Используем формулу суммы синусов
sinα + sinβ = 2 * sin
![\frac{ \alpha + \beta }{2}](https://tex.z-dn.net/?f=+%5Cfrac%7B+%5Calpha++%2B++%5Cbeta+%7D%7B2%7D+)
* cos
α = x + 30
β = x - 30
sin (x + 30) + sin (x - 30) = 2 * sin
![\frac{x + 30 + x - 30}{2}](https://tex.z-dn.net/?f=+%5Cfrac%7Bx+%2B+30+%2B+x+-+30%7D%7B2%7D+)
* cos
![\frac{x + 30 - x + 30}{2}](https://tex.z-dn.net/?f=+%5Cfrac%7Bx+%2B+30+-+x+%2B+30%7D%7B2%7D+)
= 2 √ (3cosx)
2 * sin
![\frac{2x}{2}](https://tex.z-dn.net/?f=+%5Cfrac%7B2x%7D%7B2%7D+)
* cos
![\frac{60}{2}](https://tex.z-dn.net/?f=+%5Cfrac%7B60%7D%7B2%7D+)
= 2 √(3cosx)
2 * sin x * cos 30 = 2 √(3cosx)
2 * √3/2 * cosx = 2 √(3cosx)
√3 * sinx = 2 √(3cosx)
(√3 * sinx)² = (2 √(3cosx))²
3 * sin ² x = 4 * 3 * cosx
sin²x = 1 - cos²x
3 * (1 - cos²x) = 4 * 3 * cosx
1 - cos²x = 4 *cosx
cos²x + 4cosx - 1 = 0
cosx = t
t² + 4 t - 1 = 0
D = 16 - 4 * 1 * (- 1) = 16 + 4 = 20
t ₁ = (- 4 - √20)/2 = (- 4 - 2√5)/2 = - 2 - √5
t₂ = (- 4 + √20)/2 = (- 4 + 2√5)/2 = - 2 + √5
cosx = - 2 - √5 < - 1 не удовлетворяет, т.к. значения -1 ≤ cosх ≤ 1
cos x = - 2 + √5 < 1 удовлетворяет
Используем формулу
1 + tg²x =
tg²x =
![\frac{1}{cos ^{2}x }](https://tex.z-dn.net/?f=+%5Cfrac%7B1%7D%7Bcos+%5E%7B2%7Dx+%7D+)
- 1
tg²x =
![\frac{1}{ (- 2 + \sqrt{5} )^{2} }](https://tex.z-dn.net/?f=+%5Cfrac%7B1%7D%7B+%28-+2+%2B++%5Csqrt%7B5%7D+%29%5E%7B2%7D+%7D+)
- 1 =
![\frac{1}{9 - 4 \sqrt{5} }](https://tex.z-dn.net/?f=+%5Cfrac%7B1%7D%7B9+-+4+%5Csqrt%7B5%7D+%7D+)
-1 =
![\frac{1 - 9 + 4 \sqrt{5} }{9 - 4 \sqrt{5} }](https://tex.z-dn.net/?f=+%5Cfrac%7B1+-+9+%2B+4+%5Csqrt%7B5%7D+%7D%7B9+-+4+%5Csqrt%7B5%7D+%7D+)
=
![\frac{- 8 + 4 \sqrt{5} }{9 - 4 \sqrt{5} }](https://tex.z-dn.net/?f=+++%5Cfrac%7B-+8+%2B+4+%5Csqrt%7B5%7D+%7D%7B9+-+4+%5Csqrt%7B5%7D+%7D+)
=
![\frac{(-8 + 4 \sqrt{5} ) * (9 + 4 \sqrt{5} )}{(9 - 4 \sqrt{5} ) * (9 + 4 \sqrt{5} )}](https://tex.z-dn.net/?f=+%5Cfrac%7B%28-8+%2B+4+%5Csqrt%7B5%7D+%29+%2A+%289+%2B+4+%5Csqrt%7B5%7D+%29%7D%7B%289+-+4+%5Csqrt%7B5%7D+%29+%2A+%289+%2B+4+%5Csqrt%7B5%7D+%29%7D+)
=
![\frac{-72 + 36 \sqrt{5} - 32 \sqrt{5} + 80 }{81 - 80}](https://tex.z-dn.net/?f=+%5Cfrac%7B-72+%2B+36+%5Csqrt%7B5%7D+-+32+%5Csqrt%7B5%7D+%2B+80++%7D%7B81+-+80%7D+)
= 8 + 4√5
tg²x = 8 + 4√5 = 4 (2 + √5)
tgx = 2√(2 + √5)
tgx = - 2√(2 + √5)
[ это типа корень, а то не могу напечатать)
[6*4-4[6+[6*9=2[6-4[6+3[6=[6
Ответ:
Убывает на промежутке [-3;0].