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受欢迎的三角函数 >

sec^2(x)= 1/2 csc^2(x)+tan^2(x)

解答

sec^{2} (x) = \frac{1}{2} csc^{2} (x) + tan^{2} (x)

解答

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度数

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

sec^{2} (x) = \frac{1}{2} csc^{2} (x) + tan^{2} (x)

两边减去

\frac{1}{2} csc^{2} (x) + tan^{2} (x)

sec^{2} (x) - \frac{1}{2} csc^{2} (x) - tan^{2} (x) = 0

化简

sec^{2} (x) - \frac{1}{2} csc^{2} (x) - tan^{2} (x) : \frac{2 sec ^{2} ( x ) - csc ^{2} ( x ) - 2 tan ^{2} ( x )}{2}

sec^{2} (x) - \frac{1}{2} csc^{2} (x) - tan^{2} (x)

\frac{1}{2} csc^{2} (x) = \frac{csc ^{2} ( x )}{2}

\frac{1}{2} csc^{2} (x)

分式相乘:

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

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

乘以:

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

= \frac{csc ^{2} ( x )}{2}

= sec^{2} (x) - \frac{csc ^{2} ( x )}{2} - tan^{2} (x)

将项转换为分式:

sec^{2} (x) = \frac{sec ^{2} ( x ) 2}{2}, tan^{2} (x) = \frac{tan ^{2} ( x ) 2}{2}

= \frac{sec ^{2} ( x ) \cdot 2}{2} - \frac{csc ^{2} ( x )}{2} - \frac{tan ^{2} ( x ) \cdot 2}{2}

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{sec ^{2} ( x ) \cdot 2 - csc ^{2} ( x ) - tan ^{2} ( x ) \cdot 2}{2}

\frac{2 sec ^{2} ( x ) - csc ^{2} ( x ) - 2 tan ^{2} ( x )}{2} = 0

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

2 sec^{2} (x) - csc^{2} (x) - 2 tan^{2} (x) = 0

使用三角恒等式改写

- csc^{2} (x) + 2 sec^{2} (x) - 2 tan^{2} (x)

使用毕达哥拉斯恒等式:

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

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

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

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

使用加减运算法则

- (- a) = a

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

数字相乘:

2 \cdot 1 = 2

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

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

同类项相加:

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

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

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

2 - csc^{2} (x) = 0

用替代法求解

2 - csc^{2} (x) = 0

令

: csc (x) = u

2 - u^{2} = 0

2 - u^{2} = 0 : u = 2, u = - 2

2 - u^{2} = 0

将

2

到右边

2 - u^{2} = 0

两边减去

2

2 - u^{2} - 2 = 0 - 2

化简

- u^{2} = - 2

- u^{2} = - 2

两边除以

- 1

- u^{2} = - 2

两边除以

- 1

\frac{- u ^{2}}{- 1} = \frac{- 2}{- 1}

化简

u^{2} = 2

u^{2} = 2

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = 2, u = - 2

u = csc (x)

代回

csc (x) = 2, csc (x) = - 2

csc (x) = 2, csc (x) = - 2

csc (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) = 2

csc (x) = 2

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

csc (x) = - 2 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

csc (x) = - 2

csc (x) = - 2

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

合并所有解

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

作图

流行的例子

4cos(2θ)-3cos(θ)+4=-cos(θ)+3

4 cos (2 θ) - 3 cos (θ) + 4 = - cos (θ) + 3

cos (2 x) = 0.28

cos(3x)=sqrt(3)sin(3x)

cos (3 x) = 3 sin (3 x)

- 3 cos (2 x) = 0

sec(x)=sqrt(1-tan^2(x))

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