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Questions tagged [trigonometry]

Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles and other topics relating to measuring triangles.

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0answers
26 views

A challenging trig identity

Prove that, for $g$ a nonnegative integer, \begin{eqnarray}3\left(\frac{64}{75}\sin^2\frac{\pi}{5}\sin^2\frac{2\pi}{5}\right)^{1-g}\left(\sin^{2(1-g)}\frac{\pi}{5}+\sin^{2(1-g)}\frac{2\pi}{5}\right)=\...
0
votes
2answers
28 views

Trigonometric Equation: $4\sin\theta = 3\tan\theta$

How would you find all the solutions to this question: Question Solve this equation for -180° ≤ θ ≤ 180°. Show your working. $4\sin\theta = 3\tan\theta$ My Solution $$4\sin\theta = 3\tan\theta\\ \...
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votes
2answers
24 views

Small angle approximation for $\frac{1+\sin\theta}{5+3\tan\theta-4\cos\theta}$

Please could somebody explain how the expression involving $\theta$ that $$\frac{1+\sin\theta}{5+3\tan\theta-4\cos\theta}$$ approximates to for small values of $\theta$ is $1-2\theta+4\theta^2$?
2
votes
0answers
20 views

Approximation of $\prod _{k=p+1}^{\infty } \cos \left(\frac{p \,\pi}{2 k}\right)$

After this post, I started wondering about possible approximations of the infinite product $$A_p=\prod _{k=p+1}^{\infty } \cos \left(\frac{p \,\pi}{2 k}\right)\tag 1$$ where $p$ is an integer. As far ...
1
vote
2answers
46 views

$\frac{ \sin\theta }{ \theta } = \frac{2165}{2166}$ Find the approximate value of $\theta$

$\dfrac{ \sin\theta }{ \theta }$ = $\dfrac{2165}{2166}$ Find the approximate value of $\theta$ What is the method to solve this question. (I have tried solving it by using Taylor series expansion, ...
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votes
0answers
22 views

Trigonometrical ratios [on hold]

By expressing the sin 170 in terms of trigonometrical ratios, the answer will be A) sin 10 = 0.1631 B) sin 10 = 0.1736 C) sin 10 = 0.3761 D) sin 10 = 1.7362 Please give a brief explanation for the ...
0
votes
0answers
28 views

Trigonometry equivalent

$\sin (A) = \sin 3x$ is equivalent to $\sin(A) = 3 \sin(x) - 4\sin^3(x)$, then $-\cos 3x$ is $-4\cos^3x + 3\cos$. is that correct? I just want to make sure I'm distributing the negative sign ...
1
vote
1answer
60 views

How do I integrate $4\int_0^{\pi/2}\frac{\sec^2(\theta)}{1+2\tan^2(\theta)}\,d\theta$ using symmetry?

\begin{align}\int_{-\pi}^\pi \frac{1}{1+\sin^2(\theta)}\,d\theta&=4\int_0^{\pi/2} \frac{1}{1+\sin^2(\theta)}\,d\theta\\\\&=4\int_0^{\pi/2}\frac{\sec^2(\theta)}{1+2\tan^2(\theta)}\,d\theta\\\\&...
0
votes
0answers
12 views

Given equilateral hyperbolic triangle with side “a”, angle α, and area A. [on hold]

a) compute the Taylor Series for cos(A) in powers of "a". Retain terms up to and including order a^4. b) Since cos(A) = 1- 1/2A^2, calculate area A in terms of side length "a".
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votes
3answers
47 views

Please prove this trig identity. [on hold]

Please show me how to prove this identity: Original: ((sec v - tan v)^2 +1)/csc v ( sec v - tan v) = 2tan v $$\frac{(\sec{v}-\tan{v})^2+1}{\csc{v}(\sec{v}-\tan{v})}=2\tan{v}$$ Thank you
0
votes
0answers
24 views

Solving a system of Trignometric equations

I came across this system of trigonometric equations inbetween a problem in Numerical Linear Algebra. I was required to find $p^2$ and $\cos(\theta)$ in terms of $q^{(k-1)},q^{(k)},q^{(k+1)},q^{(k+2)}$...
0
votes
1answer
19 views

