# Can you solve 9sin^2theta - 6sintheta = 1 over [0 degree, 360 degrees) ?

This is just a quadratic equation, which we can easily solve:

Then:

Using quadratic formula:

For:

and

Approximate values:

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To solve the equation (9\sin^2(\theta) - 6\sin(\theta) = 1) over the interval ([0^\circ, 360^\circ)), we can first rewrite it as a quadratic equation in terms of (\sin(\theta)):

[9\sin^2(\theta) - 6\sin(\theta) - 1 = 0]

Now, let (u = \sin(\theta)), then the equation becomes:

[9u^2 - 6u - 1 = 0]

This is a quadratic equation, which we can solve using the quadratic formula:

[u = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}]

where (a = 9), (b = -6), and (c = -1).

Plugging in the values:

[u = \frac{{-(-6) \pm \sqrt{{(-6)^2 - 4(9)(-1)}}}}{{2(9)}}]

[u = \frac{{6 \pm \sqrt{{36 + 36}}}}{{18}}]

[u = \frac{{6 \pm \sqrt{72}}}{{18}}]

[u = \frac{{6 \pm 6\sqrt{2}}}{{18}}]

[u = \frac{{1 \pm \sqrt{2}}}{{3}}]

Now, we have two possible solutions for (u):

- (u = \frac{{1 + \sqrt{2}}}{{3}})
- (u = \frac{{1 - \sqrt{2}}}{{3}})

Since (\sin(\theta) = u), we need to find the corresponding values of (\theta) within the given interval ([0^\circ, 360^\circ)).

For (u = \frac{{1 + \sqrt{2}}}{{3}}): [\theta = \sin^{-1}\left(\frac{{1 + \sqrt{2}}}{{3}}\right)]

For (u = \frac{{1 - \sqrt{2}}}{{3}}): [\theta = \sin^{-1}\left(\frac{{1 - \sqrt{2}}}{{3}}\right)]

Once we have obtained the values of (\theta), we need to ensure they fall within the interval ([0^\circ, 360^\circ)).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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