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.

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$\arctan$ of constant multiplied $\tan \varphi$

Helllo, I'm new here. I will try this great place to get my answer. I have a next problem: $$\vartheta = \arctan(K \tan \varphi)$$ $K > 0$ Where $K$ is a constant. If $K = 1$ then $\vartheta = \varphi$, but I can't find more simple answer for…
Grega
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Trigonometric functions - finding solutions

Question: Find the general solution for the equation: $$\sin x + 2\sin2x - \sin3x = 3$$ Approach: Well using the identity of $\sin a - \sin b $, I merged together $\sin x - \sin3x$ And as $\sin 2x = 2\sin x\cos x$, I got up till here: $$2\sin…
Gummy bears
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Calculate Angle between Two Intersecting Line Segments

Need some help/direction, haven't had trig in several decades. On a 2 dimensional grid, I have two line segments. The first line Segment always starts at the origin $(0,0)$, and extends to $(1,0)$ along the $X$-axis. The second line Segment…
Patrick
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Solve for $x$: $\csc^{100}x+\tan^{100}x=1$

Solve for $x$: $$\mathrm{csc}^{100}x+\tan^{100}x=1$$ I have tried it so many times but couldn't draw any conclusion. Please help.
Pratyush
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If $\left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right)$

If $\displaystyle \left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right) = $ $\bf{My\; Try::}$ Given $\displaystyle \left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right)…
juantheron
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Verifying $\sec^2x + \tan^2x = (1-\sin^4x)\sec^4x$

Verify: $$\sec^2x + \tan^2x = (1-\sin^4x)\sec^4x$$ My…
user144809
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I got a question about trig on the unit circle

The unit circle is defined to be $x^2 + y^2 = 1$. Makes sense. Its an equation of a circle. Now from here if we think about cosine as the $x$ value and sine as the $y$ value then we get a trig identity most of use know. There was something about…
Nick Freeman
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Calculate the sine of an angle in coordenate geometry

Given two lines $a_1x+b_1y+c_1 = 0$, $a_2x+b_2y+c_2 = 0$ that make an angle $\alpha$ at their intersection, show that $$\sin\alpha = \frac{a_2b_1-a_1b_2}{\sqrt{a_1^2+b_1^2}\sqrt{a_2^2+b_2^2}}$$ So I'm stuck here, I'm thinking making a circle with…
Rono
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Calculation of $ \cos\left(\frac{2\pi}{7}\right)+\cos \left(\frac{4\pi}{7}\right)+\cos \left(\frac{6\pi}{7}\right)$

Calculation of $\displaystyle \cos\left(\frac{2\pi}{7}\right)+\cos \left(\frac{4\pi}{7}\right)+\cos \left(\frac{6\pi}{7}\right)$ and $\displaystyle \cos\left(\frac{2\pi}{7}\right)\times \cos\left(\frac{4\pi}{7}\right) \times…
juantheron
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Tricky trigonometric sum evaluation

Prove that the sum $$\sum_{k=1}^{n-1} (n-k)\cdot\cos\left(\frac{2k\pi}{n}\right) $$ Is an integer for any $n\geq 3$. I found this in my textbook but am unable to evaluate this sum. Any help would be appreciated.
user34304
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Understanding the Cosine Rule

What is actually the Cosine rule? Can anyone explain it to me in way that I can understand it, explain in a simple way? that provide simple examples? (Only the Cosine rule) Thanks in advance. I tried googling online but most of it is full of…
Kyel David
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Why is the sine and cosine always between $-1$ and $1$?

Why is the sine and cosine always between $-1$ and $1$? If I would have circle with a radius other than $1$, then it wouldn't be between $-1$ and $1$ anymore, would it? This also ties in with another thing I'm getting confused about: the $x$ and…
Garth Marenghi
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Is this true? $\forall x, y \in\mathbb{Q}: (\sin(x)=\sin(y))\Rightarrow (x=y)$

I just thought about the following expression: $\forall x, y \in\mathbb{Q}: (\sin(x)=\sin(y))\Rightarrow (x=y)$ I think it is true because values of $\sin(x)$ only repeat every $\pi\times n$th time, which is never reached by any rational number. Is…
Kevin Steiger
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Trigonometry equation with arctan

Solve the following equation: $\arctan x + \arctan (x^2-1) = \frac{3\pi}{4}$. What I did Let $\arctan x = \alpha, \arctan(x^2-1) = \beta$, $\qquad\alpha+\beta = \frac{3\pi}{4}$ $\tan(\alpha+\beta) = \tan(\frac{3\pi}{4}) = -1$ $$\frac{\tan\alpha +…
B. Lee
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how to draw an arc?

How to draw an arc, I have these values x // x coordinate y // y coordinate r // Arc radius startAngle // Starting point on circle endAngle // End point on circle clockwise // clockwise or anticlockwise where x,y is the center of circle. Can…
coure2011
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