# Trig Equation

3 cos x + 4 = 0
cos x = -4/3

How do I solve this? I can't take the arcsine...

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sin (3x - 40) = 0

Is the general solution 73.3 + 120k | k E I?

robphy
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Gold Member
cscott said:
3 cos x + 4 = 0
cos x = -4/3

How do I solve this? I can't take the arcsine...

Did you try to use complex numbers or hyperbolic trig functions?

hotvette
Homework Helper
How do I solve this? I can't take the arcsine...

Could try graphing it and see where it crosses zero.

It's not going to cross the x-axis. You'll have to encorporte complex numbers like robphy said.

We don't cover complex numbers until the end of the year. Does this mean I say it has no solution?

Why can you not take the cosine inverse of x?

JoshHolloway said:
Why can you not take the cosine inverse of x?

4/3 > 1 so you get a domain error... or at least that's how I see it.

cscott said:
3 cos x + 4 = 0
cos x = -4/3
Unless I'm missing something, your given value for cos x is incorrect. Any value of the cosine function is in a way "stuck" between -1 and 1.

Use a calculator and take the cosine of any degree/radian measure you wish, it will always be between -1 and 1.

Last edited:
benjamincarson said:
Unless I'm missing something, you're given value for cos x is incorrect. Any values of the cosine function is in a way "stuck" between -1 and 1.

Use a calculator and take the cosine of any degree/radian measure you wish, it will always be between -1 and 1.

I got -4/3 from rearranging the equation. The fact that 4/3 > 1 is my problem.

Is this an equation that needs solving, or an identity that needs verifying?

"Solve for all possible values of x."

no solution

Thank you.

I have another problem

sin^2 x = 3/4

for one of the possible answers I get $\frac{\pi}{3} + 2\pi k$ but my book says it should be $\pi$ instead of $2\pi$ for the period. How come?

The sin value is squared, so the negative values square out to give a solution too.

whozum said:
The sin value is squared, so the negative values square out to give a solution too.

Yes I know. Let me clarify: $\frac{\pi}{3} + 2\pi k$ is one possible solution but my book says it should be $\frac{\pi}{3} + \pi k$ and I don't know why.

robphy
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Gold Member
$$e^{i\pi}=-1$$

robphy said:
$$e^{i\pi}=-1$$

I haven't done complex numbers so I don't really know the significance of that expression.

TD
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cscott said:
I have another problem

sin^2 x = 3/4

for one of the possible answers I get $\frac{\pi}{3} + 2\pi k$ but my book says it should be $\pi$ instead of $2\pi$ for the period. How come?
$$\sin ^2 x = \frac{3} {4} \Leftrightarrow \sin x = \pm \sqrt {\frac{3} {4}} = \pm \frac{{\sqrt 3 }} {2}$$

Now determine the solution in both cases (the + and the - case).

TD said:
$$\sin ^2 x = \frac{3} {4} \Leftrightarrow \sin x = \pm \sqrt {\frac{3} {4}} = \pm \frac{{\sqrt 3 }} {2}$$

Now determine the solution in both cases (the + and the - case).

I would say

$$\frac{\pi}{3} + 2\pi k | k \epsilon I, \frac{4\pi}{3} + 2\pi k | k \epsilon I$$

but my book says

$$\frac{\pi}{3} + \pi k | k \epsilon I, \frac{4\pi}{3} + \pi k | k \epsilon I$$

and I don't know why

TD
Homework Helper
Also remember the supplementary solution, $\sin \left( \alpha \right) = \sin \left( {\pi - \alpha } \right)$. You immediately have this if you don't multiply with 2k*pi but just with k*pi.

TD said:
Also remember the supplementary solution, $\sin \left( \alpha \right) = \sin \left( {\pi - \alpha } \right)$. You immediately have this if you don't multiply with 2k*pi but just with k*pi.

Ohhh, I see. Thanks a lot!

By the way, how do you read the <=> arrow you used and how is it different than just =>?

TD
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No problem!

Diane_
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The "=>" is an implication arrow - it means that, if the left-hand side is true, the right-hand side is also true.

The "<=>" is basically an implication arrow going both ways, and is generally read "if and only if". It's a much more powerful condition, meaning that if either side is true, the other side is also true. Outside of definitions and relatively trivial things, you don't get one of those very often. When you do, it's worth paying attention to.

Wait! Let me just check my understanding: the solution to the equation I posted can be either the one including pi/3 or the one inlcluding 4pi/3 and each solution has to encorporate the negative and positive sqrt(3)/2?

I say this because pi/3 + pi gives -sqrt(3)/2 but I got the pi/3 in the first place using sin x = +sqrt(3)/2

Is this correct? It doesn't seem like it...