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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If $\frac{\cos x}{\cos y}=\frac{a}{b}$ then $a\tan x +b\tan y$ equals

If $$\frac{\cos x}{\cos y}=\frac{a}{b}$$ Then $$a \cdot\tan x +b \cdot\tan y$$ Equals to (options below): (a) $(a+b) \cot\frac{x+y}{2}$ (b) $(a+b)\tan\frac{x+y}{2}$ (c) $(a+b)(\tan\frac{x}{2} +\tan\frac{y}{2})$ (d) …
Sachin
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Find the value of $\sin25^° \sin35^° \sin85^°$

Trigo problem : Find the value of $\sin25^° \sin35^° \sin85^°$. My approach : Using $2\sin A\sin B = \cos(A-B) -\cos(A+B)$ $$ \begin{align} & \phantom{={}}[\cos10^{°} -\cos60^°] \sin85^° \\ & = \frac{1}{2}[2\cos10^°\sin85^° -2\cos60^° \sin85^°]…
Sachin
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solving trigonometric equation $3^{\sin^2(x)}+3^{\cos^2(x)}=4$

Please help me to solve this trigonometric equation. $$3^{\sin^2(x)}+3^{\cos^2(x)}=4.$$
user97619
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Condition for $\tan A\tan B=\tan C\tan D$

Here, it is claimed that $$\tan A\tan B=\tan C\tan D$$ if one of the four following conditions holds $$\displaystyle A\pm B=C\pm D$$ If it is true, how to prove this? $\tan(x\pm y)$ did not help much. I am expecting some relation among $A,B,C,D$
lab bhattacharjee
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Prove that : $\tan 40 + \tan 60 + \tan 80 = \tan 40 \cdot \tan 60 \cdot \tan 80$

I started from Left hand side as 3^1/2 + tan 2(20) +tan 4(20). But that brought me a lot of terms to solve which ends (9 tan 20 - 48 tan^3 20 -50 tan^5 20 - 16 tan^7 20 + tan^9 20)/(1- 7 tan^2 20 + 7 tan^4 20 - tan^6 20), which is very huge to…
Jignesh Shah
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Newbee, how to solve for t?

I haven't done math in 4 years now and I'm getting into Game programming. So I can't even remember where to start with this? I would like to solve for t. $x=16\left(\sin \left(t\right)\right)^3$
ArmenB
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What is the formula to convert an angle measure in degrees to one between 0 and 360 degrees?

I know that I can find an equivalent angular measure to $460^\circ$ that's at least $0^\circ$ and less than $360^\circ$ by repeatedly subtracting $360^\circ$ till it's smaller than $360^\circ$, which comes out to be $100^\circ$. But is there a…
Muhammad Umer
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How prove this is an equilateral triangle

in $\Delta ABC$,$AB=c,AC=b,BC=a$and such $$ab^2\cos{A}=bc^2\cos{B}=ca^2\cos{C}$$ show that $\Delta ABC$ is an equilateral triangle this problem I have solution,But not nice, and I think this problem have more nice methods,Thank you everyone. my…
math110
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How prove this trigonometric identity

Show that $$\sum_{k=0}^{n-1}\dfrac{\tanh{\left(x\dfrac{1}{n\sin^2{\left(\dfrac{2k+1}{4n}\pi\right)}}\right)}}{1+\dfrac{\tanh^2{x}}{\tan^2{\left(\dfrac{2k+1}{4n}\pi\right)}}}=\tanh{(2nx)}$$ Thank you ,This problem I take some hours,and at last I…
math110
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Is it possible to adjust the sine function to pass through specific points?

I have a set of points $\left\{ \left(−\frac{6}{4}, 0\right), \left(−\frac{4}{4}, −\frac{\sqrt{6}}{3}\right), \left(−\frac{3}{4}, −1\right), \left(−\frac{2}{4}, −\frac{\sqrt{6}}{3}\right), \left(\frac{0}{4}, 0\right), \left(\frac{2}{4},…
Lawton
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Question about proof of general formula for $\sin^n \theta$

I was reading the wikipedia page about list of trigonometric identites and came across the following identities for $$\cos^n \theta $$ when n is odd:- $$\cos^n \theta = \frac{2}{2^n}\sum_{k=0}^{\frac{n-1}{2}} {n\choose k}\cos ((n-2k)\theta)$$ for…
koiboi
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Solving $\sin x= 2\sin(x-30^\circ) \sin(40^{\circ})$

To solve $\sin x= 2\sin(x-30^\circ) \sin(40^{\circ})$ I expanded RHS by using $\sin(\alpha-\beta)=\sin\alpha\cos\beta-\sin\beta\cos\alpha$, $$\sin x= 2\sin(40^{\circ})\times\left(\frac{\sqrt3}2\sin x- \frac12\cos x\right)$$ $$\left(\sqrt3…
Etemon
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Query on methods to solve the equation $\sin(\theta) = -\cos(\theta)$

in the interval of $0< \theta <360$ solve for $\theta$ , the equation $(1)$, I would just say $\tan \theta =-1$ and solve that. $$\sin\theta=-\cos\theta\tag1$$ but what if i said $\sin θ +\cos θ =0$ then $\cos θ (\tan θ +1)=0$. I'd have solutions…
j jose
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How to prove that $\tan 15^{\circ} = \tan 27^{\circ} \tan 33^{\circ} \tan 39^{\circ}$?

Found this equality $$\tan 15^{\circ} = \tan 27^{\circ} \tan 33^{\circ} \tan 39^{\circ}$$ with a random search. It checks with WA, and these kind of equalities can be proved automatically. I am looking for an elementary proof. Some similarly looking…
orangeskid
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Simplifying $\arctan\biggl(\frac{\sqrt{2+\sqrt{3}}}{\sqrt{2}}\biggr)-\arcsin\biggl(\frac{3}{2\sqrt{4+\sqrt{3}}}\biggr)$

After some calculations I have $$\arctan\biggl(\frac{\sqrt{2+\sqrt{3}}}{\sqrt{2}}\biggr)-\arcsin\biggl(\frac{3}{2\sqrt{4+\sqrt{3}}}\biggr).$$ Mathematica simplyfies this to $\pi/12$ and I wonder how it is done. Edit: The results comes from…
mf67
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