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受欢迎的三角函数 >

(sin(b))/(350)=(sin(153))/(536)

解答

\frac{sin ( b )}{350} = \frac{sin ( 15 3 ^{\circ} )}{536}

解答

b = 0.30097 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.30097 \dots + 36 0^{\circ} n

+1

弧度

b = 0.30097 \dots + 2 πn, b = π - 0.30097 \dots + 2 πn

求解步骤

\frac{sin ( b )}{350} = \frac{sin ( 15 3 ^{\circ} )}{536}

sin (15 3^{\circ}) = \frac{2 4 - 2 5 - 5}{4}

sin (15 3^{\circ})

使用三角恒等式改写:

sin (2 7^{\circ})

sin (15 3^{\circ})

使用基本三角恒等式:

sin (x) = sin (18 0^{\circ} - x)

= sin (18 0^{\circ} - 15 3^{\circ})

化简:

18 0^{\circ} - 15 3^{\circ} = 2 7^{\circ}

18 0^{\circ} - 15 3^{\circ}

将项转换为分式:

18 0^{\circ} = 18 0^{\circ}

= 18 0^{\circ} - 15 3^{\circ}

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{18 0 ^{\circ} 20 - 306 0 ^{\circ}}{20}

同类项相加:

360 0^{\circ} - 306 0^{\circ} = 54 0^{\circ}

= 2 7^{\circ}

= sin (2 7^{\circ})

= sin (2 7^{\circ})

使用三角恒等式改写:

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

sin (2 7^{\circ})

将

sin (2 7^{\circ})

写为

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

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

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

使用三角恒等式改写:

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

cos (5 4^{\circ})

使用三角恒等式改写:

sin (3 6^{\circ})

cos (5 4^{\circ})

利用以下特性:

cos (x) = sin (9 0^{\circ} - x)

= sin (9 0^{\circ} - 5 4^{\circ})

化简:

9 0^{\circ} - 5 4^{\circ} = 3 6^{\circ}

9 0^{\circ} - 5 4^{\circ}

2, 10

的最小公倍数:

10

2, 10

最小公倍数 (LCM)

2

质因数分解:

2

2

2

是质数，因此无法因数分解

= 2

10

质因数分解:

2 \cdot 5

10

10

除以

210 = 5 \cdot 2

= 2 \cdot 5

2, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 5

将每个因子乘以它在

2

或

10

中出现的最多次数

= 2 \cdot 5

数字相乘:

2 \cdot 5 = 10

= 10

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

10

对于

9 0^{\circ} :

将分母和分子乘以

5

9 0^{\circ} = \frac{18 0 ^{\circ} 5}{2 \cdot 5} = 9 0^{\circ}

= 9 0^{\circ} - 5 4^{\circ}

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{18 0 ^{\circ} 5 - 54 0 ^{\circ}}{10}

同类项相加:

90 0^{\circ} - 54 0^{\circ} = 36 0^{\circ}

= 3 6^{\circ}

约分:

2

= 3 6^{\circ}

= sin (3 6^{\circ})

= sin (3 6^{\circ})

使用三角恒等式改写:

\frac{2 5 - 5}{4}

sin (3 6^{\circ})

显示:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

使用以下积化和差公式:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

显示:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

两边除以

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

利用以下特性:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

两边除以

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

两边除以

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

代入

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

显示:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

使用因式分解法则:

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

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

整理后得

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

显示:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

两边除以

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

利用以下特性:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

两边除以

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

两边除以

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

代入

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

代入

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

整理后得

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

两边加上

\frac{1}{4}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

整理后得

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

在两侧开平方

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

不能为负

sin (1 8^{\circ})

不能为负

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

以下方程式相加

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

整理后得

cos (3 6^{\circ}) = \frac{5 + 1}{4}

两边进行平方

(cos (3 6^{\circ}))^{2} = (\frac{5 + 1}{4})^{2}

利用以下特性:

