1. 4 sin^2x – 1 = 0
2. 2 sin^2x + sin x = 1
3. 2 sin^2x + 7 sin x = 4
4. cos x sin x + sin x = 0
5. 2 sin^2x - 1 = 0
Solve for x?
1 and 5 are both difference of two squares...
5) (2sinx)^2=1
sinx= +- sqrt(1/2)
x = +- 30 deg
others are quadratics, factor. On 4 dont forget the sollution sinx=0
I'll pick another, how about #3 ?
2 sin^2x + 7 sin x = 4
2 sin^2x + 7 sin x - 4 = 0
You should recognize this as a quadratic this factors like bob suggested,
e.g.
2y^2 + 7y - 4 = 0 , in your case y = sinx
(2y - 1)(y + 4) = 0
y = 1/2 or y = -4
then (2sinx - 1)(sinx + 4) = 0
sinx = 1/2 or sinx = -4, but you know sinx has to always fall between -1 and +1
sinx = 1/2
by the CAST rule, x must be in either quadrant I or II
x = 30° OR x = 180-30 or 150°
To solve these equations, we will use basic algebraic principles and trigonometric identities. Let's go through each equation step by step:
1. 4 sin^2x - 1 = 0
We start by adding 1 to both sides of the equation:
4 sin^2x = 1
Next, we divide both sides by 4:
sin^2x = 1/4
To solve for sin(x), we take the square root of both sides:
sin x = ± √(1/4)
This simplifies to:
sin x = ± 1/2
2. 2 sin^2x + sin x = 1
We can write this quadratic equation as:
2 sin^2x + sin x - 1 = 0
To solve this equation, we can factor it or use the quadratic formula.
Factoring the equation, we have:
(2sin x - 1)(sin x + 1) = 0
Setting each factor to zero, we get:
2sin x - 1 = 0 or sin x + 1 = 0
From the first equation, we solve for sin x:
2sin x = 1
sin x = 1/2
From the second equation, we solve for sin x:
sin x = -1
3. 2 sin^2x + 7 sin x = 4
Let's move all terms to one side of the equation:
2 sin^2x + 7 sin x - 4 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula.
Factoring the equation, we have:
(2sin x - 1)(sin x + 4) = 0
Setting each factor to zero, we get:
2 sin x - 1 = 0 or sin x + 4 = 0
From the first equation, we solve for sin x:
2sin x = 1
sin x = 1/2
From the second equation, we solve for sin x:
sin x = -4
4. cos x sin x + sin x = 0
We can factor out the common factor sin x:
sin x (cos x + 1) = 0
Setting each factor to zero, we get:
sin x = 0 or cos x + 1 = 0
From the first equation, we solve for sin x:
sin x = 0
From the second equation, we solve for cos x:
cos x = -1
5. 2 sin^2x - 1 = 0
Adding 1 to both sides of the equation, we have:
2 sin^2x = 1
Dividing both sides by 2, we get:
sin^2x = 1/2
Taking the square root of both sides, we have:
sin x = ± √(1/2)
Simplifying the square root, we get:
sin x = ± (√2/2)