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Math Help - Proof sin(θ)=cos(θ) when...

  1. #1
    Junior Member maxpancho's Avatar
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    Question Proof sin(θ)=cos(θ) when...

    Proof sin(θ)=cos(θ) when...-problem185.png

    I know that adjacent side should be equal to opposite which happens at $\dfrac{\pi}{4}$ and $\dfrac{5\pi}{4}$ (plus coterminal angles); the quadrants are 1 and 3 because signs of sin and cos thus coincide. I'm not sure that proves it though? How do I go about this? I don't think the proof should be very elaborate, though, in my case.
    Last edited by maxpancho; July 10th 2014 at 04:46 PM.
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    Re: Proof sin(θ)=cos(θ) when...

    It is virtually impossible to give help on such questions because we do not know what you are permitted to use in proofs nor how formal your proofs must be. With that said, this strikes me as correct but perhaps too informal.

    I suspect what is wanted is some kind of proof based on the unit circle and basic plane geometry that in fact

    $0 \le \theta \le 2 \pi\ and\ sin( \theta ) = cos( \theta ) \implies \theta = \dfrac{ \pi }{4}\ or\ \theta = \dfrac{5 \pi }{4}.$

    What also may be wanted is a more formal way of expressing your "plus coterminal angles."

    I emphasize that your logic is fine, but it is very thinly developed.
    Last edited by JeffM; July 10th 2014 at 05:39 PM.
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    Re: Proof sin(θ)=cos(θ) when...

    A more formal way might be like this:

    $\displaystyle \begin{align*} \sin{(\theta)} &= \cos{(\theta)} \\ \frac{\sin{(\theta)}}{\cos{(\theta)}} &= 1 \\ \tan{(\theta)} &= 1 \end{align*}$

    which should be an easy equation to solve.

    Note: You can only ever divide by a nonzero quantity, so that means you have to check that values of $\displaystyle \begin{align*} \theta \end{align*}$ which make your divisor of $\displaystyle \begin{align*} \cos{(\theta)} \end{align*}$ to be equal to 0 aren't also solutions of the equation.
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    Junior Member maxpancho's Avatar
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    Re: Proof sin(θ)=cos(θ) when...

    Prove It, solve it by taking inverse tangent on calculator?
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    Junior Member maxpancho's Avatar
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    Re: Proof sin(θ)=cos(θ) when...

    JeffM, "plus coterminal angles" is easy to denote, they've shown it in the book as well: $+2\pi n$ where $n \in \mathbb{Z}$. Or in this case it's just: $\dfrac{\pi}{4}+\pi n$ where $n \in \mathbb{Z}$.
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    Re: Proof sin(θ)=cos(θ) when...

    By circular functions, I assume you mean x = r\cos \theta, y = r\sin \theta? So, \cos \theta = \sin \theta when \dfrac{x}{r} = \dfrac{y}{r}, or more simply when x=y. So, the circle interpretation tells you that the solutions to \cos \theta = \sin \theta intersect the line y=x. Those give the angles \dfrac{\pi}{4}, \dfrac{5\pi}{4} as desired.
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    Junior Member maxpancho's Avatar
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    Re: Proof sin(θ)=cos(θ) when...

    A trig function is $\sin(x)$ or $\cos(x)$ where x corresponds to radian measure on a circle. And a circlular function is $\sin(\theta)$ or $\cos(\theta)$.

    As I understood.
    Last edited by maxpancho; July 11th 2014 at 04:50 AM.
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    Junior Member maxpancho's Avatar
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    Re: Proof sin(θ)=cos(θ) when...

    And so we use values from the circle to graph trig functions.
    Last edited by maxpancho; July 11th 2014 at 04:27 AM.
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    Re: Proof sin(θ)=cos(θ) when...

    Quote Originally Posted by maxpancho View Post
    Click image for larger version. 

