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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Number of triangles with integer sides

Total number of right angle triangles whose inradius is $2013$ and sides are integer Attempt: assuming that $a,b,c>0$ are the sides of a triangle. so form a right angle triangle with sides $a,b,c$ and right angle at $C$ $\displaystyle r=(s-c)\tan…
DXT
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Trigonometry without sine and cosine

Maybe an unusual (and too easy for you) question, but my younger brother is requested to calculate the height of the Eiffel Tower: Is this possible, given that he has not learned sine and cosine yet (5th grade)? Details: A-to-B=200m, alpha=65°,…
caw
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The following equation to solve :$ \tan x+\cot x=\sqrt{2}(\cos x+\sin x)$

The following equation to solve : $$ \tan x+\cot x=\sqrt{2}(\cos x+\sin x)$$ My try: $$\frac{2}{\sin 2x}=\sqrt{2}(\cos x+\sin x)$$ $$\left(\frac{2}{\sin 2x}\right)^2=(\sqrt{2}(\cos x+\sin x))^2$$ $$\left(\frac{2}{\sin 2x}\right)^2=2(1+\sin…
Almot1960
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How to solve this trigonometrical equation $\tan 2x -\tan x=2$?

The equation to be solve, $$\tan 2x -\tan x=2$$ My Try : $$\tan 2x=\frac{2\tan x}{1-\tan^2x}$$ $$\frac{2\tan x}{1-\tan^2x} -\tan x=2$$ $$\frac{2\tan x-\tan x(1-\tan ^2 x)}{1-\tan^2x} =2$$ $$2-2\tan^2x =2\tan x-\tan x+\tan^3x…
Almot1960
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How to prove in this trig problem

i have to prove this $$\frac{\cos 3x}{\sin 2x \sin 4x}+\frac{\cos 5x}{\sin 4x \sin 6x}+\frac{\cos 7x}{\sin 6x \sin 8x}+\frac{\cos 9x}{\sin 8x \sin 10x} = \frac{1}{2}\csc x(\csc 2x - \csc 10x)$$ i tried taking lcm but does not leads to anything. i…
J. Deff
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Prove that: $\tan^6 20° - 33\tan^4 20° + 27\tan^2 20°=3$

Prove that: $\tan^6 20° - 33\tan^4 20° + 27\tan^2 20°=3$ My Attempt: $$L.H.S=\tan^6 20 - 33\tan^4 20 + 27\tan^2 20°$$ $$=\tan^2 20°(\tan^4 20° - 33\tan^2 20°+27)$$ $$=(\sec^2 20° -1)(\tan^4 20° - 33\tan^2 20° + 27)$$ Please help me to continue from…
pi-π
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To evaluate $\frac{\cot25+\cot55}{\tan25+\tan55}+ \frac{\cot55+\cot100}{\tan55+\tan100}+\frac{\cot100+\cot25}{\tan100+\tan25}$

To evaluate $$\frac{\cot25^{\circ}+\cot55^{\circ}}{\tan25^{\circ}+\tan55^{\circ}}+ \frac{\cot55^{\circ}+\cot100^{\circ}}{\tan55^{\circ}+\tan100^{\circ}}+\frac{\cot100^{\circ}+\cot25^{\circ}}{\tan100^{\circ}+\tan25^{\circ}}$$ i took lcm and after…
Gathdi
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If $(1+ \cos A) (1+ \cos B) (1+\cos C)= y = (1- \cos A) (1-\cos B) (1-\cos C)$ then prove that $y = \pm \sin A \sin B \sin C$

If $$(1+ \cos A) (1+ \cos B) (1+\cos C)= y = (1- \cos A) (1-\cos B) (1-\cos C)$$ then prove that $$y = \pm \sin A \sin B \sin C$$ My Work: If $y=\prod(1+\cos A)=\prod(1-\cos A)$ $y^2=y\cdot y=\prod(1+\cos A)\cdot\prod(1-\cos A)$ How, should I move…
pi-π
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Rotate Object About a Point With Crank Mechanism

I am trying to figure out how to create this physics simulation but I need some guidance on how to go about calculating it. Below is an image of the system I am working to solve. Here I am trying to find out how to calculate the angle at which to…
Konig
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Solving $\sin^2 x+ \sin^2 2x-\sin^2 3x-\sin^2 4x=0$

I am stuck on a trigonometric problem which is as follow: Solve $\sin^2 x+ \sin^2 2x-\sin^2 3x-\sin^2 4x=0$ for $x$. these kind of problems tease me always. Please help me. To get a rough idea of how can I approach such problems.
super saiyan
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Finding height and area of non-right triangle - Heron's Formula?

I would like to calculate the area for a triangle such that $a^2+b^2-c^2=1$ (an almost Pythagorean triple). I know that the triangle is non-right, so I would like to use $\text{Area}=\frac{1}{2}ab\sin C$... but I do not know how to represent $\sin…
Math1
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Prove $\sin\alpha \cos\alpha + \sin\beta \cos\beta + \sin\gamma \cos\gamma = 2\sin\alpha \sin\beta \sin\gamma$, where $\alpha + \beta + \gamma = \pi$

This identity stems from:$$\tan\alpha + \tan\beta + \tan\gamma = \tan\alpha \tan\beta \tan\gamma$$ (for $\alpha + \beta + \gamma = \pi$) which can easily be proven using the $\tan(\alpha + \beta)$ identity. However, I have been unable to prove…
Ujkan Sulejmani
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finding $\frac{\sin 2x}{\sin 2y}+\frac{\cos 2x}{\cos 2y}$

If: $$\frac{\cos x}{\cos y}=\frac{1}{2}$$ and $$\frac{\sin x}{\sin y}=3$$ How to find $$\frac{\sin 2x}{\sin 2y}+\frac{\cos 2x}{\cos 2y}$$
max
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Conversion from the linear-combination to the sinusoidal form of a sinusoidal function (simple problem, but I'm missing something.)

This is a standard trigonometric identity that can be easily verified: $$a\cos (x) + b\sin (x) = \sqrt{a^2+b^2}\cos (x - p),\text{ where }\tan(p)=\frac ba.$$ So for example, $$\sqrt2\cos(1)-\sqrt2\sin(1)=-0.43=2\cos(1-p_1),\text{ where…
ryang
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Need help with a trigonometry problem (w/ picture)

I have this trigonometry problem I got when programming a code library for cameras in games. I made a picture in Paint to explain the problem as simple as possible. Here's a link: The known values are random but it shouldn't be a problem in this…
JohannesW
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