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I was looking at a mark scheme for a question I was stuck on, and I came across this. You are asked to work out the value of $\tan 75^\circ$ after you've worked out $\cos 15^\circ$ and $\sin 15^\circ$. I noticed that $\tan(x)=\cot(90^\circ-x)$. I've never seen this before, and this makes no sense to me, so please could someone explain it to me? Are there any other similar trig properties that I should know about?

enter image description here

Américo Tavares
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Jim
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    Have you seen $;\sin(90-x)=\cos x;,;;\cos(90-x)=\sin x;$ ? This follows at once from the most basic definition of trigonometric functions in high school. – DonAntonio Apr 19 '14 at 15:56
  • @DonAntonio I've never seen those before, (they're probably not on my syllabus then), but I can see how they work, since you're just translating the graph, right? I went on graphing software and entered sin(90-x) and cos(x), but the sin(90-x) graph was not translated enough to become the cos(x) graph. I think I went wrong somewhere then – Jim Apr 19 '14 at 16:12
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    @Jim use degrees while plotting – evil999man Apr 19 '14 at 16:17
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    @Jim, weren't you taught the definitions of sine and cosine using a straight-angled triangle?? – DonAntonio Apr 19 '14 at 16:17
  • @Awesome I'm pretty sure I set my software to degrees, but I'll double check – Jim Apr 19 '14 at 16:21
  • @DonAntonio sin=opp/hyp yup we did all that, but this is seriously my first time seeing those identities – Jim Apr 19 '14 at 16:22
  • But @jim : in a straight-angled triangle, both acute angles sum to $;90^\circ;$ , so the formulas I mentioned in my first comment are trivial!! – DonAntonio Apr 19 '14 at 16:23
  • @DonAntonio,I am pretty sure most schools teach trig functions using a right angled triangle,not a straight angled one. – rah4927 Apr 19 '14 at 16:49
  • What is a straight angled triangle for you, @rah4927 ? – DonAntonio Apr 19 '14 at 16:49
  • @DonAntonio,I have never heard of straight angled triangles before.What is it?A degenerate triangle? – rah4927 Apr 19 '14 at 16:52
  • straight angled triangle = right angled triangle (apparently, this is the correct term in english or, at least, in some english speaking countries). – DonAntonio Apr 19 '14 at 16:58
  • @DonAntonio,really?What is reason behind calling such triangles straight angled i.e. what is the etymology behind its usage? Straight angle means $180$ degrees,so how does a triangle with a right angle become a straight angled triangle? – rah4927 Apr 19 '14 at 17:02
  • @DonAntonio: I've spent all my life in an English speaking country (US), and I've never heard "straight angled triangle" until now. – Nate Eldredge Apr 19 '14 at 18:48
  • @rah,I meant in that past comment that right angled is the correct term to what I called "straight angled" ... – DonAntonio Apr 19 '14 at 18:51
  • @NateEldredge, are you sure USA is an english speaking country? Just kidding...but not much. :) – DonAntonio Apr 19 '14 at 18:53

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enter image description here

so $\tan\theta = \frac{a}{b} = \cot(90-\theta)$.

By the way your computation above computes $\cot 75$, not $\tan 75$.

rogerl
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You might want to learn here about trigonometric identities. Note that, $\cot(90^\circ-x)$ means $x$ is reflected from angle $90^\circ$ and in this case, the result is $\tan x$. Just click the given link. I hope this helps.

Anastasiya-Romanova 秀
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This follows from simple well-known trigonometric identities:

$$\tan(90-x) = \frac{\sin(90-x)}{\cos(90-x)}=\frac{\cos(x)}{\sin(x)}=\cot(x)$$

naslundx
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