# How to solve for theta for this trig question?

## Homework Statement

cos(theta) = -3/4, pi <= theta <= 2pi

## The Attempt at a Solution

I forgot how to do this. How do I use the special triangles to do this question? Do I need to square the sqrt(3) / 2 one or something? Thanks.

SteamKing
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Try the Pythagorean theorem.

So I use Pythagorean to get the opposite length? (sqrt(7) / 4) ?

haruspex
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So I use Pythagorean to get the opposite length? (sqrt(7) / 4) ?
It depends what you're trying to find. That will help if you want some other trig function of theta, but not much use if you want theta itself.
If it's a numerical value for theta that you want, at some point you will have to use a calculator or whatever to compute an arccos. But the more interesting aspect of the question is getting an answer in the right quadrant. Do you know the relationships between cos(x), cos(-x), cos(pi+x), etc?

SteamKing
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Not all trig functions require the use of a calculator to evaluate.

haruspex
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Not all trig functions require the use of a calculator to evaluate.
True, but certainly to find arccos in this case.

SteamKing
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Not really. If you know basic trig values for 45-45-90 and 30-60-90 triangles, solving this problem should be possible without a calculator.

haruspex
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Not really. If you know basic trig values for 45-45-90 and 30-60-90 triangles, solving this problem should be possible without a calculator.
For |cos(theta)| = 3/4? Are you sure about that? If 3/5 I'd agree.

SammyS
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I'm pretty sure that zeion has left the building ...

At any rate, given a value for cos(θ), for π ≤ θ ≤ 2π, one can determine in which quadrant θ lies, and thus determine the values of all the other trig functions at angle θ.

Of course, we're still not sure about just what zeion was supposed to do in regards to this problem.

I'm pretty sure that zeion has left the building ...

At any rate, given a value for cos(θ), for π ≤ θ ≤ 2π, one can determine in which quadrant θ lies, and thus determine the values of all the other trig functions at angle θ.

Of course, we're still not sure about just what zeion was supposed to do in regards to this problem.

I'm supposed to

i) Sketch the angle and state the related acute angle
ii) Determine the exact value of each trigonometric ratio

The previous 3 questions I could do because they were all values I could find on the special triangle so they were easy. But I'm not sure how to relate 3/4 to the special triangles?

HallsofIvy
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Since $\theta$ lies between $\pi$ and $2\pi$, and $cos(\theta)$ is negative, $\theta$ is in the fourth quadrant. Use a calculator to find $cos^{-1}$ of -4/3 and add $\pi$. Equivalently, find $cos^{1}$ of 3/4 and subtract from $2\pi$.

SammyS
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Since $\theta$ lies between $\pi$ and $2\pi$, and $cos(\theta)$ is negative, $\theta$ is in the fourth quadrant. Use a calculator to find $cos^{-1}$ of -4/3 and add $\pi$. Equivalently, find $cos^{1}$ of 3/4 and subtract from $2\pi$.
That should be, θ is in the third quadrant.

cos(θ) is positive for θ in the first & fourth quadrants.

cos(θ) is negative for θ in the second & third quadrants.

Last edited:
haruspex