Finding a third point of a triangle in 3D

I have 3 vertices in 3D: C, P and W. I know: Points C and P and therefore $\overrightarrow{CP}$ and $\overline{CP}$. A direction vector collinear with $\overrightarrow{CW}$ $\overline{PW}$ I ...
0
votes
0answers
20 views

Clarification for inverse trigonometric fuctions. [duplicate]

Since sin⁻¹(x) means the inverse of sin(x), also written as arcsin(x), how would you write 1/sin(x)? Since sin²(x) actually does mean sin(x) * sin(x), can you write 1/(sinx *sinx) as sin⁻²(x)?
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votes
1answer
25 views

Water levels during global warming

Water levels near a glacier currently average 9 feet, varying seasonally by 2 inches above and below the average and reaching their highest point in January. Due to global warming, the glacier has ...
0
votes
4answers
36 views

Minimum value of $y=\sin( 2x) - x$, where $x\in [-\frac{\pi}2,\frac{\pi}2]$

I tried applying the concept that at minima, derivative of $y$ with respect to $x$ should be zero, but realised that it fails as the domain is restricted. Rightly, upon plotting the graph, we can see ...
0
votes
0answers
7 views

Spring model function with different amplitudes

A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring t seconds ...
0
votes
1answer
18 views

Exponential decrease of amplitude with time

I was wondering about a particular math problem. It says that a particular trigonometric function, $10 \cos(2\pi x)$ models a bus going over a speed bump. They say that the amplitude decreases over ...
0
votes
0answers
40 views

How to work out this finite product?

$$\prod_{k=1}^{4n-2}\left[\sin\left(a+\frac{k\pi}{4n-2}\right)+\cos\left(a+\frac{k\pi}{4n-2}\right)\right]^{(-1)^k}\tag1$$ Suppose $a\ge0 $ and $n\ge1$ How to verify that $$(1)=(-1)^n\left[\frac{1-\...
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votes
3answers
64 views

Why does $\lim\limits_{n \rightarrow \infty} 2 \int_{0}^{n} \frac {\tan^{-1} (x)} {n}\ dx = \pi$? [on hold]

Show that $$\lim\limits_{n \rightarrow \infty} 2 \int_{0}^{n} \frac {\tan^{-1} (x)} {n}\ dx = \pi.$$
2
votes
1answer
45 views

Simplifying $\sin\frac{11\pi}{12}\sin\frac{29\pi}{12}-\cos\frac{13\pi}{12}\cos\frac{41\pi}{12}$. Why do I get the wrong answer?

Can someone explain why I get wrong answer in simplifying this expression? $$\sin\frac{11\pi}{12}\sin\frac{29\pi}{12}-\cos\frac{13\pi}{12}\cos\frac{41\pi}{12}$$ If we rewrite the expression with new ...
3
votes
2answers
77 views

Calculating $\lim_{n\to\infty}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} +\cdots+\frac{\sin(2\sqrt n)}{n\sqrt n\cos\sqrt n}\right)$

Using the trigonometric identity of $\sin 2\alpha = 2\sin \alpha \cos \alpha$, I rewrote the expression to: $$\lim_{n\to\infty}\left(\frac{\sin(2\sqrt 1)}{n\sqrt 1\cos\sqrt 1} + \cdots+\frac{\sin(2\...
0
votes
0answers
18 views

Find a point on a circle which contains a rectangle using another point, the angle between the two and the rectangle's dimensions

I'm trying to construct a mathematical formula that will calculate a point (x,y) on a circle which contains a rectangle with a width ...
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votes
2answers
59 views

Trig Identity Question Finding Value of K

If $\sin(x) + \cos(x) = k$ for what value(s) of $k$ can $\sin(x)\cos(x)=1$?
0
votes
4answers
20 views

Parametrization of a line segment using angle as parameter

I know this is probably elementary level for most people here, but I've been stuck on this problem for no less than 4 hours and I am completely clueless as to how to figure this out. Is it possible ...
-1
votes
1answer
41 views

If $\sin(x)+\sin(y)\ge \cos(\alpha) \times \cos(x)$ $\forall x\in \mathbb R$, then $\sin(y)+\cos(\alpha)$ is equal to?