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

sin^{2} (3 6^{\circ}) = 1 - cos^{2} (3 6^{\circ})

代入

cos (3 6^{\circ}) = \frac{5 + 1}{4}

sin^{2} (3 6^{\circ}) = 1 - (\frac{5 + 1}{4})^{2}

整理后得

sin^{2} (3 6^{\circ}) = \frac{5 - 5}{8}

在两侧开平方

sin (3 6^{\circ}) = \pm \frac{5 - 5}{8}

sin (3 6^{\circ})

不能为负

sin (3 6^{\circ}) = \frac{5 - 5}{8}

整理后得

sin (3 6^{\circ}) = \frac{\frac{5 - 5}{2}}{2}

= \frac{\frac{5 - 5}{2}}{2}

\frac{\frac{5 - 5}{2}}{2} = \frac{2 5 - 5}{4}

\frac{\frac{5 - 5}{2}}{2}

\frac{5 - 5}{2} = \frac{5 - 5}{2}

\frac{5 - 5}{2}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{5 - 5}{2}

= \frac{\frac{5 - 5}{2}}{2}

使用分式法则:

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

= \frac{5 - 5}{2 \cdot 2}

\frac{5 - 5}{2 2}

有理化:

\frac{2 5 - 5}{4}

\frac{5 - 5}{2 2}

乘以共轭根式

\frac{2}{2}

= \frac{5 - 5 2}{2 \cdot 2 2}

2 \cdot 22 = 4

2 \cdot 22

使用指数法则:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

同类项相加:

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

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

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

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

数字相加:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= \frac{2 5 - 5}{4}

= \frac{1 - \frac{2 5 - 5}{4}}{2}

化简

\frac{1 - \frac{2 5 - 5}{4}}{2} : \frac{2 4 - 2 5 - 5}{4}

\frac{1 - \frac{2 5 - 5}{4}}{2}

\frac{1 - \frac{2 5 - 5}{4}}{2} = \frac{4 - 2 5 - 5}{8}

\frac{1 - \frac{2 5 - 5}{4}}{2}

化简

1 - \frac{2 5 - 5}{4} : \frac{4 - 2 5 - 5}{4}

1 - \frac{2 5 - 5}{4}

将项转换为分式:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{2 5 - 5}{4}

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 4 - 2 5 - 5}{4}

数字相乘:

1 \cdot 4 = 4

= \frac{4 - 2 5 - 5}{4}

= \frac{\frac{4 - 2 5 - 5}{4}}{2}

使用分式法则:

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

= \frac{4 - 2 5 - 5}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{4 - 2 5 - 5}{8}

= \frac{4 - 2 5 - 5}{8}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{4 - 2 5 - 5}{8}

8 = 22

8

8

质因数分解:

2^{3}

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

使用指数法则:

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

= 2^{2} \cdot 2

使用根式运算法则:

= 2 2^{2}

使用根式运算法则:

2^{2} = 2

= 22

= \frac{- 2 5 - 5 + 4}{2 2}

\frac{4 - 2 5 - 5}{2 2}

有理化:

\frac{2 - 2 5 - 5 + 4}{4}

\frac{4 - 2 5 - 5}{2 2}

乘以共轭根式

\frac{2}{2}

= \frac{4 - 2 5 - 5 2}{2 2 2}

222 = 4

222

使用指数法则:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

同类项相加:

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

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

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

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

数字相加:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 4 - 2 5 - 5}{4}

= \frac{2 - 2 5 - 5 + 4}{4}

= \frac{2 4 - 2 5 - 5}{4}

\frac{sin ( b )}{350} = \frac{\frac{2 4 - 2 5 - 5}{4}}{536}

在两边乘以

10

\frac{sin ( b )}{350} = \frac{\frac{2 4 - 2 5 - 5}{4}}{536}

在两边乘以

10

\frac{sin ( b )}{350} \cdot 10 = \frac{\frac{2 4 - 2 5 - 5}{4}}{536} \cdot 10

化简

\frac{sin ( b )}{350} \cdot 10 = \frac{\frac{2 4 - 2 5 - 5}{4}}{536} \cdot 10

化简

\frac{sin ( b )}{350} \cdot 10 : \frac{sin ( b )}{35}

\frac{sin ( b )}{350} \cdot 10

分式相乘:

