If I know sin/cos/tan... of an angle x, how do I know the x without look at the table? How do computers calculate the angle? How the table of sin cos tan was constructed? How the old mathematicians knew this angles without computers?
24339
Sine $\theta$ = opposite/hypotenuse   Cosine $\theta$  = adjacent/hypotenuse   Tangent $\theta$  = opposite/adjacent      In order to calculate the sine or the cosine or the tangent I need to know $3$ sides of a right triangle. $2$ for each corresponding trigonometric function. How does a calculator calculate the sine, cosine, tangent of a number (that is actually an angle ?) without knowing any sides?
77232
I need to use the fact that  $\tan 2x=\sin2x \ /\cos2x$ to prove that: $$\tan 2x=\frac{2\tan x}{1-\tan^2x}$$ I don't know where to start. Please help or hint. Thanks in advance.
115819
If an angle is the measure of distance between to points (Edit: Ok, admittedly bad phrasing. A measure of rotation between two intersecting lines, or points, etc.), is there such a thing as a zero degree angle? I asked a math nerd friend about this years ago and he still hates me for it but couldn't prove it either way at the time. Is this merely pedantic or delusional thoughts regarding definition of "angle" (Edit: based on answer below regarding Euclidian geometry, this all now seems very likely a colossal misunderstanding) or am I missing something? I do not think zero degree angles exist. Is there proof of this either way?  Edit: A better question would be why Euclidian zero degree angles were thought to not exist, or why he avoided them?
88075
Suppose we are given the free-body diagram above, with a mass on an incline at an angle $\theta$. If my coordinate axes are taken to be the regular $x-y$ plane rotated an angle of $\theta$ (i.e. the x axis is parallel to the direction of $mg \sin \theta$ and the $y$ axis is parallel to $mg \cos \theta$), then if there is no vertical motion, $N = mg \cos \theta$. Now suppose I take my coordinate axes to be the $x-y$ plane but NOT rotated by any angle (i.e. the $x$ axis is parallel to the base of the triangle and the $y$ axis is parallel to the adjacent side), then if I want to resolve my weight $mg$ in the direction of $N$, wouldn't I need to make it $\frac{mg}{\cos \theta}$? (since in this case I would have to find the force whose vertical component is $mg$) If this is the case, no vertical motion would imply that $N = \frac{mg}{\cos \theta}$. Does this mean that $N$ depends on my choice of coordinate axes? I might be making a mistake, but I'd appreciate if someone could point out the mistake in my logic.
121759
How can I find the following product using elementary trigonometry?     Suppose $0 \lt x \lt \frac{\pi}{2}$   is an angle measured in radians. Use the trigonometric circle and   show that $\cos(x) \le \frac{\sin(x)}{x} \le \frac{1}{\cos(x)}$.   I have been trying to solve this question. I can't figure out whether or not the solution requires a trigonometric circle or if it can be done using another method.
29025
Show that if $x, y,$ and $z$ are consecutive terms of an arithmetic sequence, and $\tan y$ is defined, then  $$\frac{\sin x + \sin y + \sin z }{\cos x + \cos y + \cos z} = \tan y. $$  I'm not sure what trig identities I would use and how to use them.  Could I get some help?  Thanks.
50712
In Multivariable Calculus, the professor said that in order to compute the angle $x$ between two vectors $v$ and $w$, we use the formula: $\cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot |\vec{w}|}$  He also mentioned "Projection" but I didn't understand much of it. The only thing I got was that we can create a right triangle by drawing a perpendicular line from one vector to the head of the other and compute the angle like that using $\cos$, but I don't understand why we're multiplying vectors, or dividing by their magnitudes…how do those operations make sense?  Also, in the specific example, the result of $\cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot |\vec{w}|}$ ended up being $1/2 = \pi/2$. Where does $\pi/2$ come from here? I can't understand how we go from $1/2$ to $\pi/2$.
90272
I am looking for different ways of differentiating $\sin$ and $\cos$, especially when using the geometric definition, but ways that use other defintions are also welcome.   Please include the definition you are using.   I think there are several ways to do this I didn't find, that make clever use of trigonometric identities or derivative facts. I am looking for these for several reasons. Firstly, it is just interesting to see. Secondly, there may be interesting techniques that can also be applied in a clever way to other derivative problems. It is also interesting to see how proofs can come form completely different fields of mathematics or even from physics.  I have included several solutions in the answers.
34743
If $$\sin(a)\sin(b)\sin(c)+\cos(a)\cos(b)=1,$$;where abc are the angles of the triangle.! then find the value of $\sin(c)$. By trial and error put this triangle as right angled isosceles and got the answer..!! But i want a complete proof!
110936
I'm learning about trig identities, and I'm struggling to prove that two expressions are equal:  $$ \frac{\sin(A + B)}{\sin(A - B)}=\frac{\tan A + \tan B}{\tan A - \tan B} $$  How do you go about proving this? I know about compound angles - i.e. the sine, cosine and tangent of $(A \pm B)$, but don't know how to apply it in this situation.
45200
How does that bit work?  How is  $$\cos(\arcsin(x)) = \sin(\arccos(x)) = \sqrt{1 - x^2}$$
120874
Prove the following trigonometric identity: $$1 + \tan^2\theta = \sec^2\theta$$  I'm curious to know of the different ways of proving this depending on different characterizations of tangent and secant.
11231
Another problem that I already wasted hours on. Given $$4\sinθ +3\cosθ  = 5$$  Find   $$4\cosθ -3\sinθ$$  Help me guys (PS:I'm not that good in maths)
47287
How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized)  Thinking about how to prove this in the most intuitive way resulted in proving a trigonometric identity: $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$.  But even after proving this successfully, the connection between and cosine and dot product does not immediately stick out and instead I rely on remembering that this is valid while taking comfort in the fact that I've seen the proof in the past.  My questions are:   How do you see this connection? How do you extend the notion of dot product vs. angle to higher dimensions - 4 and higher?
64473
$$\tan A + \tan 2A + \tan 3A=0$$ I tried converting these all in sin and cos and I got the answer but the answer didn't match.  So any one could just help me.  Thanks
1853
Multiple Angle Identities:  How to expand  $\cos nx$  with  $\cos x$, such as  $$\cos10x=512(\cos x)^{10}-1280(\cos x)^8+1120(\cos x)^6-400(\cos x)^4+50(\cos x)^2-1$$ See a list of trigonometric identities in /
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So in trig, say I have an acute angle $X$. And one can intuitively conclude that $\sin x + \cos x \ge 1$, but how does the fact that $$(\sin x + \cos x)^2 = 1 + 2 \sin x \cos x$$ tell me that it is true that $\sin x + \cos x \ge 1$? I don't quite see the connection. Thanks