If $\sin(x)+\sin(y)\ge \cos(\alpha) \times \cos(x)$, $\forall x\in \mathbb R$, then $\sin(y)+\cos(\alpha)$ is equal to ? My thinking:- I have break the left hand side on $sinC + sinD$ and right hand ...
36
votes
1answer
4k views

Why are the trig functions versine, haversine, exsecant, etc, rarely used in modern mathematics?

I was browsing through a Wikipedia article about the trigonometric identities, when I came across something that caught my attention, namely forgotten trigonometric functions. The versine (arguably ...
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votes
0answers
26 views

Circumference touching a sine wave

I'm trying to get the intersection points between a sine function and a circumference. So, i have this equations: $y = a\sin(bx + c) + d$ $(x-h)^2 + (y-k)^2 = r^2$ If i substitute the sine in the ...
2
votes
2answers
31 views

Two possible angles in a triangle

"Determinate a value for $\angle DBC$ if $\angle DAC = 2\angle DCA= 40º$ and $BC=\sqrt 3\space AD$. "The diagram is not to scale" After trying this problem, i ended with $\sin \angle DBC = \cos 20º$ ...
0
votes
1answer
33 views

Trigonometry: Model of snowfall

The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. a) Construct a function that models this behavior. b) During what ...
0
votes
1answer
46 views

Is $\tan^{-1}(-1) = 3\pi /4$ or $=7\pi /4$? I understand they're both valid solutions, but what about places where the value is added/subtracted?

For example: calculate $\int^4_2 \tan^{-1} x \, dx$ If $\tan^{-1}(-1) = 3\pi /4$, then the final answer is $\frac{-\pi}{2}$. But if $\tan^{-1}(-1)= 7\pi /4$, then the final answer would be $\frac{3}{-...
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0answers
21 views

How to find the center of an after-rotated rectangle in 2d-space?

I'm trying to write an algorithm that solves the following problem. And although there is a lot of rectangle geometry questions here on math.stackexchange.com, I have not yet found one that answers ...
4
votes
2answers
92 views

Simplifying $\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$

I am looking to simplify the following, without the use of capital Pi notation: $$\prod_{k=3}^{n-1}\cos\left(\frac{\pi}{k}\right)$$ Which is meant to produce the sequence: $\left[1,\ \frac{1}{2},\ \...
0
votes
0answers
26 views

Extract param of $\sin$ from expression $y=2(\sin b-\sin a)/(\sin c-\sin a)$

$y=2\frac{\sin b-\sin a}{\sin c-\sin a}$, where $a=q(n+0)$ $b=q(n+1)$ $c=q(n+2)$ $q=\frac{2 \pi f}{s}$ Is it possible to extract $n$ from this formula? I already try this on WolframAlpha but I do ...
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3answers
49 views

Solving Trigonometric Questions Without a Calculator [on hold]

How do I solve the following question without using a calculator?
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2answers
46 views

Find $\sin(\frac{x}{2})$, given $\tan(x) = 2$, with $0 < x < \frac{\pi}{2}$. [on hold]

Find $\sin(\frac{x}{2})$, given $\tan(x) = 2$, with $0 < x < \frac{\pi}{2}$. Which half-identity formular should I use and why?
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votes
1answer
42 views

Find $\tan \left(\frac{\theta}{2}\right)$, given $\sin (\theta) = \frac 35$, with $90^\circ < \theta < 180^\circ$ [on hold]

Find tan theta/2, given sin theta = 3/5, with 90^∘ < theta < 180^∘. I don't know how to solve it! Help? do I use the tangent half-identity formula?
1
vote
2answers
44 views

Find the median in a triangle with trigonometry

In a triangle $ABC$, $AB=7$, $AC=4$ and $\angle CAB=50º$. Let $M$ be the midpoint of $BC$. Determinate $AM$. My try I applied law of cosines $3$ times, first to find $BC$, then I let $\angle BCA=\...
2
votes
1answer
98 views

