类11 NCERT解决方案-第3章函数-练习3.4

找到以下方程式的主要和一般解：

问题1. tan x =√3

解决方案：

Given: tan x = √3

Here, x lies in first or third quadrant.

∴ tanx = tan60°∘ or tanx = tan(180° + 60°)

tanx = tan60°∘ or tanx = tan240°∘

Here, the principal solutions are π/3​, 4π/3​.

∴ tanx = tanπ/3

⇒ x = nπ + π​/3 where n ∈ Z​

问题2.秒x = 2

解决方案：

Given: sec x = 2

It can be written as, cos x = 1/2

Here, x lies in first or fourth quadrant.

∴ cosx = cos60°∘ or cosx = cos(360° – 60°)

cosx = cos60°∘ or cosx = cos300°∘

cosx = cosπ/3 or cosx = cos 5π/3

Here, the principal solutions are π/3, 5 π/3

∴ cosx = cosπ$\frac{ π}{3}$

x = 2nπ ± π/3​ where, n ∈ Z​

问题3. cot x = −√3

解决方案：

Given: cot x = −√3

It can be written as, tan x = -1/√3

Here, x lies in second or fourth quadrant.

∴ tanx = −tan30°∘= tan(180°– 30°) or tanx = tan(360°∘- 60°)

tanx = tan150°∘ or tanx = tan330°∘

tanx = tan5π/6​ or tanx = tan11π/6​​

Here, the principal solutions are 5π/6, 11 π/6

∴ tan x = tan 5π/6

⇒x = nπ + 5π/3​ where, n ∈ Z​

问题4。cosec x = – 2

解决方案：

Given: cosec x = – 2

It can be written as, sin x = -1/2

Here x lies in third or fourth quadrant.

∴ sinx = -sin30°∘= sin(180° + 30°) or sinx = sin(360°∘- 30°)

sinx = sin210°∘ or sinx = sin330°∘

sinx = sin7π/6​ or sinx = sin11π/6​​

Here, the principal solutions are 7π/6, 11 π/6

∴ sin x = -sin π/6

⇒ x = nπ + (−1)ⁿ7π/6​ where, n ∈ Z​

找到以下每个方程式的一般解：

问题5. cos 4x = cos 2x

解决方案：

Given: cos 4x = cos 2x

4x = 2nπ ± 2x, n ∈ Z

4x – 2x = 2nπ or 4x + 2x = 2nπ, n ∈ Z

2x = 2nπ or 6x = 2nπ, n ∈ Z

2x = 2nπ or 6x = 2nπ, n ∈ Z

x = nπ or x = 3nπ​, n ∈ Z

Therefore, the principal solutions are nπ, nπ​/3
 

问题6. cos 3x + cos x – cos 2x = 0

解决方案：

Given: cos 3x + cos x – cos 2x = 0

$2cos\Big(\frac{3x+x}{2} \Big)cos\Big(\frac{3x-x}{2} \Big)-cos2x=0$

2cos2xcosx – cos2x = 0

cos2x(2cosx – 1) = 0

cos2x = 0 or 2cosx – 1 = 0

2x = (2n + 1)π/2​ or cosx = 1/2 ​= cosπ/3​, n ∈ z

x = (2n + 1)π/4​ or x = 2nπ ± 2π​/3, n ∈ z

问题7：sin 2x + cos x = 0

解决方案：

Given: sin 2x + cos x = 0

2sinxcosx + cosx = 0

cosx(2sinx + 1) = 0

cosx = 0 or 2sinx + 1 = 0

x = (2n + 1)π/2​ or sinx = −1/2​ = −sinπ/6​, n ∈ z

x = (2n + 1)π/4​ or x = 2nπ ± 2π/3​, n ∈ z

x = (2n + 1)π/4​ or x = nπ + (-1)ⁿ-π/6​, n ∈ Z

x = (2n + 1)π/4​ or x = nπ + (-1)ⁿ7π/6​, n ∈ Z

问题8.秒² 2x = 1 –棕褐色2x

解决方案：

Given: sec²2x = 1 – tan 2x

1 + tan²2x = 1 – tan2x

tan2x(tan2x + 1) = 0

tan2x = 0 or tan2x + 1 = 0

2x = nπ or tan2x – 1 = –tanπ/4

x = nπ/2​ or x = nπ​/2 + 3π/8​, n = Z​
 

问题9. sin x + sin 3x + sin 5x = 0

解决方案：

Given: sin x + sin 3x + sin 5x = 0

$2sin\Big(\frac{5x+x}{2} \Big)cos\Big(\frac{5x-x}{2} \Big)+sin3x=0$

2sin3xcos2x + sin3x = 0

sin3x(2cos2x + 1) = 0

sin3x = 0 or 2cos2x + 1 = 1

3x = nπ or cos2x = -1/2 ​= cos2π/3​, n ∈ z

x = nπ/3​ or 2x = nπ/2 ± 2π/3​, n ∈ z

x = nπ​/3 or x = nπ ± π/3​, n ∈ z