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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Proof of $\arctan(x) = \arcsin(x/\sqrt{1+x^2})$

I've tried following this way, but I haven't succeeded. Thank you!
Alex
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Prove $\sin^2(A)+\sin^2(B)-\sin^2(C)=2\sin(A)\sin(B) \cos(C)$ if $A+B+C=180$ degrees

I most humbly beseech help for this question. If $A+B+C=180$ degrees, then prove $$ \sin^2(A)+\sin^2(B)-\sin^2(C)=2\sin(A)\sin(B) \cos(C) $$ I am not sure what trig identity I should use to begin this problem.
Fernando Martinez
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Prove trigonometry identity for $\cos A+\cos B+\cos C$

I humbly ask for help in the following problem. If \begin{equation} A+B+C=180 \end{equation} Then prove \begin{equation} \cos A+\cos B+\cos C=1+4\sin(A/2)\sin(B/2)\sin(C/2) \end{equation} How would I begin the problem I mean I think $\cos C $ can…
Phantom
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How many points to prove a trigonometric identity?

I am taking a trigonometric identity from another post, arbitrarily. $$\frac{2\sec\theta +3\tan\theta+5\sin\theta-7\cos\theta+5}{2\tan\theta +3\sec\theta+5\cos\theta+7\sin\theta+8}=\frac{1-\cos\theta}{\sin\theta}.$$ Besides the usual approach by…
user65203
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$\frac{1}{\sin 8^\circ}+\frac{1}{\sin 16^\circ}+....+\frac{1}{\sin 4096^\circ}+\frac{1}{\sin 8192^\circ}=\frac{1}{\sin \alpha}$,find $\alpha$

Let $\frac{1}{\sin 8^\circ}+\frac{1}{\sin 16^\circ}+\frac{1}{\sin 32^\circ}+....+\frac{1}{\sin 4096^\circ}+\frac{1}{\sin 8192^\circ}=\frac{1}{\sin \alpha}$ where $\alpha\in(0,90^\circ)$,then find $\alpha$(in degree.) $\frac{1}{\sin…
diya
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Find the closed form of $\cos \frac{2 \pi}{13} + \cos \frac{6 \pi}{13} + \cos \frac{8 \pi}{13}$

From the problem 1 from this link. I was trying to prove that $$\cos \dfrac{2 \pi}{13} + \cos \dfrac{6 \pi}{13} + \cos \dfrac{8 \pi}{13} = \dfrac{\sqrt{13} - 1}{4} $$ Please provide me some hint.
Harshith Sai
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Determining angle to cut two boards

I am building a small roof and I need to determine how to find the distance/angle in that I need to cut two 2x4's in order to make them fit together perfectly. I've drawn a crude image below to illustrate. Given that my base is 6', and I'd like to…
George
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Writing $3\sin(x)+4\cos(x)$ in the form of $r \sin(x-a)$

I need to write $3\sin(x) + 4\cos(x)$ in the form of $r\sin(x-a)$. Expanding $r\sin(x-a): r\sin(x)\cos(a)-r\sin(a)\cos(x)$ Comparing the two forms (The original equation and the expanded form): $3=r\cos(a)$ and $4=-r\sin(a)$. Getting…
w4j3d
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Show that $\tan 3x =\frac{ \sin x + \sin 3x+ \sin 5x }{\cos x + \cos 3x + \cos 5x}$

I was able to prove this but it is too messy and very long. Is there a better way of proving the identity? Thanks.
Keneth Adrian
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How to Solve Trigonometric Equations?

How are you supposed to go about solving equations such as: $$-\sqrt{3} = \frac{\sin{4\theta}}{\sin{7\theta}}.$$ I know that $\theta = 30^{\circ}$ is one such solution, but how do I find all solutions using algebra? Thanks Edit: I figured out one…
A is for Ambition
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Is there a reason of $\cos(11x)+\sin(11(x+1))\approx 0$

Is there a reason of $$\cos(11x)+\sin(11(x+1))\approx 0$$
user189855
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Skewed Trigonometric Function

What would be an expression for a periodic function (period $2\pi$) that essentially behaves just like a negative sine function, but it has the following quirk: It's $0$s lie on the usual places (even integer multiples of $\frac \pi 2$), but it's…
Disousa
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Adding tangents of natural numbers and getting the tangent of another natural number

Is it possible to find a set of numbers $(a_i)_{i\leq n} \in \mathbb {(N^{\star})}^n$ and another natural number $b$ such that $$\sum_{i=0}^n \tan a_i = \tan b $$? In other words, given integral, strictly positive, angles in radians, can their…
Benoit
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Relation between $\sin(\cos(\alpha))$ and $\cos(\sin(\alpha))$

If $0\le\alpha\le\frac{\pi}{2}$, then which of the following is true? A) $\sin(\cos(\alpha))<\cos(\sin(\alpha))$ B) $\sin(\cos(\alpha))\le \cos(\sin(\alpha))$ and equality holds for some $\alpha\in[0,\frac{\pi}{2}]$ C)…
User Not Found
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Prove $\sin(-x) = -\sin(x)$

I'm looking for a really basic proof of $\sin(-x) = -\sin(x)$. The proof should pretty much only employ basic trigonometry. Thanks
Kermit the Hermit
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