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受欢迎的三角函数 >

sec(x)+sin^2(x)+cos^2(x)=tan^2(x)

解答

sec (x) + sin^{2} (x) + cos^{2} (x) = tan^{2} (x)

解答

x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度数

x = 18 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

sec (x) + sin^{2} (x) + cos^{2} (x) = tan^{2} (x)

两边减去

tan^{2} (x)

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

用 sin, cos 表示

cos^{2} (x) + sec (x) + sin^{2} (x) - tan^{2} (x)

使用基本三角恒等式:

sec (x) = \frac{1}{cos ( x )}

= cos^{2} (x) + \frac{1}{cos ( x )} + sin^{2} (x) - tan^{2} (x)

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= cos^{2} (x) + \frac{1}{cos ( x )} + sin^{2} (x) - (\frac{sin ( x )}{cos ( x )})^{2}

化简

cos^{2} (x) + \frac{1}{cos ( x )} + sin^{2} (x) - (\frac{sin ( x )}{cos ( x )})^{2} : \frac{cos ^{4} ( x ) + cos ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

cos^{2} (x) + \frac{1}{cos ( x )} + sin^{2} (x) - (\frac{sin ( x )}{cos ( x )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= cos^{2} (x) + \frac{1}{cos ( x )} + sin^{2} (x) - \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

将项转换为分式:

cos^{2} (x) = \frac{cos ^{2} ( x )}{1}, sin^{2} (x) = \frac{sin ^{2} ( x )}{1}

= \frac{cos ^{2} ( x )}{1} + \frac{1}{cos ( x )} + \frac{sin ^{2} ( x )}{1} - \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

1, cos (x), 1, cos^{2} (x)

的最小公倍数:

cos^{2} (x)

1, cos (x), 1, cos^{2} (x)

最小公倍数 (LCM)

1, 1

的最小公倍数:

1

1, 1

最小公倍数 (LCM)

1

质因数分解

1

质因数分解

将每个因子乘以它在

1

或

1

中出现的最多次数

= 1

数字相乘:

1 = 1

= 1

计算出由至少在以下一个因式表达式中出现的因子组成的表达式

= cos^{2} (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

cos^{2} (x)

对于

\frac{cos ^{2} ( x )}{1} :

将分母和分子乘以

cos^{2} (x)

\frac{cos ^{2} ( x )}{1} = \frac{cos ^{2} ( x ) cos ^{2} ( x )}{1 \cdot cos ^{2} ( x )} = \frac{cos ^{4} ( x )}{cos ^{2} ( x )}

对于

\frac{1}{cos ( x )} :

将分母和分子乘以

cos (x)

\frac{1}{cos ( x )} = \frac{1 \cdot cos ( x )}{cos ( x ) cos ( x )} = \frac{cos ( x )}{cos ^{2} ( x )}

对于

\frac{sin ^{2} ( x )}{1} :

将分母和分子乘以

cos^{2} (x)

\frac{sin ^{2} ( x )}{1} = \frac{sin ^{2} ( x ) cos ^{2} ( x )}{1 \cdot cos ^{2} ( x )} = \frac{sin ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{cos ^{4} ( x )}{cos ^{2} ( x )} + \frac{cos ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} - \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{cos ^{4} ( x ) + cos ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{cos ^{4} ( x ) + cos ( x ) + sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

\frac{cos ( x ) + cos ^{4} ( x ) - sin ^{2} ( x ) + cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos (x) + cos^{4} (x) - sin^{2} (x) + cos^{2} (x) sin^{2} (x) = 0

分解

cos (x) + cos^{4} (x) - sin^{2} (x) + cos^{2} (x) sin^{2} (x) : (1 + cos (x)) (cos (x) (cos^{2} (x) + 1 - cos (x)) + sin^{2} (x) (- 1 + cos (x)))

cos (x) + cos^{4} (x) - sin^{2} (x) + cos^{2} (x) sin^{2} (x)

分解

cos (x) + cos^{4} (x) : cos (x) (cos (x) + 1) (cos^{2} (x) - cos (x) + 1)

cos (x) + cos^{4} (x)

使用指数法则:

a^{b + c} = a^{b} a^{c}

cos^{4} (x) = cos (x) cos^{3} (x)

= cos (x) + cos (x) cos^{3} (x)

因式分解出通项

cos (x)

= cos (x) (1 + cos^{3} (x))

分解

cos^{3} (x) + 1 : (cos (x) + 1) (cos^{2} (x) - cos (x) + 1)

1 + cos^{3} (x)

将

1

改写为

1^{3}

= cos^{3} (x) + 1^{3}

使用立方和公式:

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

cos^{3} (x) + 1^{3} = (cos (x) + 1) (cos^{2} (x) - cos (x) + 1)

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

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

分解

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

- sin^{2} (x) + cos^{2} (x) sin^{2} (x)

