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受欢迎的三角函数 >

cos^2(x+30)= 1/4

解答

cos^{2} (x + 3 0^{\circ}) = \frac{1}{4}

解答

x = 36 0^{\circ} n + 3 0^{\circ}, x = 36 0^{\circ} n + 27 0^{\circ}, x = 36 0^{\circ} n + 9 0^{\circ}, x = 36 0^{\circ} n + 21 0^{\circ}

+1

弧度

x = \frac{π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn

求解步骤

cos^{2} (x + 3 0^{\circ}) = \frac{1}{4}

用替代法求解

cos^{2} (x + 3 0^{\circ}) = \frac{1}{4}

令

: cos (x + 3 0^{\circ}) = u

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

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

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

对于

x^{2} = f (a)

解为

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

u = \frac{1}{4}, u = - \frac{1}{4}

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

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

使用法则

1 = 1

= \frac{1}{2}

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

- \frac{1}{4}

化简

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

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

= - \frac{1}{2}

使用法则

1 = 1

= - \frac{1}{2}

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

u = cos (x + 3 0^{\circ})

代回

cos (x + 3 0^{\circ}) = \frac{1}{2}, cos (x + 3 0^{\circ}) = - \frac{1}{2}

cos (x + 3 0^{\circ}) = \frac{1}{2}, cos (x + 3 0^{\circ}) = - \frac{1}{2}

cos (x + 3 0^{\circ}) = \frac{1}{2} : x = 36 0^{\circ} n + 3 0^{\circ}, x = 36 0^{\circ} n + 27 0^{\circ}

cos (x + 3 0^{\circ}) = \frac{1}{2}

cos (x + 3 0^{\circ}) = \frac{1}{2}

的通解

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}

x + 3 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

x + 3 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

解

x + 3 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 3 0^{\circ}

x + 3 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

将

3 0^{\circ}

到右边

x + 3 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

两边减去

3 0^{\circ}

x + 3 0^{\circ} - 3 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

化简

x + 3 0^{\circ} - 3 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

化简

x + 3 0^{\circ} - 3 0^{\circ} : x

x + 3 0^{\circ} - 3 0^{\circ}

同类项相加:

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

= x

化简

6 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ} : 36 0^{\circ} n + 3 0^{\circ}

6 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

对同类项分组

= 36 0^{\circ} n + 6 0^{\circ} - 3 0^{\circ}

3, 6

的最小公倍数:

6

3, 6

最小公倍数 (LCM)

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

将每个因子乘以它在

3

或

6

中出现的最多次数

= 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 6

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

6

对于

6 0^{\circ} :

将分母和分子乘以

2

6 0^{\circ} = \frac{18 0 ^{\circ} 2}{3 \cdot 2} = 6 0^{\circ}

= 6 0^{\circ} - 3 0^{\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} 2 - 18 0 ^{\circ}}{6}

同类项相加:

36 0^{\circ} - 18 0^{\circ} = 18 0^{\circ}

= 36 0^{\circ} n + 3 0^{\circ}

x = 36 0^{\circ} n + 3 0^{\circ}

x = 36 0^{\circ} n + 3 0^{\circ}

x = 36 0^{\circ} n + 3 0^{\circ}

解

x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 27 0^{\circ}

x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

将

3 0^{\circ}

到右边

x + 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

两边减去

3 0^{\circ}

x + 3 0^{\circ} - 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

化简

x + 3 0^{\circ} - 3 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

化简

x + 3 0^{\circ} - 3 0^{\circ} : x

x + 3 0^{\circ} - 3 0^{\circ}

同类项相加:

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

= x

化简

30 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ} : 36 0^{\circ} n + 27 0^{\circ}

30 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

对同类项分组

= 36 0^{\circ} n - 3 0^{\circ} + 30 0^{\circ}

6, 3

的最小公倍数:

6

6, 3

最小公倍数 (LCM)

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

将每个因子乘以它在

6

或

3

中出现的最多次数

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

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

6

对于

30 0^{\circ} :

将分母和分子乘以

2

30 0^{\circ} = \frac{90 0 ^{\circ} 2}{3 \cdot 2} = 30 0^{\circ}

= - 3 0^{\circ} + 30 0^{\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} + 180 0 ^{\circ}}{6}

同类项相加:

- 18 0^{\circ} + 180 0^{\circ} = 162 0^{\circ}

= 27 0^{\circ}

约分:

