Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
5 sin^2(x)- 21 sin(x)+ 4 =0
factor it
(5sinx - 1)(sinx - 4) = 0
sinx = 1/5 or sinx = 4
the latter is not possible, so
sinx = 1/5
x must be in I or II
x = 11.537° or 168.463°
if you need the answer in radians , set your calculator to RAD (DRG button) and repeat the inverse sin operation.
Thank you so much!
To solve the given equation, we can use the quadratic formula. However, the equation is in terms of sin(x), so we will replace sin(x) with a variable, let's say u.
Let's rewrite the equation using u:
5u^2 - 21u + 4 = 0
Now, we can solve this quadratic equation using the quadratic formula:
u = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 5, b = -21, and c = 4.
Plugging in these values, we get:
u = (21 ± √((-21)^2 - 4 * 5 * 4)) / (2 * 5)
u = (21 ± √(441 - 80)) / 10
u = (21 ± √361) / 10
u = (21 ± 19) / 10
Simplifying further:
u = 2 or u = 4
Now, we need to find the values of x.
Since u = sin(x), we can use the inverse sine (or arcsine) function to find the values of x.
For u = 2:
x = sin^(-1)(2)
Since the sine function only takes values between -1 and 1, there is no solution for x when u = 2.
For u = 4:
x = sin^(-1)(4)
Since the sine function only takes values between -1 and 1, there is no solution for x when u = 4.
Therefore, the equation has NO SOLUTION.