Question

Решите тригонометрические уравнения, с объяснением желательно 1.7tgx-10ctgx+9=0 2.5sin2x-14cos^2x+2=0 3.9cos2x-4cos^2x=11sin2x+9 Пожалуйста не пишите что ошиблась или вместо это напиши то, я все проверила

Answer 1

1) 7tgx -10Ctgx +9 =0 |* tgx
     7tg^2x -10 +9tgx = 0
tgx = y
7y^2 +9y -10 = 0
y1 = 10/14 = 5/7
у2 = -2
а) у = 5/7
tgx = 5/7
x = arctg5/7 + $\pi$k, k ЄZ
б) у = -2
tgx = -2
x = -arctg2 +$\pi$k, k ЄZ
2) 10SinxCosx -14Cos^2x +2*1 = 0
10SinxCosx -14Cos^2x +2(Sin^2x+Cos^2x) = 0
10SinxCosx -14Cos^2x +2Sin^2x +2Cos^2x = 0
10SinxCosx -12Cos^2x +2Sin^2x = 0 :Cos^2x
10tgx -12 +2tg^2x= 0
tgx = y
2y^2 +10y -12=0
y^2 + 5y - 6 = 0
По т. Виета у1 = - 6 и  у2 = 1
а) у = - 6
tgx = -6
x = -arctg6+$\pi$k, kЄZ 
б)у = 1
tgx = 1
x = $\pi /4$ + $\pi k, k ЄZ$
3) 9(Cos^2x - Sin^2x) -4Cos^2x = 22SinxCosx + 9*1
9Cos^2x - 9Sin^2x -4Cos^2x -22SinxCosx -9(Sin^2x+Cos^2x) = 0
9Cos^2x - 9Sin^2x -4Cos^2x -22SinxCosx -9Sin^2x - 9Cos^2x = 0
-18Sin^2x -4Cos^2x -22SinxCosx = 0
9Sin^2x +2Cos^2x +11SinxCosx = 0|:Cos^2x
9tg^2x +2 +11tgx = 0
tgx = y
9y^2 +11y +2 = 0
y1=-1,  y2 = -2/9
a) y = -1
tgx = -1
x = -$\pi /4$ + $\pi k, kЄZ$
б) у = -2/9
tgx = -2/9
x = -arctg(2/9) + $\pi k, k ЄZ$
 

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Answer 2

Solving Trigonometric Equations

Let's solve the given trigonometric equations step by step.

1. Equation 1: 7tg(x) - 10ctg(x) + 9 = 0

To solve this equation, we can use the fact that cotangent is the reciprocal of tangent: ctg(x) = 1/tg(x). Let's denote tg(x) as t for simplicity.

So, the equation becomes: 7t - 10(1/t) + 9 = 0

Multiplying through by t to clear the fraction gives us a quadratic equation: 7t^2 - 10 + 9t = 0

Solving this quadratic equation will give us the values of t, and then we can find x using the inverse tangent function.

2. Equation 2: 5sin(2x) - 14cos^2(x) + 2 = 0

To solve this equation, we can use the double angle identity for sine: sin(2x) = 2sin(x)cos(x). Let's denote sin(x) as s and cos(x) as c for simplicity.

So, the equation becomes: 5(2sc) - 14c^2 + 2 = 0

This equation involves both sine and cosine, and we can use trigonometric identities to simplify and solve for either s or c.

3. Equation 3: 9cos(2x) - 4cos^2(x) = 11sin(2x) + 9

To solve this equation, we can use the double angle identities for cosine and sine: cos(2x) = cos^2(x) - sin^2(x) and sin(2x) = 2sin(x)cos(x).

Substitute these identities into the equation and simplify to solve for either sin(x) or cos(x).

Let's proceed with solving each of these equations step by step.