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受欢迎的三角函数 >

sec^2(x)cos(x)=cos(x),0<x<360

解答

sec^{2} (x) cos (x) = cos (x), 0^{\circ} < x < 36 0^{\circ}

解答

x = 18 0^{\circ}

+1

弧度

x = π

求解步骤

sec^{2} (x) cos (x) = cos (x), 0^{\circ} < x < 36 0^{\circ}

两边减去

cos (x)

sec^{2} (x) cos (x) - cos (x) = 0

分解

sec^{2} (x) cos (x) - cos (x) : cos (x) (sec (x) + 1) (sec (x) - 1)

sec^{2} (x) cos (x) - cos (x)

因式分解出通项

cos (x)

= cos (x) (sec^{2} (x) - 1)

分解

sec^{2} (x) - 1 : (sec (x) + 1) (sec (x) - 1)

sec^{2} (x) - 1

将

1

改写为

1^{2}

= sec^{2} (x) - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

sec^{2} (x) - 1^{2} = (sec (x) + 1) (sec (x) - 1)

= (sec (x) + 1) (sec (x) - 1)

= cos (x) (sec (x) + 1) (sec (x) - 1)

cos (x) (sec (x) + 1) (sec (x) - 1) = 0

分别求解每个部分

cos (x) = 0 or sec (x) + 1 = 0 or sec (x) - 1 = 0

cos (x) = 0, 0 < x < 36 0^{\circ} : x = 9 0^{\circ}, x = 27 0^{\circ}

cos (x) = 0, 0 < x < 36 0^{\circ}

cos (x) = 0

的通解

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

在

0 < x < 36 0^{\circ}

范围内的解

x = 9 0^{\circ}, x = 27 0^{\circ}

sec (x) + 1 = 0, 0 < x < 36 0^{\circ} : x = 18 0^{\circ}

sec (x) + 1 = 0, 0 < x < 36 0^{\circ}

将

1

到右边

sec (x) + 1 = 0

两边减去

1

sec (x) + 1 - 1 = 0 - 1

化简

sec (x) = - 1

sec (x) = - 1

sec (x) = - 1

的通解

sec (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 18 0^{\circ} + 36 0^{\circ} n

x = 18 0^{\circ} + 36 0^{\circ} n

在

0 < x < 36 0^{\circ}

范围内的解

x = 18 0^{\circ}

sec (x) - 1 = 0, 0 < x < 36 0^{\circ} :

无解

sec (x) - 1 = 0, 0 < x < 36 0^{\circ}

将

1

到右边

sec (x) - 1 = 0

两边加上

1

sec (x) - 1 + 1 = 0 + 1

化简

sec (x) = 1

sec (x) = 1

sec (x) = 1

的通解

sec (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 36 0^{\circ} n

x = 0 + 36 0^{\circ} n

解

x = 0 + 36 0^{\circ} n : x = 36 0^{\circ} n

x = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

x = 36 0^{\circ} n

x = 36 0^{\circ} n

在

0 < x < 36 0^{\circ}

范围内的解

无解

合并所有解

x = 9 0^{\circ}, x = 27 0^{\circ}, x = 18 0^{\circ}

因为方程对以下值无定义:

9 0^{\circ}, 27 0^{\circ}

x = 18 0^{\circ}

作图

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