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受欢迎的三角函数 >

2csc^2(x)=3cot^2(x)-1

解答

2 csc^{2} (x) = 3 cot^{2} (x) - 1

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

2 csc^{2} (x) = 3 cot^{2} (x) - 1

两边减去

3 cot^{2} (x) - 1

2 csc^{2} (x) - 3 cot^{2} (x) + 1 = 0

使用三角恒等式改写

1 + 2 csc^{2} (x) - 3 cot^{2} (x)

使用毕达哥拉斯恒等式:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

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

化简

1 + 2 csc^{2} (x) - 3 (csc^{2} (x) - 1) : - csc^{2} (x) + 4

1 + 2 csc^{2} (x) - 3 (csc^{2} (x) - 1)

乘开

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

- 3 (csc^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = - 3, b = csc^{2} (x), c = 1

= - 3 csc^{2} (x) - (- 3) \cdot 1

使用加减运算法则

- (- a) = a

= - 3 csc^{2} (x) + 3 \cdot 1

数字相乘:

3 \cdot 1 = 3

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

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

化简

1 + 2 csc^{2} (x) - 3 csc^{2} (x) + 3 : - csc^{2} (x) + 4

1 + 2 csc^{2} (x) - 3 csc^{2} (x) + 3

同类项相加:

2 csc^{2} (x) - 3 csc^{2} (x) = - csc^{2} (x)

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

对同类项分组

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

数字相加:

1 + 3 = 4

= - csc^{2} (x) + 4

= - csc^{2} (x) + 4

= - csc^{2} (x) + 4

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

用替代法求解

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

令

: csc (x) = u

4 - u^{2} = 0

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

4 - u^{2} = 0

将

4

到右边

4 - u^{2} = 0

两边减去

4

4 - u^{2} - 4 = 0 - 4

化简

- u^{2} = - 4

- u^{2} = - 4

两边除以

- 1

- u^{2} = - 4

两边除以

- 1

\frac{- u ^{2}}{- 1} = \frac{- 4}{- 1}

化简

u^{2} = 4

u^{2} = 4

对于

x^{2} = f (a)

解为

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

u = 4, u = - 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= - 2

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}{6} + 2 πn, x = \frac{5 π}{6} + 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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

作图

流行的例子

cot^{2} (x) = 3

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

3 cos (x) = sin (x)

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

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

sin(x)tan(x)=2tan(x)

sin (x) tan (x) = 2 tan (x)

tan^2(x)-2tan(x)=0

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