Inverse of $\frac{\sin(x)}{x}$

How would one find the inverse of the function $y=\frac{\sin(x)}{x}$? Here are my steps: $y=\frac{\sin(x)}{x}$, $x=\frac{\sin(y)}{y}$, $xy=\sin(y)$, $\arcsin(xy)=y$, After that step, I can’t find a ...
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votes
0answers
19 views

find the curve of best fit of the type $y= a \sin(bx)$ by the method of least squares [on hold]

Find the curve of best fit of the type $y= a \sin(bx)$ by the method of least squares?
0
votes
0answers
14 views

Proving $\sum_{j=0}^{N-1}\cos\frac{\left(2j+1\right)\pi}{2N}=0$ [duplicate]

Let $l\in\mathbb{Z}$ and $N\in\mathbb{N}$. I need to prove the following: \begin{equation} \sum_{j=0}^{N-1}\cos\left(l\frac{\left(2j+1\right)\pi}{2N} \right)=0 \end{equation} I tried to use Euler ...
0
votes
2answers
24 views

A puzzle about replacing $v_1$, $v_2$, $v_3$ while retaining the linear independence of the resulting set.

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section3.5 Exercise 5, I was puzzled at some of it. Here is the problem description: Exercise 5. ...
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vote
0answers
16 views

What would the law of the tangents be for a tetrahedron?

In generalised trigonometry, a corollary for the law of sines in a tetrahedron with vertices A,B,C and D is defined as being: sin (angle DAB) multiplied by sin (angle DBC) multiplied by sin (angle DCA)...
0
votes
0answers
23 views

How to find the location of a point in 3D space from projected 2D angle

I have points $A,B,C$ in 3D space and I know the position of $A=(x_1, y_1, z_1)$ and $B=(x_2, y_2, z_2)$. I want to find the location for $C$ given that $AB$ and $BC$ is perpendicular in 3D space but ...
0
votes
1answer
36 views

Find the general value of $\theta$

Find the general value of $\theta$ which satisfies the equation $(\cos\theta+i\sin\theta)(\cos3\theta+i\sin3\theta)\dots \{\cos(2n-1)\theta+i\sin(2n-1)\theta\}=1$. My attempt: $(\cos\theta+i\sin\...
1
vote
3answers
41 views

Prove $ \frac{\sin\theta}{1-\cos\theta} - \frac{\sin\theta}{1+\cos\theta} = 2\cot \theta$

Prove $$ \frac{\sin\theta}{1-\cos\theta} - \frac{\sin\theta}{1+\cos\theta} = 2\cot \theta$$ So I started by combining the two fractions, which gave me: $$ \frac{\sin\theta(1+\cos\theta) - \sin\theta(...
0
votes
2answers
25 views

Exponential double angle formula

My question is whether someone could provide a proof for the following identity: $$ \frac{1 - e^{int}}{1 - e^{it}} = e^{i(n-1)t/2} \frac{\sin(nt/2)}{\sin(t/2)} $$ Motivation: The left hand side is ...
1
vote
4answers
53 views

Find the value of $\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$

Find the value of $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ My apporach:- $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ $$=\log_2 (\sin(36^{\circ}))+\log_2 (\sin(2*36^{\circ}))+\log_2 (\sin(3*...
4
votes
1answer
65 views

Can one simplify $\arctan(a\tan(x))$?

We know that $\arctan(\tan(x))=x$ when $x$ lies between $-\pi/2$ and $+\pi/2$; but do you know a way to transform the expression $\arctan(a\tan(x))$, where $a$ is a real number between $0$ and $1$? I ...
2
votes
4answers
66 views

If $\tan 9\theta = 3/4$, then find the value of $3\csc 3\theta - 4\sec 3\theta$.

If $\tan9\theta=\dfrac{3}{4}$, where $0<\theta<\dfrac{\pi}{18}$, then find the value of $3\csc 3\theta - 4\sec 3\theta$. My approach:- $$\begin{align*} \tan9\theta &=\frac{3}{4} \\[6pt] \...