Linked Questions

0
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0answers
53 views

In trigonometry, how can we find the sine/cosine of an angle larger than 90 degrees, if sine and cosine only work for right triangles? [duplicate]

In trigonometry, how can there be angle thetas larger than 90 degrees, if sine and cosine only work for right triangles? I realize what we do is that we take the sine or cosine of the angle ...
0
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1answer
24 views

In Unit circle, in second quadrant, why is X taken negative? [duplicate]

Cos theta = base / hypotenuse Base and hypotenuse are lengths, so they should always be positive. So why is cos 3π / 4 negative ?
86
votes
17answers
56k views

What is the most elegant proof of the Pythagorean theorem? [closed]

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
88
votes
10answers
38k views

How can I understand and prove the “sum and difference formulas” in trigonometry?

The "sum and difference" formulas often come in handy, but it's not immediately obvious that they would be true. \begin{align} \sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \...
23
votes
16answers
6k views

Why is $n$ divided by $n$ equal to $1$?

I've understood how to apply this operation for many years but I recently was reminded that: $n^0 = 1$ Which in high school I accepted and just "solved for", but I'm now curious. Why is that true? ...
22
votes
9answers
4k views

Different definitions of trigonometric functions

In school, we learn that sin is "opposite over hypotenuse" and cos is "adjacent over hypotenuse". Later on, we learn the power series definitions of sin and cos. How can one prove that these two ...
14
votes
6answers
3k views

Why does $\sin(0)$ exist?

I can't understand why should $\sin(0)$ exist, because if an angle is $0^{\circ}$, then the triangle doesn't exist i.e. there is no perpendicular or hypotenuse. However, if we take $\lim_{x \to 0} \...
19
votes
4answers
470 views

How is $\sin 90° = 1$ possible?

How can two angles of a triangle be equal to $90°$? If two angles were $90°$, this would mean that the two sides would be parallel and the angle of the third side would be equal to 0. Thus, there ...
4
votes
4answers
519 views

A trigonometric identity: $(\sin x)^{-2}+(\cos x)^{-2}=(\tan x+\cot x)^2$

I've been trying to prove it for a while, but can't seem to get anywhere. $$\frac{1}{\sin^2\theta} + \frac{1}{\cos^2\theta} = (\tan \theta + \cot \theta)^2$$ Could someone please provide a valid ...
4
votes
2answers
2k views

Why are the Cosine and Sine of obtuse angles defined differently? If by convention, please explain the logic behind.

(I already know the unit circle) Why is it that the sine of an obtuse angle is the sine of its supplementary angle but the cosine of an obtuse angle is the negative of the cosine of its supplementary ...
2
votes
3answers
849 views

Why are trig functions defined for the unit circle?

Why did we ever need to define the trig functions of angles greater than 90 degrees or less than 0 degrees? What is the use of applying trig functions to such angles? If we apply the trig functions ...
1
vote
8answers
772 views

Why is $\cos135^{\circ}$ negative when length is always positive?

Consider the following diagram. I am told that $\cos 45^{\circ}$ = $\frac{1}{\sqrt 2}$. I understand this. I am next told taught that $\cos 135^{\circ}$ = $\cos 45^{\circ}$ in 2nd quadrant. And ...
3
votes
2answers
2k views

Trigonometric Ratios for angles greater than 90 degrees and the Unit Circle

I am confused about the Unit Circle explanation for the trigonometric ratios for angles greater than 90 degrees. It seems that for the first (top right) quadrant, $\sin(\theta)$ is equivalent to the ...
0
votes
2answers
187 views

How can $\sin(\pi)$ = 0 without breaking a bunch of math rules?

If I have an angle on a right triangle that is $\pi$ degrees, that would mean that the length of the opposite side would have to equal $0$, because: $\sin(\pi) = \frac {\text{opposite}} {\text{...
1
vote
3answers
409 views

What could be the minimum and maximum value of angle in a triangle?

Sum of angles of a triangle is 180 degress. So while studying trigonometric ratios , I got surprised at cos0 and cos 180 values. Although cosine is ratio of adjecent and hypotenuse ,in case of cos0 ...

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