MHB Finding Period, Amplitude, and Phase Shift in Trig Expressions

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The discussion centers on solving the equation cot(x) + 2csc(x) - 3 = 0 using various methods, including manipulating trigonometric identities and quadratic equations. Participants explore the relationship between sine and cosine, expressing sine in terms of cosine and simplifying the equation. They also discuss the challenges of finding precise values for θ, suggesting approximate solutions instead. Additionally, there is curiosity about determining the period, amplitude, and phase shift of trigonometric expressions, indicating a need for graphical analysis or further exploration of trigonometric properties. Overall, the conversation highlights different approaches to solving trigonometric equations and understanding their characteristics.
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first not completely convinced that these answers are correct
I assume the x-axis is in radians.

also if $$2csc(\theta) = 5.633$$ but it does not cross the $$x$$ axis why then does it work in the equation

also, sure there is a better ways to solve this. but using the triangle was the just one way I could think of.
 
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Re: cot(x)+2csc(x)-3=0

I would write the equation using sine and cosine, simplify as much as possible, then express sine as 1 minus cosine squared and solve the quadratic equation to find cosine.

I suspect θ cannot be found precisely, only approximately.
 
Re: cot(x)+2csc(x)-3=0

Evgeny.Makarov said:
I would write the equation using sine and cosine, simplify as much as possible, then express sine as 1 minus cosine squared and solve the quadratic equation to find cosine.

isn't $\displaystyle\sin^2(\theta) = 1-\cos^2(\theta)$

then $\displaystyle\sin(\theta) = \sqrt{1-\cos^2(\theta)}$

if so then

$
\displaystyle
\frac{\cos{\theta}}{\sqrt{1-\cos^2{\theta}}}
+
\frac{2}{\sqrt{1-\cos^2{\theta}}}
=
\frac{\cos{\theta}+2}{\sqrt{1-\cos^2{\theta}}}
=3
$

this still looks a little daunting
 
Re: cot(x)+2csc(x)-3=0

karush said:
isn't $\displaystyle\sin^2(\theta) = 1-\cos^2(\theta)$
Yes, sorry.

karush said:
$
\displaystyle
\frac{\cos{\theta}}{\sqrt{1-\cos^2{\theta}}}
+
\frac{2}{\sqrt{1-\cos^2{\theta}}}
=
\frac{\cos{\theta}+2}{\sqrt{1-\cos^2{\theta}}}
=3
$

this still looks a little daunting
Don't be daunted. :) Multiply both sides by the denominator, take the square of both sides, denote $\cos\theta$ by $x$ and solve the resulting quadratic equation for $x$.
 
Re: cot(x)+2csc(x)-3=0

Another approach would be to multiply through by $\sin(x)$ and arrange as:

$$3\sin(x)-\cos(x)=2$$

Using a linear combination identity, we may write:

$$\sqrt{10}\sin\left(x-\tan^{-1}\left(\frac{1}{3} \right) \right)=2$$

$$\sin\left(x-\tan^{-1}\left(\frac{1}{3} \right) \right)=\frac{2}{\sqrt{10}}=\frac{\sqrt{10}}{5}$$

Now, using the identity $\sin(\pi-\theta)=\sin(\theta)$, can you find the solutions?
 
Re: cot(x)+2csc(x)-3=0

$$10cos^2\theta+4cos\theta-5=0$$

so from quadratic formula
$$
\cos\theta
= \frac{-2\pm 3 \sqrt{6}}{10}
=-0.934847 \text{ or }0.534847
$$

so

$$\cos^{-1}(-0.934847)=2.77862 \text{ rad} =159.20^o$$

and

$$\cos^{-1}(0.534847)=1.00647 \text{ rad} =57.67^o$$

check

$$\cot(2.77862)+2\csc(2.77862)-3=0$$
$$\cot(1.00647)+2\csc(1.00647)-3=0$$
 
Last edited:
Re: cot(x)+2csc(x)-3=0

Using the method I gave, you find:

$$x=\sin^{-1}\left(\frac{\sqrt{10}}{5} \right)+\tan^{-1}\left(\frac{1}{3} \right)\approx1.00646975739893$$

$$x=\pi-\sin^{-1}\left(\frac{\sqrt{10}}{5} \right)+\tan^{-1}\left(\frac{1}{3} \right)\approx2.77862400498415$$
 
Re: cot(x)+2csc(x)-3=0

karush said:
$$10cos^2\theta+4cos\theta-5=0$$

so from quadratic formula
$$
\cos\theta
= \frac{-2\pm\sqrt{6}}{10}
=-0.934847 \text{ or }0.534847
$$
This should be
\[
\cos\theta
= \frac{-2\pm3\sqrt{6}}{10}
\]
but the numerical answers are correct.
 
Re: cot(x)+2csc(x)-3=0

thanks for the solutions. learned a lot from this one:cool:

one question I have these expressions with multiple trig functions in them... how do you find out the period, amplitude and phase shift on these. or do you just look at a plot..:confused:
 
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