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.

29665 questions
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$\tan^2 10^\circ+\tan^2 50^\circ+\tan^2 70^\circ=9$

Strangest thing...*: $$\tan^2 10^\circ+\tan^2 50^\circ+\tan^2 70^\circ=9\tag{1}$$ The trick, as always, is how to prove it. My idea was to add a "missing" tangent and analyze a similar expression: $$\tan^2 10^\circ+\tan^2 30^\circ+\tan^2…
Saša
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Does $\sin^2 x - \cos^2 x = 1-2\cos^2 x$?

I am finishing a proof. It seems like I can use $\cos^2 + \sin^2 = 1$ to figure this out, but I just can't see how it works. So I've got two questions. Does $\sin^2 x - \cos^2 x = 1-2\cos^2 x$? And if it does, then how?
KKendall
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Why is $\tan$ so different from $\sin$ & $\cos$?

I'm just curious, considering how similar the graphs of the sine & cosine functions are in shape, why is the shape of the tangent's graph so different, despite being used in very similar types of problem? Thanks!
Danny King
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If $x=\cos a + i \sin a$, $y=\cos b + i \sin b$, $z=\cos c +i \sin c$, and $x+y+z = xyz$, then show that $\cos(a-b) + \cos(b-c) + \cos(c-a) + 1=0$

If $x=\cos a + i \sin a$, $y=\cos b + i \sin b$, $z=\cos c +i \sin c$, and $x+y+z = xyz$, then show that $$\cos(a-b) + \cos(b-c) + \cos(c-a) + 1=0$$ Here's how I tried it $$x+y+z=xyz $$ So, by De Moivre's Theorem, $$(\cos a + \cos b + \cos c) +…
Pratim Das
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Solve trig function for $x$

For what value(s) of $x$ does the following function satisfy? $$ \dfrac{-16}{9} = \dfrac{\cos(18x)}{\cos (24x)} $$ Unsure if I need an identity to solve or it's just basic.
maths101
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Finding relationships between angles, a, b and c when $\sin a - \sin b - \sin c = 0$

while I was doing some calculations, I came across the condition where $$ \sin a - \sin b - \sin c = 0$$ where $0 < b, c < a < \dfrac{\pi}{2}$. Is there a way to find a relationship between the angles without using the trigonometric functions above?…
Isamu Isozaki
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Help Solving Trigonometry Equation

I am having difficulties solving the following equation: $$4\cos^2x=5-4\sin x$$ Hints on how to solve this equation would be helpful.
Bilbo
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How to prove $\cos(n!) \neq 1$ without using $\pi$ is irrational

Prove $[\forall n \in \mathbb{N}, \cos(n!) \neq 1]$, without using $\pi$ is irrational. Using $\pi \in (\mathbb R-\mathbb Q)$, I can prove it... Thanks everyone!
user515454
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Simplifying $\sum 2^k \tan(2^k x)$

Simplify $\sum\limits_{k = 0}^n {{2^k}\tan ({2^k}x)}$ which $k \in \{ 0,1,...,n + 1\} ,{2^k}x \notin \{ 0,\frac{\pi }{2}\}$
fidele007
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Determine sinus and cosinus : $2\sin x + 3\cos x = 3$

$$2\sin x + 3\cos x = 3$$ Seemed easy at first but I have no idea how to determine them. I tried replacing the 3 with $3(\sin^2 x + \cos x^2x)$ But it doesnt work. I also tried switching sides from almost everything and no luck. Please help!
RiktasMath
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If $\cos(z-x) + \cos(y-z) + \cos(x-y) = -\frac{3}{2}$, then $\sin x + \sin y + \sin z = 0 = \cos x + \cos y + \cos z $.

If $$\cos(z-x) + \cos(y-z) + \cos(x-y) = -\frac{3}{2}$$ then how can I show that the sum of cosines of each angle ($x$, $y$, $z$) and sines of each angle sum up to zero? i.e. $$\sin x + \sin y + \sin z = 0 = \cos x + \cos y + \cos z $$ I tried:…
Fghj
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Proving the product expansion of $\sin\theta$. Where did I go wrong?

You can prove the expansion$$\frac {\sin\theta}{\theta}=\left\{1-\left(\frac {\theta}{\pi}\right)^2\right\}\left\{1-\left(\frac {\theta}{2\pi}\right)^2\right\}\ldots$$By taking the expansion$$\sin n\phi=2^{n-1}\sin\phi\cos\phi\left(\sin^2\frac…
Frank
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Please prove $\frac{1 + \sin\theta - \cos\theta}{1 + \sin\theta + \cos\theta} = \tan \left(\frac \theta 2\right)$

Prove that $\dfrac{1 + \sin\theta - \cos\theta}{1 + \sin\theta + \cos\theta} = \tan\left(\dfrac{\theta}{2}\right)$ Also it is a question of S.L. Loney's Plane Trignonometry What I've tried by now: \begin{align} &…
TheMathGuy
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Number of integral values of p

Let $p=144^{\sin ^2 x}+144^{\cos ^2x}$, then we have to find the number of integral value of p . Using the concept of inequlitiy of AM ,GM I got its minimum value as 24 . But how could we find the maximum value .
Koolman
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Why is the period of $\sin(2x)$ is $\pi$?

In my textbook, it is stated that the period of $\sin(2x)$ is $\pi$. The author justifies this using a mathematical statement which I cannot understand. He writes that, since $\sin(2x) = \sin(2x+2\pi) = \sin(2(x+\pi))$ the period of $\sin(2x)$ is…
MrAP
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