Given TaNx = 2, find (1) (2sinx - cos x) / (cosx + SiNx) (2) The value of sin ^ 2 x + 2sinx cos x | Math enthusiast | Mathematical art

Given TaNx = 2, find (1) (2sinx - cos x) / (cosx + SiNx) (2) The value of sin ^ 2 x + 2sinx cos x

Because TaNx = SiNx / cosx = 2, SiNx = 2cosx
(1) The original formula = [2 (2cosx) - cosx)] / (cosx + SiNx) = (4cosx cosx) / (cosx + 2cosx) = 3cosx / 3cosx = 1
(2) The original formula = (sin? X + 2sinxcosx) / 1 = (sin? X + 2sinxcosx) / (sin? X + cos? X) = (4cos? X + 2 * 2cos? X) / (4cos? X + cos? X)
=(8cos²x)/(5cos²x)=8/5

If TaNx / tanx-1 = - 1, find the value of sin (π / 2 + x) cos (3 π - x)

tanx/tanx-1=-1
tanx=-tanx+1
tanx=1/2
sinx/cosx=1/2
2sinx = cosx squared
4sin^2x=4(1-cos^2x)=4-4cos^2x=cos^2x
cos^2x=4/5
sin(π/2+x)cos(3π-x)
=cosx*(-cosx)
=-cos^2x
=-4/5 .zhu,ni haohao xuexi ,shangahnag

Given the function f (x) = tanxtan2x / tan2x TaNx + root 3 (sin ^ 2x cos ^ 2x), find the definition domain sum of function f (x) Known function f (x) = tanxtan2x / tan2x TaNx + radical 3 (sin ^ 2x cos ^ 2x) (1) Find the definition domain and maximum value of function f (x) (2) given the inner angles a, B, C of △ ABC, the opposite edges are a, B, C respectively. If B = 2A, find the value range of F (a)

Known function f (x) = tanxtan2x / tan2x TaNx + radical 3 (sin ^ 2x cos ^ 2x)
(1) Find the definition domain and maximum value of function f (x) (2) given the inner angles a, B, C of △ ABC, the opposite edges are a, B, C respectively. If B = 2A, find the value range of F (a)
(1) Analysis: ∵ f (x) = Tan (x) Tan (2x) / (Tan (2x) - Tan (x)) + √ 3 ((sin (x)) ^ 2 - (COS (x) ^ 2))
=1/(cot(x)-cot(2x))-√3cos(2x) =sin(2x)-√3cos(2x)=2sin(2x-π/3)
∴f(x)=2sin(2x-π/3)
The definition domain is r, the value domain is [- 2,2], and the maximum value is 2
(2) Analysis: ∵ ∵ the inner angles of ABC a, B, C are opposite to the edges of a, B, C, B = 2A
From the sine theorem, Sina = 1 / 2 * SINB is obtained
Let g (x) = arcsin (1 / 2 * SiNx) x ∈ (0, π)
Let g '(x) = 1 / √ [1 - (1 / 2 * SiNx) ^ 2] * (1 / 2 * SiNx)' = cosx / √ (4 - (SiNx) ^ 2) = 0 = = > x = π / 2
When x = π / 2, the maximum value of G (x) is 1 / 2
The maximum value of sina is 1 / 2, that is, a ∈ (0, π / 6]
∴f(A)=2sin(2A-π/3)==>f(A)∈(-√3,0]

Known function f (x) = tanxtan2x / (tan2x TaNx) + √ 3 (sin2 ^ x-cos2 ^ x)

Cos (x + π) = 3 / 5, and X is the angle of the third quadrant. Find the value of (sin2x + 2Sin ^ 2x) / (1 + TaNx) emergency

Since x is the angle of the third quadrant, so x + π is the angle of the first quadrant, so it can be expressed as that x + π belongs to (2k π, 2K π + π / 2). If an acute angle a satisfies cosa = 3 / 5, then x + π can be expressed as 2K π + A, x = 2K π - π + a
Firstly, TaNx = tan2k π - π + a = Tana = 4 / 3 is obtained
sinx=sin2kπ-π+a=-sina=-4/5
2x=4kπ-2π+2a
So sin2x = sin4k π - 2 π + 2A = sin2a = 2sinacosa = 2 * 4 / 5 * 3 / 5 = 24 / 25
So:
(sin2x+2sin^2x)/（1+tanx）
=（24/25+2*（-4/5）^2)/(1+4/3)
=24/25

Because 'square of X' = x, and it makes sense for X to take any real number, the condition that makes the fraction 'square of X' meaningful is that x is any real number. Do you think that's right? Why?

incorrect
X is not equal to 0
When x is equal to 0, the fraction 'x squared by X' has no meaning

If any value of X is always meaningful, then the value range of M is------

M>1

The square of X is X-1. When x is the value, the fraction is meaningless When x and y are the values, the fraction x + y + 1 / x + 2 is zero Observe the following regular numbers: one-fifth, thirty-seven, fifty-nine, seventy-one, nine Please use a fraction to express the nth number A goes AKM / h, B km / h. If the distance from the starting point to the end point is MKM, and the speed of a is higher than that of B, then a will arrive at the destination a few hours earlier than B

If the X-1 / x? Fraction is meaningless, then x? = 0  x = 0
x+2/x+y+1=0
∴ x+2=0 x+y+1≠0
The solution is x = - 2, y ≠ 1
The nth number is 2N-1 / 2n + 1

What condition does x satisfy in Square-1 of fraction (x + 2) (x-1) is meaningful

When x ≠ - 2 and X ≠ 1, the fraction is meaningful

Fraction x − y X2 + Y2 is meaningful if () A. x≠0 B. y≠0 C. X ≠ 0 or Y ≠ 0 D. X ≠ 0 and Y ≠ 0

As long as X and y are not both 0, the denominator x2 + Y2 must not be equal to 0
Therefore, C

Given TaNx = 2, find (1) (2sinx - cos x) / (cosx + SiNx) (2) The value of sin ^ 2 x + 2sinx cos x