Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [trigonometry]

Questions about trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles and other topics relating to measuring triangles.

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.

Trigonometry is most simply associated with planar right-angle triangles. The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles.

One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.

Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles.

See Wikipedia's list of trigonometric identities.

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Explanation of formula given on Wikipedia (for sign in half-angle formula for sine)

This Wikipedia article listing trigonometric identities, states the following identity under half angles: $$ \sin{\frac{\theta}{2}} = \text{sgn}\bigg(2\pi-\theta+4\pi\bigg\lfloor\frac{\theta}{4\pi}\bigg\rfloor…
Aadi Prasad
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What stops me from making this conclusion?

Suppose I want to find $\sin^6x+\cos^6x$. What stops me from saying that $\sin^2t=\sin^6x$, and $\cos^2t=\cos^6x$? Of course this is wrong because $\sin^2t+\cos^2t=1$ and $\sin^6x+\cos^6x$ does not equal 1. So what stops me from making this…
Ovi
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The behavior of $\sin(\sin(\sin(\cdots)))$, $\cos(\cos(\cos(\cdots)))$, and $\sin(\cos(\sin(\cos(\cdots))))$

I was playing around with desmos, when i saw this I think that: $\sin(\sin(\sin(\cdots)))$ will approach zero veeeery slowly, also approaching a square wave. $\cos(\cos(\cos(\cdots)))$ will approach $0.7389$. $\sin(\cos(\sin(\cos(\cdots))))$…
gdor11
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What am I doing wrong with solving $2\tanh^2x-\text{sech}~x=1$?

$2\tanh^2(x)-\text{sech}(x)=1$ $\tanh^2(x)=1-\text{sech}^2(x)$ $2(1-\text{sech}^2(x))-\text{sech}(x)=1$ $2\text{sech}^2(x)+\text{sech}(x)-1=0$ $\text{sech}(x)=\frac{1}{2} $ Not possible. And $\text{sech}(x)=-1$ Also not possible What am I doing…
maxmitch
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Trigonometric Manipulation with inequality:$x^2 \cos \theta - x(1 - x) + (1 - x)^2 \sin \theta > 0.$

Find all angles $\theta,$ $0 \le \theta \le 2 \pi,$ with the following property: For all real numbers $x,$ $0 \le x \le 1,$ $$x^2 \cos \theta - x(1 - x) + (1 - x)^2 \sin \theta > 0.$$ I am not exactly sure how to solve this problem. A friend gave…
djdumpling
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Simple trigonometry question

I am just wondering how can you get from $\cos(\pi t)=1/2 $ or $\cos(\pi t)=-1$ for $0
Moses
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Prove that sin A/2 * sin B/2 * sin C/2 = r/4R

The other day I came across an identity in the book "Problems from the Book" and it was presented as well known: $$\sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2} = \frac{r}{4R}$$ However I wasn't familiar with thee identity so I…
Rio
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What is the actual geometric meaning of trigonometric operations such as adding cos,sine,tan

$$\sin(\pi/4)+\cos(\pi/4)=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}= \frac{2\sqrt{2}}{2}=\sqrt{2}$$ Thinking of trig components (cosine, sine) that I used to produce the result using the mechanics of algebra, makes me wonder what is the geometric…
themhz
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Deriving what $\cos(A+2B)$ is

We know that $\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$ , but I don't understand with the $2B$. Would it just become $\cos(A+2B) = \cos(A)\cos(2B) - \sin(A)\sin(2B)$? Thank you for any help you can give me.
Tryingtolearnpython
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Find constants $a$, $b$, $c$, $d$ and $e$ such that $\cos4x=a\sin^4x+b\sin^3x+c\sin^2x+d\sin x+e$ for all angles $x$

Basically, write $\cos4x$ as a polynomial in $\sin x$. I've tried the double angles theorem and $\cos2x=\cos^2x-\sin^2x$. I'm still having trouble right now though. Please help! Thanks!
riorpewruiowe
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If $\frac{\cos x+\cos y+\cos z}{\cos(x+y+z)}=\frac{\sin x+\sin y+\sin z}{\sin(x+y+z)}=k$, then find $\cos(x+y)+\cos(y+z)+\cos(z+x)$.

If $x,y,z \in\mathbb{R}$ and $$\frac{\cos x+\cos y+\cos z}{\cos \left(x+y+z\right)}=\frac{\sin x+\sin y+\sin z}{\sin \left(x+y+z\right)}=k$$ Find the value of $\cos(x+y)+\cos(y+z)+\cos(z+x)$. I was working through this problem and found an…
V.G
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How to solve $\sqrt3\sin3x -\cos x =\sqrt2$

$\sqrt3\sin3x -\cos x =\sqrt2$ Does $\sin3x = 3\sin x - 4\sin^3x$ work? $=3\sin x\cos^2x - \sin^3x$ ??? I don't see any factor and next step?
Solitarie
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Can we write $\sin(\frac{\pi}{14})$ as an finite expression using only basic operations?

First, let us recall some trigonometric values $\sin(0)=0$ $\sin(\pi/6)=\frac{1}{2}$ $\sin(\pi/3)=\frac{\sqrt{3}}{2}$ $\sin(\pi/2)=1$ $\sin(\pi/10)=\frac{\sqrt{5}-1}{4}$ $\sin(\pi/12)=\frac{\sqrt{3}-1}{2\sqrt{2}}$ Here, we can observe that for some…
Mathematical Curiosity
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Find the angle between the 2 points (50.573,-210.265) and (117.833,-80.550)

I am attempting to find the angle between the 2 points (50.573,-210.265) and (117.833,-80.550). Is my calculation correct because a program is giving me a different answer? It says the angle is 27'24'27.27 DMS dx = x2 - x1; dy = y2 - y1; angle =…
sazr
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Find, in radians the general solution of cos 3x = sin 5x

I am studying maths as a hobby. I have come across this problem: Find a general solution for the equation cos 3x = sin 5x I have said, $\sin 5x = \cos(\frac{\pi}{2} - 5x)$ so $\cos 3x = \sin 5x \implies 3x = 2n\pi\pm(\frac{\pi}{2} - 5x)$ When I add…
Steblo
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