Name:	problem185.PNG 
Views:	16 
Size:	33.8 KB 
ID:	31250

    I know that adjacent side should be equal to opposite which happens at $\dfrac{\pi}{4}$ and $\dfrac{5\pi}{4}$ (plus coterminal angles); the quadrants are 1 and 3 because signs of sin and cos thus coincide. I'm not sure that proves it though? How do I go about this? I don't think the proof should be very elaborate, though, in my case.
    I am puzzled as to why you are talking about "adjacent side" and "opposite". There is NO right triangle and so no "adjacent side" or "opposite side" in this problem. The problem says "use the unit circle interpretation". Sin(t) is the y component and cos(t) is the x component of the (x, y) point on the unit circle at distance (measured around the circumference of the unit circle) t from (1, 0). Saying that sin(t)= cos(t) means that x= y. The points such that x= y lie on the line y= x which has slope 1 so has distance from (1, 0) equal to 1/8 the circumference: 2\pi/8= \pi/4 with the x-axis. That line is a diameter of the unit circle so the additional distance is half the circumference: \pi/4+ \pi= 5\pi/4.
    Last edited by HallsofIvy; July 11th 2014 at 06:39 AM.
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    Junior Member maxpancho's Avatar
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    Re: Proof sin(θ)=cos(θ) when...

    Great. I knew my logic wasn't good enough and that's why I posted this question, to learn what would be the correct way to prove this.

    Thank you.
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    Re: Proof sin(θ)=cos(θ) when...

    $\sin\theta = \cos\theta = u$.

    $\sin^2\theta + \cos^2\theta = 1$ (because these lie on the unit circle).

    $2u^2 = 1$

    $u = \pm \dfrac{\sqrt{2}}{2} = \pm a$

    Thus the points on the unit circle where this occurs are: $\left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2} \right), \left(-\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2}\right)$

    If we form a triangle with the points $(0,0), (a,0) (a,a)$, we see it is an isosceles right-triangle, hence $\theta = \dfrac{\pi}{4} + 2k\pi$ (= 45 degrees (+360 degrees times an integer)).

    Clearly, since $(a,a)$ and $(-a,-a)$ lie on the same line $y = x$ the angle between then is $\pi$ (180 degrees), so $(-a,-a)$ gives $\theta = \dfrac{5\pi}{4} + 2k\pi$

    We can combine these as: $\theta = \dfrac{\pi}{4} + k\pi,\ k \in \Bbb Z$ <---the last summand is often omitted (except for $k = 1$, here), as we only bother to "go around the circle once".

    A word about this: it assumes it is known that the sum of angles of an interior triangle sum to $\pi$ (or 180 degrees). This, actually, is a non-trivial assertion. HallsOfIvy's post assumes it is known that the radial length of 1/8-th of a circle is $\pi/4$. This, too, is also a non-trivial assertion (both length and angle are concepts that are harder to define than it might first appear).

    It is unknown to me which non-trivial assertion you are assumed to "know" (it is unlikely you will see a "proof" of either of these facts in a trigonometry class; I presume the former, since it is possible you have been exposed to geometry, but perhaps unlikely you have been exposed to the tools (from calculus) necessary to compute arc-length). Nevertheless, both facts are "intuitively obvious", and you might be allowed to appeal to either one.
    Last edited by Deveno; July 11th 2014 at 08:17 AM.
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    Junior Member maxpancho's Avatar
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    Question Re: Proof sin(θ)=cos(θ) when...

    Do you mean it can't be proved that a circumference of a circle is $2\pi r$ at this level which I'd then use to determine the angles in radians?
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    Re: Proof sin(θ)=cos(θ) when...

    Quote Originally Posted by Deveno View Post
    $\sin\theta = \cos\theta = u$.

    $\sin^2\theta + \cos^2\theta = 1$ (because these lie on the unit circle).

