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受欢迎的三角函数 >

((1+cos^2(a)))/(sin^2(a))= 5/3

解答

\frac{( 1 + cos ^{2} ( a ) )}{sin ^{2} ( a )} = \frac{5}{3}

解答

a = \frac{π}{3} + 2 πn, a = \frac{2 π}{3} + 2 πn, a = \frac{4 π}{3} + 2 πn, a = \frac{5 π}{3} + 2 πn

+1

度数

a = 6 0^{\circ} + 36 0^{\circ} n, a = 12 0^{\circ} + 36 0^{\circ} n, a = 24 0^{\circ} + 36 0^{\circ} n, a = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

\frac{( 1 + cos ^{2} ( a ) )}{sin ^{2} ( a )} = \frac{5}{3}

两边减去

\frac{5}{3}

\frac{1 + cos ^{2} ( a )}{sin ^{2} ( a )} - \frac{5}{3} = 0

化简

\frac{1 + cos ^{2} ( a )}{sin ^{2} ( a )} - \frac{5}{3} : \frac{3 ( 1 + cos ^{2} ( a ) ) - 5 sin ^{2} ( a )}{3 sin ^{2} ( a )}

\frac{1 + cos ^{2} ( a )}{sin ^{2} ( a )} - \frac{5}{3}

sin^{2} (a), 3

的最小公倍数:

3 sin^{2} (a)

sin^{2} (a), 3

最小公倍数 (LCM)

计算出由出现在

sin^{2} (a)

或

3

中的因子组成的表达式

= 3 sin^{2} (a)

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

3 sin^{2} (a)

对于

\frac{1 + cos ^{2} ( a )}{sin ^{2} ( a )} :

将分母和分子乘以

3

\frac{1 + cos ^{2} ( a )}{sin ^{2} ( a )} = \frac{( 1 + cos ^{2} ( a ) ) \cdot 3}{sin ^{2} ( a ) \cdot 3}

对于

\frac{5}{3} :

将分母和分子乘以

sin^{2} (a)

\frac{5}{3} = \frac{5 sin ^{2} ( a )}{3 sin ^{2} ( a )}

= \frac{( 1 + cos ^{2} ( a ) ) \cdot 3}{sin ^{2} ( a ) \cdot 3} - \frac{5 sin ^{2} ( a )}{3 sin ^{2} ( a )}

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 + cos ^{2} ( a ) ) \cdot 3 - 5 sin ^{2} ( a )}{3 sin ^{2} ( a )}

\frac{3 ( 1 + cos ^{2} ( a ) ) - 5 sin ^{2} ( a )}{3 sin ^{2} ( a )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 (1 + cos^{2} (a)) - 5 sin^{2} (a) = 0

使用三角恒等式改写

(1 + cos^{2} (a)) \cdot 3 - 5 sin^{2} (a)

使用毕达哥拉斯恒等式:

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

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

= (1 + 1 - sin^{2} (a)) \cdot 3 - 5 sin^{2} (a)

化简

(1 + 1 - sin^{2} (a)) \cdot 3 - 5 sin^{2} (a) : - 8 sin^{2} (a) + 6

(1 + 1 - sin^{2} (a)) \cdot 3 - 5 sin^{2} (a)

数字相加:

1 + 1 = 2

= 3 (- sin^{2} (a) + 2) - 5 sin^{2} (a)

乘开

3 (- sin^{2} (a) + 2) : - 3 sin^{2} (a) + 6

3 (- sin^{2} (a) + 2)

使用分配律:

a (b + c) = ab + a c

a = 3, b = - sin^{2} (a), c = 2

= 3 (- sin^{2} (a)) + 3 \cdot 2

使用加减运算法则

+ (- a) = - a

= - 3 sin^{2} (a) + 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= - 3 sin^{2} (a) + 6

= - 3 sin^{2} (a) + 6 - 5 sin^{2} (a)

化简

- 3 sin^{2} (a) + 6 - 5 sin^{2} (a) : - 8 sin^{2} (a) + 6

- 3 sin^{2} (a) + 6 - 5 sin^{2} (a)

对同类项分组

= - 3 sin^{2} (a) - 5 sin^{2} (a) + 6

同类项相加:

- 3 sin^{2} (a) - 5 sin^{2} (a) = - 8 sin^{2} (a)

= - 8 sin^{2} (a) + 6

= - 8 sin^{2} (a) + 6

= - 8 sin^{2} (a) + 6

6 - 8 sin^{2} (a) = 0

用替代法求解

6 - 8 sin^{2} (a) = 0

令

: sin (a) = u

6 - 8 u^{2} = 0

6 - 8 u^{2} = 0 : u = \frac{3}{2}, u = - \frac{3}{2}

6 - 8 u^{2} = 0

将

6

到右边

6 - 8 u^{2} = 0

两边减去

6

6 - 8 u^{2} - 6 = 0 - 6

化简

- 8 u^{2} = - 6

- 8 u^{2} = - 6

两边除以

- 8

- 8 u^{2} = - 6

两边除以

- 8

\frac{- 8 u ^{2}}{- 8} = \frac{- 6}{- 8}

化简

\frac{- 8 u ^{2}}{- 8} = \frac{- 6}{- 8}

化简

\frac{- 8 u ^{2}}{- 8} : u^{2}

\frac{- 8 u ^{2}}{- 8}

使用分式法则:

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

= \frac{8 u ^{2}}{8}

数字相除:

\frac{8}{8} = 1

= u^{2}

化简

\frac{- 6}{- 8} : \frac{3}{4}

\frac{- 6}{- 8}

使用分式法则:

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

= \frac{6}{8}

约分:

2

= \frac{3}{4}

u^{2} = \frac{3}{4}

u^{2} = \frac{3}{4}

u^{2} = \frac{3}{4}

对于

x^{2} = f (a)

解为

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

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{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{3}{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{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

化简

\frac{3}{4} : \frac{3}{2}

\frac{3}{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{3}{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{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

u = sin (a)

代回

sin (a) = \frac{3}{2}, sin (a) = - \frac{3}{2}

sin (a) = \frac{3}{2}, sin (a) = - \frac{3}{2}

sin (a) = \frac{3}{2} : a = \frac{π}{3} + 2 πn, a = \frac{2 π}{3} + 2 πn

sin (a) = \frac{3}{2}

sin (a) = \frac{3}{2}

的通解

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}

a = \frac{π}{3} + 2 πn, a = \frac{2 π}{3} + 2 πn

a = \frac{π}{3} + 2 πn, a = \frac{2 π}{3} + 2 πn

sin (a) = - \frac{3}{2} : a = \frac{4 π}{3} + 2 πn, a = \frac{5 π}{3} + 2 πn

sin (a) = - \frac{3}{2}

sin (a) = - \frac{3}{2}

的通解

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}

a = \frac{4 π}{3} + 2 πn, a = \frac{5 π}{3} + 2 πn

a = \frac{4 π}{3} + 2 πn, a = \frac{5 π}{3} + 2 πn

合并所有解

a = \frac{π}{3} + 2 πn, a = \frac{2 π}{3} + 2 πn, a = \frac{4 π}{3} + 2 πn, a = \frac{5 π}{3} + 2 πn

作图

流行的例子

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