它们从何而来。有些数字给了我完全正确的答案，而另一些则相差甚远。那么你能告诉我我做错了什么吗？

#include <iostream>
#include <cmath>
using namespace std;

double bernoulli_numbers[] = { 1, -1/2.0, 1/6.0, -1/30.0, 5/66.0, -691/2730.0, 7/6.0, -3617/510.0, 43867/798.0, -174611/330.0, 854513/138.0};

int angleToRadian(int angle) {
    float rad_angle;
    rad_angle = angle * (M_PI / 180);
    return rad_angle;
}

int calculatingExponents(int num, int power) {
    long double result = 1;
    for (int i = 0; i < power ; ++i) {
        result *= num;
    }
    return result;
}

unsigned long long calculatingFactorials(int n) {
    unsigned long long factorial = 1;
    if ( n < 0 ) {
        cout << "Can't compute factorials for negative numbers" << endl;
    }
    else if ( n < 2 ) {
        return 1;
    }
    else {
        for (int i = n; (i >= 2) ; i--) {
            factorial = factorial * i;
        }
    }
    return factorial;
}

void calculatingCos(double angle) {
    long double numerator;
    long double last_term;
    unsigned long long denominator;
    long double result;
    long double final_result = 0;
    for (int i = 0; i < 15; ++i) {
        numerator   = calculatingExponents(-1, i);
        denominator = calculatingFactorials(2 * i);
        last_term   = calculatingExponents(angle, (2 * i));
        result =  (numerator / denominator) * last_term;
        final_result += result;
    }
    cout << "The Cosine of the angle = " << final_result << endl;
}

void calculatingSin(double angle) {
    long double numerator;
    long double last_term;
    unsigned long long denominator;
    long double result;
    long double final_result = 0;
    for (int i = 0; i < 15; ++i) {
        numerator   = calculatingExponents(-1, i);
        denominator = calculatingFactorials((2 * i) + 1);
        last_term   = calculatingExponents(angle, ((2 * i) + 1));
        result =  (numerator / denominator) * last_term;
        final_result += result;
    }
    cout << "The Sine of the angle = " << final_result << endl;
}

void calculatingTan(double angle) {
    int bernoulli_index;
    long double bernoulli_number;
    long double numerator;
    long double last_term;
    unsigned long long denominator;
    long double result;
    long double final_result = 0;
    for (int i = 0; i < 15; ++i) {
        bernoulli_index = (2 * i) + 2;
        bernoulli_number = bernoulli_numbers[bernoulli_index];
        numerator   =
        calculatingExponents(-1, i)
        *
        calculatingExponents(2, (2 * i) + 2)
        *
        ( calculatingExponents(2, (((2 * i) + 2) * 1)) - 1) * bernoulli_number;
        denominator = calculatingFactorials((2 * i) + 2);
        last_term   = calculatingExponents(angle, (2 * i) + 1);
        result =  (numerator / denominator) * last_term;
        final_result += result;
    }
    cout << "The Tan of the angle = " << final_result << endl;
}

int main() {
    int degree_angle;
    int x;
    cout << "Please input an angle in degrees:" << endl;
    cin >> degree_angle;
    x = angleToRadian(degree_angle);
    
    calculatingCos(x);
    calculatingSin(x);
    calculatingTan(x);
}

我遇到的一些错误是：

Input: 180
Output: 
The Cosine of the angle = -0.989992
The Sine of the angle = 0.14112
The Tan of the angle = 3.49908e+07

Expected Outputs:
The Cosine of the angle = -1
The Sine of the angle = 0
The Tan of the angle = 0

Input: 60
Output:
The Cosine of the angle = 0.540302
The Sine of the angle = 0.841471
The Tan of the angle = 1705.3

Expected Outputs:
The Cosine of the angle = 0.5
The Sine of the angle = 0.86602540378
The Tan of the angle = 1.73205080757

Answer 1

很多参数和变量 int 只有作为双精度值才有意义。

#include <iostream>
#include <cmath>
using namespace std;

double bernoulli_numbers[] = { 1, -1/2, 1/6, -1/30, 5/66, -691/2730, 7/6, -3617/510, 43867/798, -174611/330, 854513/138};

double angleToRadian(int angle) {
    double rad_angle = angle * (M_PI / 180);
    return rad_angle;
}

angleToRadian 之前 return 编辑了 int，圆中有 pi 弧度，因此 360 度以下的角度将捕捉到 0、1、2 或 3 弧度。您需要 return 一位小数以获得正确的转换。

long double calculatingExponents(double num, int power) {
    long double result = 1;
    for (int i = 0; i < power ; ++i) {
        result *= num;
    }
    return result;
}

calculatingExponents中的num参数取小数角作为angleToRadians的结果，所以num、result和return calculatingExponents 的类型也必须是浮点数。

unsigned long long calculatingFactorials(int n) {
    unsigned long long factorial = 1;
    if ( n < 0 ) {
        cout << "Can't compute factorials for negative numbers" << endl;
    }
    else if ( n < 2 ) {
        return 1;
    }
    else {
        for (int i = n; (i >= 2) ; i--) {
            factorial = factorial * i;
        }
    }
    return factorial;
}

如果您使用的是 unsigned long long 但 return 使用的是 int、you're relying on implementation defined behaviour，因此无法保证合理的答案。

int main() {
    int degree_angle = 45;
    cout << "Angle in degrees: " << degree_angle << endl;
    double x = angleToRadian(degree_angle);
    
    calculatingCos(x);
    calculatingSin(x);
    calculatingTan(x);
}

同样 angleToRadian return 是一个浮点数，所以 x 需要是一个浮点类型。

这些修改似乎修正了 -90 到 90 度之间的正弦和余弦。

我还不确定切线计算中发生了什么错误。

在不使用外部库的情况下使用 C++ 中的伯努利级数展开计算角度的 Cosine/Sine/Tangent

Calculating the Cosine/Sine/Tangent of an angle using Bernoulli Series Expansion in C++ without using outside libraries

c++

bernoulli-numbers