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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Calculate the angle between the lines

I need to calculate the angle x. I feel like I'm missing something basic since this is supposed to be an easy exercise but I can't seem to get it. It's under the chapter trigonometry. AB is 8 units long, the radius is 5. I was looking for right…
Mixoftwo
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If $ \cos x +2 \cos y+3 \cos z=0 , \sin x+2 \sin y+3 \sin z=0$ and $x+y+z=\pi$. Find $\sin 3x+8 \sin 3y+27 \sin 3z$

Problem : If $ \cos x +2 \cos y+3 \cos z=0 , \sin x+2 \sin y+3 \sin z=0$ and $x+y+z=\pi$. Find $\sin 3x+8 \sin 3y+27 \sin 3z$ Solution: Adding $ \cos x +2 \cos y+3 \cos z=0$ and $\sin x+2 \sin y+3 \sin z=0$,we get $ (\cos x+\sin x) +2(\cos y+\sin…
rst
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What does $\tan x = 1$ mean?

I can solve for $x$ by taking the arctangent of both sides but I'm not understanding what the equation means. Does the equation represent the interesection between $y = \tan x$ and $y =1$? Likewise is $\sin x = 2$ said to be undefined as the two…
salman
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How can I determine the value of $\theta$

If $\frac{\sin\theta-\cos\theta}{\sin\theta+\cos\theta}=\sqrt3-2$, then determine the value of $\theta$. Help appreciated
user88232
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Solving the equation $8\Delta = \left( {b + c} \right)\left( {bc + 1} \right)$

In $\Delta ABC$, $8\Delta = \left( {b + c} \right)\left( {bc + 1} \right)$ then circumradius of is $\Delta ABC$ is ( where $\Delta$ denotes area of triangle and b, c are length of sides AC and AB respectively) (1) $\sqrt \Delta $ (2) …
Samar Imam Zaidi
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$\sec^2\theta+\csc^2\theta=\sec^2\theta\csc^2\theta$

I was playing around with trigonometric functions when I stumbled across this $$\sec^2\theta+\csc^2\theta=\sec^2\theta\csc^2\theta$$ Immediately I checked it to see if it was flawed so I devised a…
Ali Caglayan
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Finding $2\sin x + 4\sin y$, given $\sin^2x+\cos^2y = \frac{11}{16}$ and $\sin\frac12(x+y) \cos\frac12(x-y) = \frac{5}{8}$

I think my basics are pretty weak, I am not able to solve this question. $$\begin{align} \sin^2x+\cos^2y &= \frac{11}{16} \tag1 \\[4pt] \sin\frac12(x+y) \cos\frac12(x-y) &= \frac{5}{8} \tag2\\ \\ \end{align}$$ Find the value of $$2\sin x + 4\sin…
NotRamanujan
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Another proof for $\,\cos(\beta-\alpha)=\sin\alpha\sin\beta+\cos\alpha\cos\beta\,$ formula

I was solving a vector’s problem : The question was: Find the length of $\vec{a} + \vec{b}$ vector in terms of $a,\,b,\,\alpha,\,\beta\,.$ Well, we know $\vec{a}$ equals to $\,(a\cos\beta,a\sin\beta)\,$ and $\vec{b}$ equals to…
forough m
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How can I transform a point on a square to a point on a circle?

Given a unit square of bounds (-1, -1), (-1, 1), (1, 1), (1, -1) how can I find a point on a circle of radius 1 inside the square such that each increment on the square represent an evenly spaced incremented point on the circle. I tried the…
Bugbeeb
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Trig identity $1+\tan x \tan 2x = \sec 2x.$

I need to prove that: $$1+\tan x \tan 2x = \sec 2x.$$ I started this by making sec 1/cos and using the double angle identity for that and it didn't work at all in any way ever. Not sure why I can't do that, but something was wrong. Anyways I looked…
Adam
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About the bounds of accuracy for $\sum_{r=0}^a\binom{a}{r}\sum_{m=0}^r\sum_{k=0}^m\binom{m}{k}\binom{r}{m}(-1)^{m-k}2^{m-k}\sin(k)=\sin(a)$

I noticed that the following sine formula: $$\sum_{r=0}^{a} \binom{a}{r} \sum_{m=0}^{r} \sum_{k=0}^{m} \binom{m}{k} \binom{r}{m} (-1)^{m-k} 2^{m-k} \sin(k) = \sin(a)$$ seem to hold for integer values of 'a' but only numerically up to 7. 8 and above,…
Ryan Kane Harlow
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Height of an object given two projections

So have to measure the height of an object $c$ given two known projections ($a$ and $b$). The angle $\phi$ between the two projections is known (it is 85.5°) What is the length of $c$? (in terms of $\phi$, $a$ and $b$) Before I was using the…
Manuel Ruiz
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What happened to the domain here?

If we start with: $$\dfrac{1-\cos(2\theta)}{\sin^2(2\theta)}\;,$$ we can simplify that to , $$\dfrac{1-(1-2\sin^2(\theta)}{4\sin^2(\theta)\cos^2(\theta)}\;,$$ which simplifies further…
Nav Bhatthal
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Maximum value of $y=\tan^{-1}x- \tan^{-1}\frac{x}{3}$

Let's suppose $f(x)=\tan^{-1}x$ and $g(x)=- \tan^{-1}\frac{x}{3}$ I know that the range of $f(x)$ is $$-\frac{\pi }{2} < \tan^{-1}x < \frac{\pi }{2}$$ and the range of $g(x)$ is the same. I used desmos to simulate the graph of $y$ and found that the…
kauselis3000
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