Question

4tgx+3ctgx=7 sin3x+sin7x=-2

Answer 1

4tgx+3ctgx=7

tgx=t

4t+3/t-7=0  4t^2-7t+3=0  t=1  t2=3/4

x=П/4+Пn

x=arctg(3/4)+Пn

sin3x+sin7x=-2

sinx>-1;

чтобы сумма синусов была равна -2,необходимо, чтобы

sin3x=-1  

sin7x=-1   x=П/2+2Пn

Answer 2

To solve the system of trigonometric equations:

1. 4tg(x) + 3ctg(x) = 7
2. sin(3x) + sin(7x) = -2

Let's start by simplifying the equations.

1. 4tg(x) + 3ctg(x) = 7

Recall that "ctg(x)" represents the cotangent of x, which is the reciprocal of the tangent. You can rewrite this equation in terms of the tangent function:

4tg(x) + 3(1/tg(x)) = 7

Now, common denominator can be used:

4tg(x) + 3/tg(x) = 7

To clear the fractions, multiply through by tg(x):

4(tg(x))^2 + 3 = 7(tg(x))

Rearrange the equation:

4(tg(x))^2 - 7(tg(x)) + 3 = 0

Now, you have a quadratic equation in terms of the tangent function. You can solve this quadratic equation, and once you find the values of tg(x), you can find the corresponding values of x.

2. sin(3x) + sin(7x) = -2

This equation involves the sum of two trigonometric functions. There isn't a simple algebraic way to solve it. You'll likely need to use trigonometric identities to simplify it and then attempt to solve it.

One possible approach is to use the product-to-sum identities to express the sum of sines as a product of sines:

sin(3x) + sin(7x) = 2sin((3x + 7x)/2)cos((7x - 3x)/2)

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

Now you have:

2sin(5x)cos(2x) = -2

Divide both sides by 2:

sin(5x)cos(2x) = -1

Now, you have a product of sine and cosine. You might want to use trigonometric identities like the double-angle identities and sum-to-product identities to further simplify and solve this equation. It can get quite complex, and it may not have simple solutions. Numerical methods or graphing software might be needed to find approximate solutions.