SOLUTION: Use the Quadratic Formula to solve the equation in the interval [0, 2π). Then use a graphing utility to approximate the angle x. (Enter your answers as a comma-separated list.

Algebra ->  Trigonometry-basics -> SOLUTION: Use the Quadratic Formula to solve the equation in the interval [0, 2π). Then use a graphing utility to approximate the angle x. (Enter your answers as a comma-separated list.      Log On


   

Question 1038839: Use the Quadratic Formula to solve the equation in the interval [0, 2π). Then use a graphing utility to approximate the angle x. (Enter your answers as a comma-separated list. Round each answer to four decimal places.)
15 sin^2x - 22 sinx + 8 = 0
Answer by ikleyn(52794) About Me  (Show Source):
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Use the Quadratic Formula to solve the equation in the interval [0, 2pi). Then use a graphing utility to approximate the angle x.
15 sin^2x - 22 sinx + 8 = 0
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let u = sin(x) be the new variable.
Then your equation takes the form

15u%5E2+-+22u+%2B+8 = 0.

Solve it for  "u"  using the quadratic formula:

u%5B1%2C2%5D = %2822+%2B-+sqrt%2822%5E2-4%2A15%2A8%29%29%2F30 = %2822+%2B-+sqrt%284%29%29%2F30 = %2822+%2B-+2%29%2F30,

u%5B1%5D = 24%2F30 = 4%2F5  and  u%5B2%5D = 20%2F30 = 2%2F3.

Then you have two equations:

1) sin(x) = 4%2F5 --->  x = arcsin(4/5),  and

2) sin(x) = 2%2F3 --->  x = arcsin(2/3).

Next use your calculator.


The way of introducing new variable is the standard method for solving equations like this one.