What's the general solution for 3sin^2x+sinxcosx=2
3sin^2(x)+sin(x)cos(x)=2sin^2(x)+2cos^2(x)
sin^2(x)+sin(x)cos(x)-2cos^2(x)=0
Divide by cos^2(x)
tan^2(x)+tan(x)-2=0
(tan(x)+2)(tan(x)-1)=0
tan(x)=-2 => x=-Arctan(2)+pi*n
tan(x)=1 => x=pi/4+pi*n
Thnx hey next culd plz explain the steps step by step thnx onc agn
To find the general solution for the equation 3sin^2x + sinxcosx = 2, we can follow these steps:
Step 1: Rearrange the equation.
Start by subtracting 2 from both sides of the equation to get:
3sin^2x + sinxcosx - 2 = 0
Step 2: Factor the equation.
Now we need to factor the left side of the equation. Let's rewrite sinxcosx as 2sinxcosx/2:
3sin^2x + (2sinxcosx - 2) = 0
Step 3: Simplify the equation.
Factor out sinx from the first term:
sinx (3sinx + 2cosx - 2) = 0
Step 4: Solve for sinx and cosx separately.
Now we have two possible scenarios:
Case 1: sinx = 0
If sinx = 0, then x must be a multiple of π (pi):
x = nπ, where n is an integer.
Case 2: 3sinx + 2cosx - 2 = 0
We need to solve this equation separately to find the values of x.
To solve this equation, we can use the fact that cosx = √(1 - sin^2x). Substituting that into the equation, we get:
3sinx + 2√(1 - sin^2x) - 2 = 0
Simplify the equation by squaring both sides to eliminate the square root:
(3sinx - 2)^2 = 4(1 - sin^2x)
Expand and solve for sinx:
9sin^2x - 12sinx + 4 = 4 - 4sin^2x
13sin^2x - 12sinx = 0
Factor out sinx:
sinx(13sinx - 12) = 0
From here, we have two possibilities for sinx:
a) sinx = 0
If sinx = 0, then x must be a multiple of π:
x = nπ, where n is an integer.
b) 13sinx - 12 = 0
Solve this equation separately:
13sinx = 12
sinx = 12/13
The range of the sine function is -1 to 1, so for sinx to be equal to 12/13, it must be approximately 0.923.
To find the corresponding values of x, we need to use inverse sine (also known as arcsine) function:
x = arcsin(12/13) ≈ 0.950
In this case, x is approximately 0.950.
Therefore, the general solution for the equation 3sin^2x + sinxcosx = 2 is:
x = nπ, where n is an integer,
and x ≈ 0.950.