Mathematics Stack Exchange
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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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Proving trigonometric equation $\cos(36^\circ) - \cos(72^\circ) = 1/2$

Please help me to prove this trigonometric equation. $\cos \left( 36^\circ \right)-\cos \left( 72^\circ \right) = \frac{1}{2}$ Thank you.
xzatulx
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$ a \cos A + b \cos B + c \cos C = \dfrac{a+b+c}2 $ $\implies$ the triangle is equilateral?

If in a triangle $ a \cos A + b \cos B + c \cos C = \dfrac{a+b+c}2 $ , then is the triangle equilateral ?
user123733
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Solutions of $\cos(ax)+\cos(bx)+\cos(cx)=0$

These are easy to solve: $$\cos(ax)=0,\qquad \cos(ax)+\cos(bx)=0$$ Any insight into $$\cos(ax)+\cos(bx)+\cos(cx)=0$$ Can't find any information on the web about this last one. I know solutions can be approximated via series but I'm interested in…
user151781
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$\tan(x)=\cot(90^\circ-x)$??

I was looking at a mark scheme for a question I was stuck on, and I came across this. You are asked to work out the value of $\tan 75^\circ$ after you've worked out $\cos 15^\circ$ and $\sin 15^\circ$. I noticed that $\tan(x)=\cot(90^\circ-x)$. I've…
Jim
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How prove this $\cos{x}+\cos{y}+\cos{z}=1$

Question: let $x,y,z\in R$ and such $x+y+z=\pi$,and such $$\tan{\dfrac{y+z-x}{4}}+\tan{\dfrac{x+z-y}{4}}+\tan{\dfrac{x+y-z}{4}}=1$$ show that $$\cos{x}+\cos{y}+\cos{z}=1$$ My idea: let…
math110
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An expression equal to its reciprocal (but not really)

"Everybody knows" that \begin{align} \tan\theta & =\frac{2\tan\frac\theta2}{1 - \tan^2\frac\theta2} = \frac{2\sin\frac\theta2\cos\frac\theta2}{\cos^2\frac\theta2-\sin^2\frac\theta2} \tag 1 \\[10pt] & = \dfrac{\text{2 times a product}}{\text{a…
Michael Hardy
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What are the uses of trigonometry in architecture?

After some searching I found: Triangulation Using theta and "b" to find out the height of a building. Thanks in advance.
aviraldg
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Is the following trigonometric equation solvable?

I have the following equation, and I want to solve for $\theta$ : $$f(x,y,\theta) = \frac{x \cos(\theta) - y \sin(\theta)}{x \sin(\theta) + y \cos(\theta)}$$ It seems to me this equation should be easily solvable given known $x$ and $y$, however I…
levesque
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Find $\sum\limits_{k=1}^{12}\tan \frac{k\pi}{13}\cdot \tan \frac{3k\pi}{13}$

Find $\sum\limits_{k=1}^{12}\tan \frac{k\pi}{13}\cdot \tan \frac{3k\pi}{13}$. I tried some elementary ways while all failed.
Steven Sun
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Another Verify the identity: $\sec^2 \frac{x}{2} = \frac{2}{1+\cos x}$

Another Verify the identity that I can't get: $$\sec^2 \frac{x}{2} = \frac{2}{1+\cos x}$$ $$ = \frac{1 + \left(\frac{1}{\cos x}\right)}{2}$$ $$ = \frac{\cos x + 1}{2 \cos x}$$
KKendall
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solve a trigonometric equation $\sqrt{3} \sin(x)-\cos(x)=\sqrt{2}$

$$\sqrt{3}\sin{x} - \cos{x} = \sqrt{2} $$ I think to do : $$\frac{(\sqrt{3}\sin{x} - \cos{x} = \sqrt{2})}{\sqrt{2}}$$ but i dont get anything. Or to divied by $\sqrt{3}$ : $$\frac{(\sqrt{3}\sin{x} - \cos{x} = \sqrt{2})}{\sqrt{3}}$$
dona12
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How show this $\Delta ABC\sim\Delta A'B'C'$

Assmue that:if $\Delta ABC$ and $\Delta A'B'C'$ are not right triangle,and such $$\sin{(2A)}:\sin{(2A')}=\sin{(2B)}:\sin{(2B')}=\sin{(2C)}:\sin{(2C')}$$ show that $$\Delta ABC\sim\Delta A'B'C'$$ My idea: if…
user94270
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Which one is greater, $\sin(\sin(\sin(1)))$ or $\cos(\cos(\cos(1)))$

I know I asked a similar question sometime before, and the thing is I need them for a proof. So, please help, I promise this is the last one. What is the simplest way we can find which one of $\sin(\sin(\sin(1)))$and $\cos(\cos(\cos(1)))$ [in…
Sawarnik
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Calculating edge coordinates of arrowhead. (pretty basic trigonometry)

What would be the most efficient (or right..) way to determine edge coordinates of both sides of arrowhead? Picture is worth a thousand words so I'll explain this with one. I know coordinates A and B, and will need coordinates C and D. Side of an…
Juha
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How to use atan2?

I've looked at Wikipedia and don't really understand what I'm reading or how to implement atan2(). I was hoping someone could point me to a better resource or give me an example of how to use atan2(). For some context, I'm asking about atan2()…
Nick
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