Question

1) ctg(3pi/2-x)-ctg^2(x)-2=0 2) 2(tg(x/2)-1)=cosx

Answer 1

Ответ:

6 cos ^2 x+5 cos ( π/2 -x)=7

6 cos ^2 x+5sinx-7=0

6-6sin^2x+5sinx-7=0

-6sin^2 x+5sinx-1=0

6sin^2 x- 5 sinx+1=0

D= b^2 -4 ac

y1,2= 1/2 , 1/3

sin x=(-1)^n arcsin 1/2 πn , n ∈ Z ; x= (-1)^n arcsin 1/3 +πn

x= (-1)^n π/ 6 + πn,

Answer 2

To solve the given trigonometric equations, let's address them one by one:

1. ctg(3π/2 - x) - ctg^2(x) - 2 = 0

To simplify the equation, let's rewrite ctg(3π/2 - x) as tan(x).

So the equation becomes:

tan(x) - ctg^2(x) - 2 = 0

Now, let's replace ctg(x) with 1/tan(x):

tan(x) - (1/tan(x))^2 - 2 = 0

tan(x) - 1/tan^2(x) - 2 = 0

To eliminate the fraction, let's multiply the entire equation by tan^2(x):

tan^3(x) - 1 - 2tan^2(x) = 0

Rearranging the terms:

tan^3(x) - 2tan^2(x) - 1 = 0

Now we have a cubic equation in terms of tan(x). Unfortunately, there isn't a straightforward algebraic method to solve cubic equations. You can approximate the solutions using numerical methods or use a calculator to find the approximate values.

2. 2(tan(x/2) - 1) = cos(x)

First, let's simplify the left side of the equation:

2(tan(x/2) - 1) = cos(x)

2tan(x/2) - 2 = cos(x)

Next, let's replace tan(x/2) with sin(x)/(1 + cos(x)) using the half-angle formula for tangent:

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

Now, let's simplify further:

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

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

Rearranging the terms:

cos^2(x) + 3cos(x) - 2sin(x) - 2 = 0

This equation doesn't have a simple algebraic solution. You can use numerical methods or approximations to find the solutions.