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受欢迎的三角函数 >

7sin^2(x)+3cos^2(x)=5

解答

7 sin^{2} (x) + 3 cos^{2} (x) = 5

解答

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度数

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

7 sin^{2} (x) + 3 cos^{2} (x) = 5

两边减去

5

7 sin^{2} (x) + 3 cos^{2} (x) - 5 = 0

使用三角恒等式改写

- 5 + 3 cos^{2} (x) + 7 sin^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= - 5 + 3 (1 - sin^{2} (x)) + 7 sin^{2} (x)

化简

- 5 + 3 (1 - sin^{2} (x)) + 7 sin^{2} (x) : 4 sin^{2} (x) - 2

- 5 + 3 (1 - sin^{2} (x)) + 7 sin^{2} (x)

乘开

3 (1 - sin^{2} (x)) : 3 - 3 sin^{2} (x)

3 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (x)

= 3 \cdot 1 - 3 sin^{2} (x)

数字相乘:

3 \cdot 1 = 3

= 3 - 3 sin^{2} (x)

= - 5 + 3 - 3 sin^{2} (x) + 7 sin^{2} (x)

化简

- 5 + 3 - 3 sin^{2} (x) + 7 sin^{2} (x) : 4 sin^{2} (x) - 2

- 5 + 3 - 3 sin^{2} (x) + 7 sin^{2} (x)

同类项相加:

- 3 sin^{2} (x) + 7 sin^{2} (x) = 4 sin^{2} (x)

= - 5 + 3 + 4 sin^{2} (x)

数字相加/相减:

- 5 + 3 = - 2

= 4 sin^{2} (x) - 2

= 4 sin^{2} (x) - 2

= 4 sin^{2} (x) - 2

- 2 + 4 sin^{2} (x) = 0

用替代法求解

- 2 + 4 sin^{2} (x) = 0

令

: sin (x) = u

- 2 + 4 u^{2} = 0

- 2 + 4 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

- 2 + 4 u^{2} = 0

将

2

到右边

- 2 + 4 u^{2} = 0

两边加上

2

- 2 + 4 u^{2} + 2 = 0 + 2

化简

4 u^{2} = 2

4 u^{2} = 2

两边除以

4

4 u^{2} = 2

两边除以

4

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

化简

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

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

对于

x^{2} = f (a)

解为

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

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

u = sin (x)

代回

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{1}{2}

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

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

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

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

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

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

合并所有解

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

作图

流行的例子

sin (x) = 0.3746

tan (a) = 0.49

sin(x)=1.6cos(x)

sin (x) = 1.6 cos (x)

8sin(x)+4sqrt(3)=0

8 sin (x) + 43 = 0

tan(θ)=5,sin(θ)

tan (θ) = 5, sin (θ)