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

= \frac{sin ( b ) \cdot 10}{350}

约分:

10

= \frac{sin ( b )}{35}

化简

\frac{\frac{2 4 - 2 5 - 5}{4}}{536} \cdot 10 : \frac{5 2 - 2 5 - 5 + 4}{1072}

\frac{\frac{2 4 - 2 5 - 5}{4}}{536} \cdot 10

\frac{\frac{2 4 - 2 5 - 5}{4}}{536} = \frac{2 4 - 2 5 - 5}{2144}

\frac{\frac{2 4 - 2 5 - 5}{4}}{536}

使用分式法则:

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

= \frac{2 4 - 2 5 - 5}{4 \cdot 536}

数字相乘:

4 \cdot 536 = 2144

= \frac{2 - 2 5 - 5 + 4}{2144}

= 10 \cdot \frac{2 - 2 5 - 5 + 4}{2144}

分式相乘:

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

= \frac{2 4 - 2 5 - 5 \cdot 10}{2144}

约分:

2

= \frac{5 2 - 2 5 - 5 + 4}{1072}

\frac{sin ( b )}{35} = \frac{5 2 - 2 5 - 5 + 4}{1072}

\frac{sin ( b )}{35} = \frac{5 2 - 2 5 - 5 + 4}{1072}

\frac{sin ( b )}{35} = \frac{5 2 - 2 5 - 5 + 4}{1072}

在两边乘以

35

\frac{sin ( b )}{35} = \frac{5 2 - 2 5 - 5 + 4}{1072}

在两边乘以

35

\frac{35 sin ( b )}{35} = \frac{35 \cdot 5 2 - 2 5 - 5 + 4}{1072}

化简

\frac{35 sin ( b )}{35} = \frac{35 \cdot 5 2 - 2 5 - 5 + 4}{1072}

化简

\frac{35 sin ( b )}{35} : sin (b)

\frac{35 sin ( b )}{35}

数字相除:

\frac{35}{35} = 1

= sin (b)

化简

\frac{35 \cdot 5 2 - 2 5 - 5 + 4}{1072} : \frac{175 2 - 2 5 - 5 + 4}{1072}

\frac{35 \cdot 5 2 - 2 5 - 5 + 4}{1072}

数字相乘:

35 \cdot 5 = 175

= \frac{175 2 - 2 5 - 5 + 4}{1072}

sin (b) = \frac{175 2 - 2 5 - 5 + 4}{1072}

sin (b) = \frac{175 2 - 2 5 - 5 + 4}{1072}

sin (b) = \frac{175 2 - 2 5 - 5 + 4}{1072}

使用反三角函数性质

sin (b) = \frac{175 2 - 2 5 - 5 + 4}{1072}

sin (b) = \frac{175 2 - 2 5 - 5 + 4}{1072}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

b = arcsin \frac{175 2 - 2 5 - 5 + 4}{1072} + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin \frac{175 2 - 2 5 - 5 + 4}{1072} + 36 0^{\circ} n

b = arcsin \frac{175 2 - 2 5 - 5 + 4}{1072} + 36 0^{\circ} n, b = 18 0^{\circ} - arcsin \frac{175 2 - 2 5 - 5 + 4}{1072} + 36 0^{\circ} n

以小数形式表示解

b = 0.30097 \dots + 36 0^{\circ} n, b = 18 0^{\circ} - 0.30097 \dots + 36 0^{\circ} n

作图

流行的例子

-1-2cos^2(x)+3cos(x)=0

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

sin(θ)=(10sin(105))/(24.6)

0.175=0.34sin(93.4x)