Answer:
x = 1.249, 2.82, 4.391, 5.961
Explanation:
The given equation is
3csc²(x) - 6 = -8cot(x)
First, we will use the trigonometric identity
csc²(x) = 1 + cot²(x)
So, replacing csc²(x), we get:
3(1 + cot²(x)) - 6 = -8cot(x)
3(1) + 3cot²(x) - 6 = -8cot(x)
3 + 3cot²(x) - 6 = -8cot(x)
3cot²(x) - 3 = -8cot(x)
Then, add 8cot(x) to both sides, so
3cot²(x) - 3 + 8cot(x) = -8cot(x) + 8cot(x)
3cot²(x) + 8cot(x) - 3 = 0
Now, we can make a sustitution of a = cot(x), so the equation is
3a² + 8a - 3 = 0
(3a - 1)(a + 3) = 0
Solving for a, we get:
3a - 1 = 0
3a = 1
a = 1/3
or
a + 3 = 0
a = -3
Therefore, we need to find the solutions of the equations
cot(x) = 1/3
and
cot(x) = -3
Using the inverse trigonometric functions, we get:
[tex]\begin{gathered} x=\cot ^{-1}(\frac{1}{3})=1.249 \\ x=\cot ^{-1}(-3)=2.82 \end{gathered}[/tex]
Adding π = 3.14 radians to each answer, we also get
1.249 + 3.14 = 4.391
2.82 + 3.14 = 5.961
Therefore, the answers are
x = 1.249, 2.82, 4.391, 5.961
