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受欢迎的三角函数 >

cos^2(22.5)

解答

cos^{2} (22. 5^{\circ})

解答

\frac{2 + 2}{4}

+1

十进制

0.85355 \dots

求解步骤

cos^{2} (22. 5^{\circ})

使用三角恒等式改写:

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

cos^{2} (22. 5^{\circ})

利用以下特性:

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

使用倍角公式

cos (2 θ) = 2 cos^{2} (θ) - 1

交换两边

2 cos^{2} (θ) - 1 = cos (2 θ)

两边加上

1

2 sin^{2} (θ) = 1 + cos (2 θ)

两边除以

2

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

= \frac{1 + cos ( 2 \cdot 22. 5 ^{\circ} )}{2}

化简

= \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}{4}

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

化简

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

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 + 1 )}{2}

消掉

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

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

使用根式运算法则:

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

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

数字相减:

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

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

使用根式运算法则:

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

= \frac{2 + 1}{2}

= \frac{2 + 1}{2}

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

使用分式法则:

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

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

\frac{2 + 1}{2 2}

有理化:

\frac{2 + 2}{4}

\frac{2 + 1}{2 2}

乘以共轭根式

\frac{2}{2}

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

(2 + 1) 2 = 2 + 2

(2 + 1) 2

= 2 (2 + 1)

使用分配律:

a (b + c) = ab + a c

a = 2, b = 2, c = 1

= 22 + 2 \cdot 1

= 22 + 1 \cdot 2

化简

22 + 1 \cdot 2 : 2 + 2

22 + 1 \cdot 2

使用根式运算法则:

a a = a

22 = 2

= 2 + 1 \cdot 2

乘以:

1 \cdot 2 = 2

= 2 + 2

= 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 + 2}{4}

= \frac{2 + 2}{4}

= \frac{2 + 2}{4}

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