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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Solve: $3\sin{2x}+4\cos{2x}-2\cos{x}+6\sin{x}-6=0$

Solve: $3\sin{2x}+4\cos{2x}-2\cos{x}+6\sin{x}-6=0$ My Try $6\sin{x}\cos{x}+4(\cos^2{x}-\sin^2{x})-2\cos{x}+6\sin{x}-6=0$ I have expanded the equation, But I cannot proceed further, Any hint would be appreciated. Thank you!
emil
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How to simplify $\frac {\sin 3A - \cos 3A}{\sin A + \cos A} + 1$?

So I started by using $\sin 3A$ and $\cos 3A$ identities and then I added the lone $1$ to the trigonometric term. (Done in the picture below) But after this I don't have any clue on how to proceed. $$=\frac{3 \sin \theta-4 \sin ^{3} \theta-\left(4…
Shogun BS
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what is the value of this trigonometric expression

I want to find out value of this expression $$\cos^2 48°-\sin^2 12°$$ Just hint the starting step.Is there any any formula regarding $\cos^2 A-\sin^2 B$?
iostream007
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How to derive $\cos\frac{n\pi}{3}=\frac{1+3(-1)^{[\frac{n+1}{3}]}}{4}$

Consider the following formula to calculate a trigonometric function: $$\cos\frac{n\pi}{3}=\frac{1+3(-1)^{[\frac{n+1}{3}]}}{4}$$ $[x]$ denotes the integer part of $x$. The formula is valid for $n=0,2,4,6,...$ I'm curious how this formula is…
Martin Gales
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Question in a trigonometric equation.

I tried it a lot but am not able to get this.Pls help in how should I think when solving this type of question and which side is better to try to simplify first (LHS or RHS).Please share the solution in that way. $$ \frac{1-\sin A}{1+\sin A} = 1 + 2…
user821898
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Where are these additional solutions coming from?

Solve for $x$: $2\sin(2x)-\sqrt{2} = 0$ in interval $[0,2\pi)$ Step $1$: Add $\sqrt{2}$ and divide by $2$ to get $\sin(2x) = \dfrac{\sqrt{2}}{2}$ Step $2$: Set $2x$ equal to the angles where $\sin(x) = \dfrac{\sqrt{2}}{2}$: $2x = \dfrac{\pi}{4}$ …
Matt
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What is $\tan \alpha$, if $(a+2)\sin\alpha +(2a - 1)\cos\alpha =2a + 1$?

I tried the following: $$\begin{aligned}a\sin\alpha +2\sin\alpha + 2a\cos\alpha - \cos\alpha &= 2a+1\\ a(\sin\alpha +2\cos\alpha)+(2\sin\alpha-\cos\alpha)&=2a+1\end{aligned}$$ Therefore, $$\sin\alpha +2 \cos\alpha=2$$ $$2\sin\alpha -…
user832407
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What is the physical meaning of sine, cosine and tangent of an obtuse angle?

I have quite a few questions. First of all, for an angle $90^\circ\lt\theta\lt 180^\circ$, what would the sine/cosine/tangent of this angle be? What I'm saying is that a right angled triagle will always have all other angles acute. How is this…
Eyy boss
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Is there an identity to combine a sum of more than two sines; eg $\sin(a)+\sin(b)+\sin(c)+\sin(d)$?

I get these trigonometric product to sum formulas like: $$\sin(a)+\sin(b)=2\sin\frac12(a+b)\cos\frac12(a-b)$$ And that's useful, but I'm not too sure what to do if I need to turn a product into a sum if there's more than two variables. What would I…
Alice T
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Any "shortcuts" to proving that $\frac{\sin(x)}2+\sin^2(\frac x2)\tan(\frac x2)\to\tan(\frac x2)$

I was working on simplifying some trig functions, and after a while of playing with them I simplified $$\frac{\sin(x)}{2}+\sin^2\left(\frac{x}{2}\right)\tan\left(\frac{x}{2}\right) \rightarrow \tan\left(\frac{x}{2}\right)$$ The way I got that…
Esteban
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Prove that $\sin(n x) + \sin((n+2) x) = 2\cos(x)\sin((n+1) x)$?

I need to prove that $\sin(n x) + \sin((n+2) x) = 2\cos(x)\sin((n+1) x)$. I have already checked that this is correct for $n=1$ and $n=2$, but I'm not able to prove this identity by induction. Now I was thinking of making a shift to the imaginary…
user54297
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Solve $\frac{\sin (10^\circ) \sin (30^\circ)}{\sin 40^\circ \sin (80^\circ-x^\circ)} = \frac{\sin 20^\circ}{\sin x}$

The context to this is trivial I think, I was solving a geometry problem using the trigonometric version of Ceva, I got here and I was stuck, I tried using the sum-difference, product to sums, sums to products identities but my attempts failed and…
dude076
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For $\alpha\in(0^\circ;90^\circ)$ simplify $\sin^2\alpha+\tan^2\alpha+\sin^2\alpha.\cos^2\alpha+\cos^4\alpha$

For $\alpha\in(0^\circ;90^\circ)$ simplify $\sin^2\alpha+\tan^2\alpha+\sin^2\alpha\cdot\cos^2\alpha+\cos^4\alpha.$ My try:…
10th grade
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Prove that $\tan(x)\tan(x+\frac{\pi}{3})+\tan(x)\tan(\frac{\pi}{3}-x)+\tan(x+\frac{\pi}{3})\tan(x-\frac{\pi}{3}) = -3$

Let's assume that $\tan(x) = y$. So, $\tan\Big(x+\dfrac{\pi}{3}\Big) = \dfrac{\tan(x) + \tan\Big(\dfrac{\pi}{3}\Big)}{1-\tan(x)\tan\Big(\dfrac{\pi}{3}\Big)} = \dfrac{y+\sqrt{3}}{1-\sqrt{3}y}$ Similarly, $\tan\Big(x-\dfrac{\pi}{3}\Big) =…
Rajdeep Sindhu
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Number of solutions of the equation: $\tan x=\cos2x$ in $[-π, π]$

I know that this question can be solved by using graphs, but in our examinations, we are not allowed to use calculators, or any digital devices. I have a doubt that how to check whether the graphs will intersect or not at the points which are marked…
UM Desai
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