MHB Solve sin(x)-1=cos(x): Step-by-Step Guide

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The discussion focuses on solving the equation sin(x) - 1 = cos(x) by using trigonometric identities. Participants suggest squaring both sides of the equation and applying the identity cos²(x) = 1 - sin²(x) to eliminate cosines. One user expresses confusion about how to apply the Pythagorean identity effectively. The conversation emphasizes the importance of understanding the relevant trigonometric formulas and identities for simplifying and solving the equation. Overall, the solution process involves transforming the equation to a purely sine-based form for easier resolution.
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So the problem is sin(x)-1=cos(x) and I don't know how to do this one.
 
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Which formulas do you have available to add/subtract sine and/or cosine?
 
Klaas van Aarsen said:
Which formulas do you have available to add/subtract sine and/or cosine?

I don't know what you mean.
 
Elissa89 said:
I don't know what you mean.

Well... erm... I'm a bit at a loss of the formulas you can use or not...

See for instance the wiki page of Trigonometric Identities for a list of such formulas...
This may be a bit overwhelming, but one of the formulas in that page is:
$$a\sin x+b\cos x=c\sin(x+\varphi)$$
where $c = \sqrt{a^2 + b^2}$ and $\varphi = \operatorname{atan2} \left( b, a \right)$.

To be fair, there's a good chance that you haven't been taught this formula... but what have you been taught?
Or what are you otherwise supposed to know and be able to apply?
 
I don't think it has to be too complicated. :)

Square both sides of the given equation (post back if you don't know how to do that) and use the identity $\cos^2(x)=1-\sin^2(x)$. Simplify and solve the resulting equation, then check your results with the given equation.
 
greg1313 said:
I don't think it has to be too complicated. :)

Square both sides of the given equation (post back if you don't know how to do that) and use the identity $\cos^2(x)=1-\sin^2(x)$. Simplify and solve the resulting equation, then check your results with the given equation.

I did that but I'm still lost. My professor emailed me back, said to square both sides the squared cos can be turned into sines using the pythagorean theorem identity. Which doesn't make sense to me because the pythagorean theorem identity still has cosines in it?

Sorry I'm replying so late.
 
Elissa89 said:
I did that but I'm still lost. My professor emailed me back, said to square both sides the squared cos can be turned into sines using the pythagorean theorem identity. Which doesn't make sense to me because the pythagorean theorem identity still has cosines in it?

Sorry I'm replying so late.

Let's start with squaring both sides as your professor said:
$$\sin(x)-1=\cos(x) \\
(\sin(x)-1)^2=\cos^2(x) \\
\sin^2 x - 2\sin x + 1 = \cos^2x
$$
Now we can turn the $\cos^2x$ into sines by using $\cos^2x=1-\sin^2x$, as greg1313 suggested, can't we?
Then there will be no cosines left. (Thinking)
 
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