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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Can cos(n!) in degrees tend to one if n>6?

does cos(n!) in degrees tend to 1. consider cos(n!)=cos(n*...*6!),6!=720=360*2.So this is like rotating on the plane n*...7*2 times so cos(n!)=1,When n>6 .Does this proof hold even when n tends to infinity.please give a reply which suits the mind…
Thomas Augustin
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Finding the value of $\frac{\cos^4\beta}{\cos^2\alpha} + \frac{\sin^4\beta}{\sin^2\alpha}$.

Trigonometry $\dfrac{\cos^4 \alpha}{\cos^2 \beta}+ \dfrac{\sin^4\alpha}{\sin^2\beta} = 1$ then the value of $\dfrac{\cos^4\beta}{\cos^2\alpha}+ \dfrac{\sin^4\beta}{\sin^2\alpha}$ is? NOTE: can somebody help me $\cos^2\alpha \left(\frac{\cos^2…
jyothika1
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Triangle with two angles separated by side length

If a triangle has two angles (30◦ and 50◦ respectively) separated by a side of length 8, is it possible to find the lengths of the other two sides using Sine Law or Cosine Law? If not, why not? If one of those Laws makes it possible, which one and…
Jason
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Find the maximum and minimum value of $ \sin ^4 \theta + \cos ^4 \theta $?

Please suggest suitable approach for this problem
Trewick Marian
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Evaluating an inv. tan function

The problem: Evaluate the inv. function by sketching a unit circ., finding the angle, and eval. the correct pair on the circle. Function: $\tan^{-1}(-1)$ I found a solution for this, but my teacher told me he'd prefer that I draw my conclusion by…
user2451412
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Simplifying a trigonometric expression: $\cos(\tan^{-1}x)$

The problem: Simplify the expression. Specify the range of $x$ for which the simplification holds: $\cos(\tan^{-1}x)$. So we know that, $\tan^{-1}x$ is the angle $\theta$ for which $\tan\theta = x$. So I sketched a triangle, much like I would as…
user2451412
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Trigonometric equation $2\sin x+\cos x+1=0$

I have to calculate $\dfrac{d}{dx}\dfrac{1+\cos x}{2+\sin x}=0$. I have already simplified to: $2\sin x+\cos x+1=0$, but I have no idea how to go further.. Could someone give a hint?
rae306
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How can I bring $\sin(x)$ to the following form?

What steps do we take for the following? $$\sin x = \frac{{2\tan\frac{x}{2}}}{1+\tan^2\frac{x}{2}}$$
user3601507
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How to prove that $\tan^2(\frac\theta2)= \tan^2(\frac\alpha2)\tan^2(\frac\beta2)$?

I'm unable to solve this question: $\cos(\theta)=\dfrac{\cos(\alpha)+\cos(\beta)}{1+\cos(\alpha) \cos(\beta)}$ Prove: $\tan^2\left(\frac\theta2\right)= \tan^2\left(\frac\alpha2\right)\tan^2\left(\frac\beta2\right)$ I have tried the…
Harshal Gajjar
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Finding the value of trigonometric function of any angle?

Whenever I have to calculate the value of a given trigonometric function for an angle, I always refer to a table similar to this: But what if I want to find the value for sin$\theta$, where $\theta$ = 32$^{\circ}$ or $\theta$ = 49$^{\circ}$ or for…
Always Learning Forever
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Ranges in trigonometry

How to find the range of the sum or difference of two trigonometric functions? $2\sin x-3\cos x$ Before this whenever the question of range i have solved they were either single trigonometric function or if they were in pair then they were in form…
RutvikSutaria
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How to evaluate $\sum_{k=1}^n\ln\left(2\cos\left(\frac{2\pi\cdot3^k}{3^n+1}\right)+1\right)$

By using wolfram alpha, it seems like that $$\sum_{k=1}^n\ln\left(2\cos\left(\frac{2\pi\cdot3^k}{3^n+1}\right)+1\right)=0 \text{ for all }n\in\mathbb{N}.$$ But I don't know how to prove this identity. Thank you very much.
kong
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Help showing equality involving $\tan$ function

$$\Large\frac{\left(\frac{\tan \frac \pi 4+\tan x}{1-\tan\frac \pi 4\tan x}\right)}{\left(\frac{\tan \frac \pi 4-\tan x}{1+\tan\frac \pi 4\tan x}\right)}=\frac{\left(\frac{1+\tan x}{1-\tan x}\right)}{\left(\frac{1-\tan x}{1+\tan x}\right)}$$ I am…
Always Learning Forever
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Does Niven's theorem apply to cosine function?

Niven's theorem says that if $\theta$ is a rational multiple of $\pi$ and $\sin \theta$ is rational then $\sin \theta = 0, -\frac12, \frac12, -1, 1$. But is this theorem applicable to cosine function?
Yes
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Simplifying a trigonometric identity

Simplify $1 + \tan^2x$ My attempt: $$\begin{align}1 + \tan^2x&\\ &= \frac{1}{1} + \frac{\sin^2x}{\cos^2x}\\ &= \frac{1(\cos^2x)}{1(\cos^2x)} +\frac{\sin^2x}{\cos^2x}\\ &=\frac{\cos^2x}{\cos^2x}+\frac{\sin^2x}{\cos^2x}\\ &=\frac{\cos^2x +…
Qwerty
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