solve the equation

2sin^2X - 3sinX= -1

2 sin^2 x - 3sinx + 1 = 0

(2sinx - 1)(sinx -1) = 0
sinx = 1/2 or sinx = 1

x = 30 degrees or 150 degrees or x = 90 degrees

in radians x = π/6, 5π/6, π/2

Why did the sine wave bring a calculator to the party? Because it wanted to solve all the equations! Let's simplify and solve this equation step by step:

2sin^2X - 3sinX = -1

First, let's rearrange the equation a bit:

2sin^2X - 3sinX + 1 = 0

It looks like a quadratic equation in terms of sinX. To solve it, we can use substitution. Let's substitute sinX with a variable, say, "t":

2t^2 - 3t + 1 = 0

Now, we can factor the quadratic equation or use the quadratic formula. In this case, factoring is a bit tricky, so let's go with the quadratic formula:

t = (-(-3) ± √((-3)^2 - 4(2)(1))) / (2 * 2)

Simplifying this:

t = (3 ± √(9 - 8)) / 4

t = (3 ± √1) / 4

t = (3 ± 1) / 4

Now we have two possible values for t:

t₁ = (3 + 1) / 4 = 4 / 4 = 1
t₂ = (3 - 1) / 4 = 2 / 4 = 1/2

Remember, t is just a substitution for sinX. So, we can equate t to sinX:

sinX = 1 or sinX = 1/2

Now we need to find the angles where sinX equals these values. For sinX = 1:

X = π/2 + nπ, where n is an integer.

And for sinX = 1/2:

X = π/6 + 2nπ or X = 5π/6 + 2nπ, where n is an integer.

So, the solutions to the equation are:

X = π/2 + nπ, π/6 + 2nπ, or 5π/6 + 2nπ, where n is an integer.

To solve the equation 2sin^2X - 3sinX = -1, we can rearrange it into a quadratic equation in terms of sinX. Let's go step-by-step.

1. Start with the given equation: 2sin^2X - 3sinX = -1.

2. Rewrite sin^2X as (sinX)^2: 2(sinX)^2 - 3sinX = -1.

3. Move all terms to one side to obtain a quadratic equation: 2(sinX)^2 - 3sinX + 1 = 0.

Now, we can solve this quadratic equation for sinX.

4. Factor the quadratic equation: (2sinX - 1)(sinX - 1) = 0.

5. Set each factor equal to zero and solve for sinX:

a) 2sinX - 1 = 0:
2sinX = 1
sinX = 1/2

b) sinX - 1 = 0:
sinX = 1

Therefore, the two solutions for sinX are sinX = 1/2 and sinX = 1.

To solve the equation 2sin^2X - 3sinX = -1, we can rewrite it as a quadratic equation by moving all terms to one side:

2sin^2X - 3sinX + 1 = 0

Let's denote sinX as a variable, say t. Then the equation becomes:

2t^2 - 3t + 1 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, factoring is the most convenient method:

2t^2 - 3t + 1 = 0

To factorize this equation, we need to find two numbers whose product is 2 * 1 = 2 and whose sum is -3. These numbers are -2 and -1.

Rewriting the equation:

2t^2 - 2t - t + 1 = 0

Factor by grouping:

(2t^2 - 2t) - (t - 1) = 0

2t(t - 1) - (t - 1) = 0

(t - 1)(2t - 1) = 0

Now, using the zero-product property, we set each factor equal to zero:

t - 1 = 0 or 2t - 1 = 0

Solving these equations:

t = 1 or t = 1/2

Since we substituted t as sinX, we can conclude:

sinX = 1 or sinX = 1/2

To find the values of X, we look for the angles whose sine values are 1 and 1/2. We can use the inverse sine function (sin^(-1)) or refer to the unit circle or trigonometric table.

Using the inverse sine function:

sinX = 1
=> X = sin^(-1)(1)
=> X = π/2 + 2πk, where k is an integer.

sinX = 1/2
=> X = sin^(-1)(1/2)
=> X = π/6 + 2πk or X = 5π/6 + 2πk, where k is an integer.

Therefore, X can be any of the following values:

X = π/2 + 2πk or X = π/6 + 2πk or X = 5π/6 + 2πk,

where k is an integer.