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受欢迎的三角函数 >

sqrt(2)csc(-765)+2csc(742.5)

解答

2 csc (- 76 5^{\circ}) + 2 csc (742. 5^{\circ})

解答

- 2 + 2 (2 2 - 2 + 2 2 - 2)

+1

十进制

3.22625 \dots

求解步骤

2 csc (- 76 5^{\circ}) + 2 csc (742. 5^{\circ})

利用以下特性:

csc (- x) = - csc (x)

csc (- 76 5^{\circ}) = - csc (76 5^{\circ})

= 2 (- csc (76 5^{\circ})) + 2 csc (742. 5^{\circ})

csc (76 5^{\circ}) = csc (4 5^{\circ})

csc (76 5^{\circ})

将

76 5^{\circ}

改写为

36 0^{\circ} \cdot 2 + 4 5^{\circ}

= csc (36 0^{\circ} 2 + 4 5^{\circ})

使用周期

csc

:

csc (x + 36 0^{\circ} \cdot k) = csc (x)

csc (36 0^{\circ} \cdot 2 + 4 5^{\circ}) = csc (4 5^{\circ})

= csc (4 5^{\circ})

= 2 (- csc (4 5^{\circ})) + 2 csc (742. 5^{\circ})

csc (742. 5^{\circ}) = csc (22. 5^{\circ})

csc (742. 5^{\circ})

将

742. 5^{\circ}

改写为

36 0^{\circ} \cdot 2 + 22. 5^{\circ}

= csc (36 0^{\circ} 2 + 22. 5^{\circ})

使用周期

csc

:

csc (x + 36 0^{\circ} \cdot k) = csc (x)

csc (36 0^{\circ} \cdot 2 + 22. 5^{\circ}) = csc (22. 5^{\circ})

= csc (22. 5^{\circ})

= 2 (- csc (4 5^{\circ})) + 2 csc (22. 5^{\circ})

化简

= - 2 csc (4 5^{\circ}) + 2 csc (22. 5^{\circ})

使用以下普通恒等式:

csc (4 5^{\circ}) = 2

csc (4 5^{\circ})

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

= 2

使用三角恒等式改写:

csc (22. 5^{\circ}) = 2 2 - 2 + 2 2 - 2

csc (22. 5^{\circ})

使用三角恒等式改写:

\frac{1}{sin ( 22. 5 ^{\circ} )}

csc (22. 5^{\circ})

使用基本三角恒等式:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( 22. 5 ^{\circ} )}

= \frac{1}{sin ( 22. 5 ^{\circ} )}

使用三角恒等式改写:

sin (22. 5^{\circ}) = \frac{2 - 2}{2}

sin (22. 5^{\circ})

使用三角恒等式改写:

\frac{1 - cos ( 4 5 ^{\circ} )}{2}

sin (22. 5^{\circ})

将

sin (22. 5^{\circ})

写为

sin (\frac{4 5 ^{\circ}}{2})

= sin (\frac{4 5 ^{\circ}}{2})

使用半角公式:

sin (\frac{θ}{2}) = \frac{1 - cos ( θ )}{2}

使用倍角公式

cos (2 θ) = 1 - 2 sin^{2} (θ)

用

\frac{θ}{2}

替代

θ

cos (θ) = 1 - 2 sin^{2} (\frac{θ}{2})

交换两边

2 sin^{2} (\frac{θ}{2}) = 1 - cos (θ)

两边除以

2

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

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

:

range [0, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] [18 0^{\circ}, 27 0^{\circ}] [27 0^{\circ}, 36 0^{\circ}] quadrant I II III I V sin positive positive negative negative cos positive negative negative positive

sin (\frac{θ}{2}) = \frac{( 1 - cos ( θ ) )}{2}

= \frac{1 - cos ( 4 5 ^{\circ} )}{2}

= \frac{1 - cos ( 4 5 ^{\circ} )}{2}

使用以下普通恒等式:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

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}

= \frac{2}{2}

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

化简

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

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

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

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

化简

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

1 - \frac{2}{2}

将项转换为分式:

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

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

数字相乘:

1 \cdot 2 = 2

= \frac{2 - 2}{2}

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

使用分式法则:

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

= \frac{2 - 2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2 - 2}{4}

= \frac{2 - 2}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{2 - 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

= \frac{2 - 2}{2}

= \frac{2 - 2}{2}

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

化简

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

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

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{2 - 2}

\frac{2}{2 - 2}

有理化:

2 2 - 2 + 2 2 - 2

\frac{2}{2 - 2}

乘以共轭根式

\frac{2 - 2}{2 - 2}

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

2 - 2 2 - 2 = 2 - 2

2 - 2 2 - 2

使用根式运算法则:

a a = a

2 - 2 2 - 2 = 2 - 2

= 2 - 2

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

分解

2 - 2 : 2 (2 - 1)

2 - 2

2 = 22

= 22 - 2

因式分解出通项

2

= 2 (2 - 1)

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

消掉

\frac{2 2 - 2}{2 ( 2 - 1 )} : \frac{2 2 - 2}{2 - 1}

\frac{2 2 - 2}{2 ( 2 - 1 )}

使用根式运算法则:

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

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

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

使用指数法则:

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

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

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

数字相减:

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

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

使用根式运算法则:

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

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

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

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

乘以共轭根式

\frac{2 + 1}{2 + 1}

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

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

2 2 - 2 (2 + 1)

= 2 (2 + 1) 2 - 2

使用分配律:

a (b + c) = ab + a c

a = 2 2 - 2, b = 2, c = 1

= 2 2 - 2 2 + 2 2 - 2 \cdot 1

= 22 2 - 2 + 1 \cdot 2 2 - 2

化简

22 2 - 2 + 1 \cdot 2 2 - 2 : 2 2 - 2 + 2 2 - 2

22 2 - 2 + 1 \cdot 2 2 - 2

使用根式运算法则:

a a = a

22 = 2

= 2 2 - 2 + 1 \cdot 2 2 - 2

乘以:

1 \cdot 2 = 2

= 2 2 - 2 + 2 2 - 2

= 2 2 - 2 + 2 2 - 2

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

(2 - 1) (2 + 1)

使用平方差公式:

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

a = 2, b = 1

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

化简

(2)^{2} - 1^{2} : 1

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

使用法则

1^{a} = 1

1^{2} = 1

= (2)^{2} - 1

(2)^{2} = 2

(2)^{2}

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 2

= 2 - 1

数字相减:

2 - 1 = 1

= 1

= 1

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

使用法则

\frac{a}{1} = a

= 2 2 - 2 + 2 2 - 2

= 2 2 - 2 + 2 2 - 2

= 2 2 - 2 + 2 2 - 2

= - 22 + 2 (2 2 - 2 + 2 2 - 2)

化简

- 22 + 2 (2 2 - 2 + 2 2 - 2) : - 2 + 2 (2 2 - 2 + 2 2 - 2)

- 22 + 2 (2 2 - 2 + 2 2 - 2)

使用根式运算法则:

a a = a

22 = 2

= - 2 + 2 (2 2 - 2 + 2 2 - 2)

= - 2 + 2 (2 2 - 2 + 2 2 - 2)

流行的例子

arccos((13)/(sqrt(6)*\sqrt{86)})

arccos (\frac{13}{6 \cdot 86})

sec (1 0^{\circ})

1 - sin (1)

arcsin (\frac{2}{4})

cos (32 0^{\circ})