3

= 36 0^{\circ} n + 27 0^{\circ}

x = 36 0^{\circ} n + 27 0^{\circ}

x = 36 0^{\circ} n + 27 0^{\circ}

x = 36 0^{\circ} n + 27 0^{\circ}

x = 36 0^{\circ} n + 3 0^{\circ}, x = 36 0^{\circ} n + 27 0^{\circ}

cos (x + 3 0^{\circ}) = - \frac{1}{2} : x = 36 0^{\circ} n + 9 0^{\circ}, x = 36 0^{\circ} n + 21 0^{\circ}

cos (x + 3 0^{\circ}) = - \frac{1}{2}

cos (x + 3 0^{\circ}) = - \frac{1}{2}

的通解

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}

x + 3 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n, x + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

x + 3 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n, x + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

解

x + 3 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 9 0^{\circ}

x + 3 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

将

3 0^{\circ}

到右边

x + 3 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

两边减去

3 0^{\circ}

x + 3 0^{\circ} - 3 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

化简

x + 3 0^{\circ} - 3 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

化简

x + 3 0^{\circ} - 3 0^{\circ} : x

x + 3 0^{\circ} - 3 0^{\circ}

同类项相加:

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

= x

化简

12 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ} : 36 0^{\circ} n + 9 0^{\circ}

12 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

对同类项分组

= 36 0^{\circ} n - 3 0^{\circ} + 12 0^{\circ}

6, 3

的最小公倍数:

6

6, 3

最小公倍数 (LCM)

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

将每个因子乘以它在

6

或

3

中出现的最多次数

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

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

6

对于

12 0^{\circ} :

将分母和分子乘以

2

12 0^{\circ} = \frac{36 0 ^{\circ} 2}{3 \cdot 2} = 12 0^{\circ}

= - 3 0^{\circ} + 12 0^{\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} + 72 0 ^{\circ}}{6}

同类项相加:

- 18 0^{\circ} + 72 0^{\circ} = 54 0^{\circ}

= 9 0^{\circ}

约分:

3

= 36 0^{\circ} n + 9 0^{\circ}

x = 36 0^{\circ} n + 9 0^{\circ}

x = 36 0^{\circ} n + 9 0^{\circ}

x = 36 0^{\circ} n + 9 0^{\circ}

解

x + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 21 0^{\circ}

x + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

将

3 0^{\circ}

到右边

x + 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

两边减去

3 0^{\circ}

x + 3 0^{\circ} - 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

化简

x + 3 0^{\circ} - 3 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

化简

x + 3 0^{\circ} - 3 0^{\circ} : x

x + 3 0^{\circ} - 3 0^{\circ}

同类项相加:

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

= x

化简

24 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ} : 36 0^{\circ} n + 21 0^{\circ}

24 0^{\circ} + 36 0^{\circ} n - 3 0^{\circ}

对同类项分组

= 36 0^{\circ} n - 3 0^{\circ} + 24 0^{\circ}

6, 3

的最小公倍数:

6

6, 3

最小公倍数 (LCM)

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

将每个因子乘以它在

6

或

3

中出现的最多次数

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

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

6

对于

24 0^{\circ} :

将分母和分子乘以

2

24 0^{\circ} = \frac{72 0 ^{\circ} 2}{3 \cdot 2} = 24 0^{\circ}

= - 3 0^{\circ} + 24 0^{\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} + 144 0 ^{\circ}}{6}

同类项相加:

- 18 0^{\circ} + 144 0^{\circ} = 126 0^{\circ}

= 36 0^{\circ} n + 21 0^{\circ}

x = 36 0^{\circ} n + 21 0^{\circ}

x = 36 0^{\circ} n + 21 0^{\circ}

x = 36 0^{\circ} n + 21 0^{\circ}

x = 36 0^{\circ} n + 9 0^{\circ}, x = 36 0^{\circ} n + 21 0^{\circ}

合并所有解

x = 36 0^{\circ} n + 3 0^{\circ}, x = 36 0^{\circ} n + 27 0^{\circ}, x = 36 0^{\circ} n + 9 0^{\circ}, x = 36 0^{\circ} n + 21 0^{\circ}

作图

流行的例子

cos (5 7^{\circ}) = sin (x)

3cos(x)=2-cos(x)

3 cos (x) = 2 - cos (x)

2cos^2(3x)-3cos(3x)=-1,0<= x<= 2pi

2 cos^{2} (3 x) - 3 cos (3 x) = - 1, 0 \leq x \leq 2 π

solvefor x,z=y^{sin(x)}

so l v e f or x, z = y^{s i n (x)}

solvefor t,10=14+8sin((pit)/(12))

so l v e f or t, 10 = 14 + 8 sin (\frac{π t}{12})