Find all solutions if 0 degrees is less than or equal to x and x is less than 360 degrees.
2 sin^2(x)+ sin(x)=1
hint:
treat it like a quadratic.
Some students like to change it using something like this:
let sin(x) = y , then
2y^2 + y - 1 = 0
this one factors nicely, once you found y, sub that back into sinx
To solve the equation 2sin^2(x) + sin(x) = 1, we can rearrange it to the quadratic form. Let's substitute sin(x) with a variable, let's say y.
So the equation becomes:
2y^2 + y = 1
Now, let's rewrite it in standard quadratic form:
2y^2 + y - 1 = 0
To solve this quadratic equation, we can use factoring or the quadratic formula. Let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 2, b = 1, and c = -1. Plugging these values into the formula, we have:
y = (-1 ± √(1^2 - 4(2)(-1))) / (2(2))
= (-1 ± √(1 + 8)) / 4
= (-1 ± √9) / 4
= (-1 ± 3) / 4
We get two possible values for y:
1. y = (-1 + 3) / 4 = 2 / 4 = 1/2
2. y = (-1 - 3) / 4 = -4 / 4 = -1
Now, let's substitute y back with sin(x):
1. sin(x) = 1/2
To find the solutions for sin(x) = 1/2, we can recall the trigonometric values for 30 degrees and 150 degrees in the unit circle. So the solutions for this equation are:
x = 30 degrees and x = 150 degrees
2. sin(x) = -1
To find the solutions for sin(x) = -1, we can recall the trigonometric value for 270 degrees in the unit circle. So the solution for this equation is:
x = 270 degrees
Therefore, the solutions for the equation 2sin^2(x) + sin(x) = 1, where 0 degrees ≤ x < 360 degrees, are:
x = 30 degrees, 150 degrees, and 270 degrees.
To solve the equation 2sin^2(x) + sin(x) = 1, you can use the substitution rule.
Let's substitute sin(x) with a variable, say a. The equation becomes:
2a^2 + a = 1
Now, rearrange the equation to get a quadratic equation:
2a^2 + a - 1 = 0
To solve for a, you can use the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 1, and c = -1. Plugging the values into the quadratic formula:
a = (-(1) ± √((1)^2 - 4(2)(-1))) / (2(2))
Simplifying further:
a = (-1 ± √(1 + 8)) / 4
a = (-1 ± √9) / 4
Now, we have two possible solutions for a:
a = (-1 + 3) / 4 = 2 / 4 = 0.5
a = (-1 - 3) / 4 = -4 / 4 = -1
Since a is equal to sin(x), we can find the values of x by taking the inverse sine (or arcsine) of a.
For a = 0.5:
x = sin^(-1)(0.5)
Using a calculator, you can find that sin^(-1)(0.5) is approximately 30 degrees (or π/6 radians).
For a = -1:
x = sin^(-1)(-1)
Using a calculator, you can find that sin^(-1)(-1) is approximately 270 degrees (or 3π/2 radians).
So, the solutions for the given equation in the specified range are x = 30 degrees and x = 270 degrees.