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受欢迎的三角函数 >

(sqrt(2)(cos(pi/2)+isin(pi/2)))/(1/2 (cos(pi/4)+isin(pi/4)))

解答

\frac{2 ( cos ( \frac{π}{2} ) + i sin ( \frac{π}{2} ) )}{\frac{1}{2} ( cos ( \frac{π}{4} ) + i sin ( \frac{π}{4} ) )}

解答

2 + 2 i

求解步骤

\frac{2 ( cos ( \frac{π}{2} ) + i sin ( \frac{π}{2} ) )}{\frac{1}{2} ( cos ( \frac{π}{4} ) + i sin ( \frac{π}{4} ) )}

使用以下普通恒等式:

cos (\frac{π}{2}) = 0

cos (\frac{π}{2})

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}

= 0

使用以下普通恒等式:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

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}

= \frac{2}{2}

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{2}{2}

sin (\frac{π}{4})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{2 ( 0 + i 1 )}{\frac{1}{2} ( \frac{2}{2} + i \frac{2}{2} )}

化简

\frac{2 ( 0 + i 1 )}{\frac{1}{2} ( \frac{2}{2} + i \frac{2}{2} )} : 2 + 2 i

\frac{2 ( 0 + i 1 )}{\frac{1}{2} ( \frac{2}{2} + i \frac{2}{2} )}

2 (0 + i 1) = 2 i

2 (0 + i 1)

乘以:

i 1 = i

= 2 (0 + i)

0 + i = i

= 2 i

= \frac{2 i}{\frac{1}{2} ( i \frac{2}{2} + \frac{2}{2} )}

乘

\frac{1}{2} (\frac{2}{2} + i \frac{2}{2}) : \frac{1 + i}{2 2}

\frac{1}{2} (\frac{2}{2} + i \frac{2}{2})

分式相乘:

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

= \frac{1 \cdot ( \frac{2}{2} + i \frac{2}{2} )}{2}

1 \cdot (\frac{2}{2} + i \frac{2}{2}) = \frac{2}{2} + i \frac{2}{2}

1 \cdot (\frac{2}{2} + i \frac{2}{2})

乘以:

1 \cdot (\frac{2}{2} + i \frac{2}{2}) = (\frac{2}{2} + i \frac{2}{2})

= (\frac{2}{2} + i \frac{2}{2})

去除括号:

(a) = a

= \frac{2}{2} + i \frac{2}{2}

= \frac{\frac{2}{2} + i \frac{2}{2}}{2}

\frac{2}{2} = \frac{1}{2}

\frac{2}{2}

消掉

\frac{2}{2} : \frac{1}{2}

\frac{2}{2}

使用根式运算法则:

n a = a^{\frac{1}{n}}

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

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

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

数字相减:

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

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

使用根式运算法则:

a^{\frac{1}{n}} = n a

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

= \frac{1}{2}

= \frac{1}{2}

i \frac{2}{2} = \frac{i}{2}

i \frac{2}{2}

\frac{2}{2} = \frac{1}{2}

\frac{2}{2}

使用根式运算法则:

n a = a^{\frac{1}{n}}

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

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

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

数字相减:

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

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

使用根式运算法则:

a^{\frac{1}{n}} = n a

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

= \frac{1}{2}

= i \frac{1}{2}

分式相乘:

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

= \frac{1 i}{2}

乘以:

1 i = i

= \frac{i}{2}

= \frac{\frac{1}{2} + \frac{i}{2}}{2}

合并分式

\frac{1}{2} + \frac{i}{2} : \frac{1 + i}{2}

使用法则

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

= \frac{1 + i}{2}

= \frac{\frac{1 + i}{2}}{2}

使用分式法则:

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

= \frac{1 + i}{2 \cdot 2}

= \frac{2 i}{\frac{1 + i}{2 2}}

使用分式法则:

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

= \frac{2 i 2 \cdot 2}{1 + i}

2 i 2 \cdot 2 = 4 i

2 i 2 \cdot 2

使用根式运算法则:

a a = a

22 = 2

= 2 \cdot 2 i

数字相乘:

2 \cdot 2 = 4

= 4 i

= \frac{4 i}{1 + i}

\frac{4 i}{1 + i}

有理化:

2 (1 + i)

\frac{4 i}{1 + i}

乘以共轭根式

\frac{1 - i}{1 - i}

= \frac{4 i ( 1 - i )}{( 1 + i ) ( 1 - i )}

化简

4 i (1 - i) : 4 + 4 i

4 i (1 - i)

使用复数算数法则:

(ai) (b + c i) = - a c + abi

a = 4, b = 1, c = - 1

= - 4 (- 1) + 4 \cdot 1 i

- 4 (- 1) = 4

- 4 (- 1)

使用法则

- (- a) = a

= 4 \cdot 1

数字相乘:

4 \cdot 1 = 4

= 4

4 \cdot 1 = 4

4 \cdot 1

数字相乘:

4 \cdot 1 = 4

= 4

= 4 + 4 i

(1 + i) (1 - i) = 2

(1 + i) (1 - i)

使用复数算数法则:

(a + bi) (a - bi) = a^{2} + b^{2}

a = 1, b = 1

= 1^{2} + 1^{2}

1^{2} + 1^{2} = 2

1^{2} + 1^{2}

使用法则

1^{a} = 1

1^{2} = 1

= 1 + 1

数字相加:

1 + 1 = 2

= 2

= 2

= \frac{4 + 4 i}{2}

分解

4 + 4 i : 4 (1 + i)

4 + 4 i

改写为

= 4 \cdot 1 + 4 i

因式分解出通项

4

= 4 (1 + i)

= \frac{4 ( 1 + i )}{2}

数字相除:

\frac{4}{2} = 2

= 2 (1 + i)

= 2 (1 + i)

将

2 (1 + i)

改写成标准复数形式:

2 + 2 i

2 (1 + i)

使用分配律:

a (b + c) = ab + a c

a = 2, b = 1, c = i

= 2 \cdot 1 + 2 i

数字相乘:

2 \cdot 1 = 2

= 2 + 2 i

= 2 + 2 i

= 2 + 2 i

流行的例子

cos^{2} (\frac{π}{20})

- cos (15 0^{\circ})

cos (5. 4^{\circ})

2 \cdot sin (1 0^{\circ})

(sec(32)+csc(58))/(3csc(58)+sec(32))

\frac{sec ( 3 2 ^{\circ} ) + csc ( 5 8 ^{\circ} )}{3 csc ( 5 8 ^{\circ} ) + sec ( 3 2 ^{\circ} )}