    $2u^2 = 1$

    $u = \pm \dfrac{\sqrt{2}}{2} = \pm a$

    Thus the points on the unit circle where this occurs are: $\left(\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2} \right), \left(-\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2}\right)$

    If we form a triangle with the points $(0,0), (a,0) (a,a)$, we see it is an isosceles right-triangle, hence $\theta = \dfrac{\pi}{4} + 2k\pi$ (= 45 degrees (+360 degrees times an integer)).

    Clearly, since $(a,a)$ and $(-a,-a)$ lie on the same line $y = x$ the angle between then is $\pi$ (180 degrees), so $(-a,-a)$ gives $\theta = \dfrac{5\pi}{4} + 2k\pi$

    We can combine these as: $\theta = \dfrac{\pi}{4} + k\pi,\ k \in \Bbb Z$ <---the last summand is often omitted (except for $k = 1$, here), as we only bother to "go around the circle once".

    A word about this: it assumes it is known that the sum of angles of an interior triangle sum to $\pi$ (or 180 degrees). This, actually, is a non-trivial assertion. HallsOfIvy's post assumes it is known that the radial length of 1/8-th of a circle is $\pi/4$. This, too, is also a non-trivial assertion (both length and angle are concepts that are harder to define than it might first appear).

    It is unknown to me which non-trivial assertion you are assumed to "know" (it is unlikely you will see a "proof" of either of these facts in a trigonometry class; I presume the former, since it is possible you have been exposed to geometry, but perhaps unlikely you have been exposed to the tools (from calculus) necessary to compute arc-length). Nevertheless, both facts are "intuitively obvious", and you might be allowed to appeal to either one.
    I would expect that, since the original problem said "use the unit circle interpretation of the circular functions" we would be expected to know, and able to use, properties of the unit circle.
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    Re: Proof sin(θ)=cos(θ) when...

    Quote Originally Posted by HallsofIvy View Post
    I would expect that, since the original problem said "use the unit circle interpretation of the circular functions" we would be expected to know, and able to use, properties of the unit circle.
    That seems reasonable enough, but what these properties are (that is what the original poster has been told he can use without proof) is unknown to me.

    Quote Originally Posted by maxpancho View Post
    Do you mean it can't be proved that a circumference of a circle is $2\pi r$ at this level which I'd then use to determine the angles in radians?
    It can, but finding the length of a curve is not as straight-forward as you might suspect, consider this:

    Best WTF: Explaining Why Pi Is 4!

    Basically, we can only "measure" curved lengths that are "smooth" enough, and doing this requires notions of limits. Fortunately, the circle is indeed "smooth enough", and it can be shown that the length of an arc that subtends an angle $\theta$ on a circle of radius 1 is indeed $\theta$ (so the entire circle has circumference $2\pi$).

    Understand, HallsOfIvy is perfectly "correct" and the "facts" you've been told you can use are indeed TRUE: the conflict is at the level of: when one "proves" something, one has to do one of two things:

    1) show it is always true, no matter what (a tautology)
    2) show it is true, if certain "other things" (the assumptions) are true

    (2) is the tricky business, here-if you've never seen any proof that the circumference of a circle of radius $r$ is need $2\pi r$, then instead of proving your original statement, you've proved something like this:

    "If the circumference of a circle of radius $r$ is $2\pi r$, then $\theta = \dfrac{pi}{4} + 2k\pi$ with $k$ any integer"

    which is a slightly DIFFERENT statement.

    Your teacher probably won't care-at the point in time that trigonometry is taught ALL SORTS of unproven statements are taken as fact (like the formula for an area of a triangle, or the pythagorean theorem, or that similarity is an equivalence relation that preserves ratios of sides of triangles). It's not that these things CAN'T be proven, or even that the proof of them is "difficult", it's usually that they involve more sophisticated ideas than you have been exposed to, and this will be explained to you later.

    It turns out that the most BASIC facts about mathematics, are actually surprisingly difficult to prove. Why should this be so? Part of the problem is avoiding "using things we haven't proved yet, to prove something else". If we start with very FEW assumptions, we have, as a result, very FEW tools to use. Building complexity out of simplicity is...complicated.
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