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受欢迎的三角函数 >

4-7sin^2(θ)=3cos^2(θ)

解答

4 - 7 sin^{2} (θ) = 3 cos^{2} (θ)

解答

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

+1

度数

θ = 3 0^{\circ} + 36 0^{\circ} n, θ = 15 0^{\circ} + 36 0^{\circ} n, θ = 21 0^{\circ} + 36 0^{\circ} n, θ = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

4 - 7 sin^{2} (θ) = 3 cos^{2} (θ)

两边减去

3 cos^{2} (θ)

4 - 7 sin^{2} (θ) - 3 cos^{2} (θ) = 0

使用三角恒等式改写

4 - 3 cos^{2} (θ) - 7 sin^{2} (θ)

使用毕达哥拉斯恒等式:

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

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

= 4 - 3 (1 - sin^{2} (θ)) - 7 sin^{2} (θ)

化简

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

4 - 3 (1 - sin^{2} (θ)) - 7 sin^{2} (θ)

乘开

- 3 (1 - sin^{2} (θ)) : - 3 + 3 sin^{2} (θ)

- 3 (1 - sin^{2} (θ))

使用分配律:

a (b - c) = ab - a c

a = - 3, b = 1, c = sin^{2} (θ)

= - 3 \cdot 1 - (- 3) sin^{2} (θ)

使用加减运算法则

- (- a) = a

= - 3 \cdot 1 + 3 sin^{2} (θ)

数字相乘:

3 \cdot 1 = 3

= - 3 + 3 sin^{2} (θ)

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

化简

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

4 - 3 + 3 sin^{2} (θ) - 7 sin^{2} (θ)

同类项相加:

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

= 4 - 3 - 4 sin^{2} (θ)

数字相减:

4 - 3 = 1

= - 4 sin^{2} (θ) + 1

= - 4 sin^{2} (θ) + 1

= - 4 sin^{2} (θ) + 1

1 - 4 sin^{2} (θ) = 0

用替代法求解

1 - 4 sin^{2} (θ) = 0

令

: sin (θ) = u

1 - 4 u^{2} = 0

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

1 - 4 u^{2} = 0

将

1

到右边

1 - 4 u^{2} = 0

两边减去

1

1 - 4 u^{2} - 1 = 0 - 1

化简

- 4 u^{2} = - 1

- 4 u^{2} = - 1

两边除以

- 4

- 4 u^{2} = - 1

两边除以

- 4

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

化简

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

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 = sin (θ)

代回

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

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

sin (θ) = \frac{1}{2} : θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

sin (θ) = \frac{1}{2}

sin (θ) = \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}

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

sin (θ) = - \frac{1}{2} : θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

sin (θ) = - \frac{1}{2}

sin (θ) = - \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}

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

合并所有解

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

作图

流行的例子

csc(3x)=-(2sqrt(3))/3

csc (3 x) = - \frac{2 3}{3}

7sin^2(x)-14sin(x)+2=-5

7 sin^{2} (x) - 14 sin (x) + 2 = - 5

sin(x)=-sqrt(3)

sin (x) = - 3

-cos^2(x)+sin^2(x)= 1/2

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

solvefor y,z=e^xsin(xy)

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