因式分解出通项

sin^{2} (x)

= sin^{2} (x) (- 1 + cos^{2} (x))

分解

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

- 1 + cos^{2} (x)

将

1

改写为

1^{2}

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

使用平方差公式:

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

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

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

= sin^{2} (x) (cos (x) + 1) (cos (x) - 1)

= cos (x) (cos (x) + 1) (cos^{2} (x) - cos (x) + 1) + sin^{2} (x) (cos (x) + 1) (cos (x) - 1)

因式分解出通项

(1 + cos (x))

= (1 + cos (x)) (cos (x) (cos^{2} (x) + 1 - cos (x)) + sin^{2} (x) (- 1 + cos (x)))

(1 + cos (x)) (cos (x) (cos^{2} (x) + 1 - cos (x)) + sin^{2} (x) (- 1 + cos (x))) = 0

分别求解每个部分

1 + cos (x) = 0 or cos (x) (cos^{2} (x) + 1 - cos (x)) + sin^{2} (x) (- 1 + cos (x)) = 0

1 + cos (x) = 0 : x = π + 2 πn

1 + cos (x) = 0

将

1

到右边

1 + cos (x) = 0

两边减去

1

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

化简

cos (x) = - 1

cos (x) = - 1

cos (x) = - 1

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

cos (x) (cos^{2} (x) + 1 - cos (x)) + sin^{2} (x) (- 1 + cos (x)) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) (cos^{2} (x) + 1 - cos (x)) + sin^{2} (x) (- 1 + cos (x)) = 0

使用三角恒等式改写

(- 1 + cos (x)) sin^{2} (x) + (1 - cos (x) + cos^{2} (x)) cos (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

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

= (- 1 + cos (x)) sin^{2} (x) + (1 - cos (x) + 1 - sin^{2} (x)) cos (x)

化简

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

(- 1 + cos (x)) sin^{2} (x) + (1 - cos (x) + 1 - sin^{2} (x)) cos (x)

化简

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

1 - cos (x) + 1 - sin^{2} (x)

对同类项分组

= - cos (x) - sin^{2} (x) + 1 + 1

数字相加:

1 + 1 = 2

= - sin^{2} (x) - cos (x) + 2

= sin^{2} (x) (cos (x) - 1) + cos (x) (- sin^{2} (x) - cos (x) + 2)

= sin^{2} (x) (- 1 + cos (x)) + cos (x) (- sin^{2} (x) - cos (x) + 2)

乘开

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

sin^{2} (x) (- 1 + cos (x))

使用分配律:

a (b + c) = ab + a c

a = sin^{2} (x), b = - 1, c = cos (x)

= sin^{2} (x) (- 1) + sin^{2} (x) cos (x)

使用加减运算法则

+ (- a) = - a

= - 1 \cdot sin^{2} (x) + sin^{2} (x) cos (x)

乘以:

1 \cdot sin^{2} (x) = sin^{2} (x)

= - sin^{2} (x) + sin^{2} (x) cos (x)

= - sin^{2} (x) + sin^{2} (x) cos (x) + (- sin^{2} (x) - cos (x) + 2) cos (x)

乘开

cos (x) (- sin^{2} (x) - cos (x) + 2) : - sin^{2} (x) cos (x) - cos^{2} (x) + 2 cos (x)

cos (x) (- sin^{2} (x) - cos (x) + 2)

打开括号

= cos (x) (- sin^{2} (x)) + cos (x) (- cos (x)) + cos (x) \cdot 2

使用加减运算法则

+ (- a) = - a

= - sin^{2} (x) cos (x) - cos (x) cos (x) + 2 cos (x)

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

= - sin^{2} (x) cos (x) - cos^{2} (x) + 2 cos (x)

= - sin^{2} (x) + sin^{2} (x) cos (x) - sin^{2} (x) cos (x) - cos^{2} (x) + 2 cos (x)

同类项相加:

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

= - sin^{2} (x) - cos^{2} (x) + 2 cos (x)

= - sin^{2} (x) - cos^{2} (x) + 2 cos (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

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

= 2 cos (x) - 1

2 cos (x) - 1 = 0

将

1

到右边

2 cos (x) - 1 = 0

两边加上

1

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

化简

2 cos (x) = 1

2 cos (x) = 1

两边除以

2

2 cos (x) = 1

两边除以

2

\frac{2 cos ( x )}{2} = \frac{1}{2}

化简

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

合并所有解

x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

作图

流行的例子

4sin^2(x)-3=4,sin(2x)-3=0

4 sin^{2} (x) - 3 = 4, sin (2 x) - 3 = 0

sin(2x)=5sin(x)

sin (2 x) = 5 sin (x)

tan(x)=-sqrt(3)+2

tan (x) = - 3 + 2

3tan(x)-cot(x)=0

3 tan (x) - cot (x) = 0

cos (2 x) - 0.25 = 0