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

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2
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1answer
170 views

Equation for a squashed circle with exponential corners

I'm trying to define a circle that is sort of squashed in to a square-like shape with exponential(?) curved corners. Here is an image showing what I'm describing: Notice it is a circle at a small ...
1
vote
2answers
181 views

Using Taylor Series for $\sin x$ and $\cos x$ to derive $\cos{(x-a)}$ and $\sin{(x-a)}$

My professor had this problem on our last problem set but got rid of it as it was "more involved" than he thought but I am still curious as to how it would be done (Its good that he ditched it because ...
4
votes
1answer
321 views

Find a translation that fixes the graph of cosine function

Find a translation that fixes $y=\cos x$ That is, the goal is a translation of the plane that fixes this curve: I have tried this numerous times but can't seem to find a method of doing it. ...
2
votes
1answer
437 views

Finding Bearings

I have a trigonometry problem that I can't figure out. It goes like this: Points M and P are the same distance from a third point, O. The bearing of M from O is o38 degrees and the bearing of P from ...
2
votes
1answer
59 views

Trigonometric Identity $\frac{1}{1-\cos t} + \frac{1}{1+\cos t}$

I am just learning about trig identities, and after doing a few, I am stuck on this one: $$ \frac{1}{1-\cos t} + \frac{1}{1+\cos t}. $$ The only way to start, that I can think of is this: $$ ...
3
votes
1answer
400 views

How is the following function an odd function? $S(x) = \sin x/x$, $x \neq 0$

How is the following function an odd function? $S(x) = \frac{\sin x}{x}$, $x \neq 0$ I get $$\frac{\sin(-x)}{-x} = \frac{\sin x}{x}$$ which is even right? because $S(-x) = S(x)$? So unless the ...
1
vote
2answers
91 views

Help with basic trigonometry

it's been many years since I was at school and I never did algebra so I'm having a real hard time understanding trigonometry again. ALL the sites just say use this easy formula to calculate it: ...
0
votes
2answers
475 views

Sines and cosines of angles in arithmetic progression

Prove that if $\phi$ is not equal to $2k\pi$ for any integer $k$, then $$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$ ...
1
vote
1answer
57 views

An arc of one-sixth of the circumference subtends a central angle of how many degrees?

How do I find the central angle with the following information: An arc of one-sixth of the circumference subtends a central angle of how many degrees? Do I just use the formula to find the angle even ...
2
votes
3answers
52 views

Why does $\cosh (x+y)$ has a plus in its resultant contrary to $\cos (x+y)$

ie. $\cos (x+y)=\cos x \cos y- \sin x \sin y$ but $\cosh (x+y) = \cosh x \cosh y+ \sinh x \sinh y$
3
votes
1answer
106 views

Why does $b^2 = c^2 + a^2 - 2ca\cos(B)$ in trigonometry?

http://i.stack.imgur.com/l0Dw7.png I have a (what I believe to be an isosceles) triangle and the formula $b^2 = c^2 + a^2 - 2ca \cos(B)$ and I just have to "prove it". Now this really confused me as ...
8
votes
4answers
3k views

Solving 4 unknown angles in quadrilateral possible?

Given a quadrilateral with 4 fixed lengths, is there a way to solve for its angles? For example, I have a quadrilateral with 4 sides: 698.8m 512.5m 511.9m 695.8m How do we solve for its unknown ...
2
votes
1answer
104 views

Find points on a triangle

In the diagram below, I have all the points working except c,d,e,f. I need to find these points. cp and ep are known points. I dynamically calculate a and b with cp -> ep vector's perpendicular ...
0
votes
1answer
539 views

Given one endpoint on an arc of a circle and the radius and arc angle, how to calculate the other endpoint of the arc?

I have a circle with an arc beginning at point $(x,y)$. The radius is $r$, the arc angle(w/ respect to center) is $\theta$. How do I calculate the end point of the arc $(a,b)$ ? I know that the ...
2
votes
1answer
170 views

Maclaurin expansion of $\sqrt{\cos 2x}$ and $\tan^2 x$ up to degree 4

Find the Maclaurin expansion $\sqrt{\cos(2x)}$ and $\tan^2x$ up to degree $4$. I tried differentiation but it gives me something really horrible.
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vote
3answers
136 views

Converting $x^2 + 6y - 9 = 0$ to polar.

So far I got here \begin{align} (r\cos\phi)^2 & + 6 r \sin\phi- 9 = 0\\ (r\cos\phi)^2 & = 9 - 6r \sin\phi \end{align}
2
votes
1answer
158 views

Converting polar to cartesian?

So far I got \begin{align} r & = 7 / (4 - 2 \cos\theta) \\ r (4 & - 2\cos\theta) = 7 \\ r (4 & - 2( x / r ) ) = 7 \end{align} I apologize in advance for the bad formatting.
2
votes
2answers
2k views

Convert $ x^2 - y^2 -2x = 0$ to polar?

So far I got $$r^2(\cos^2{\phi} - \sin^2{\phi}) -2 r\cos{\phi} = 0$$ $$r^2 \cos{(2\phi)} -2 r \cos{\phi} = 0$$
0
votes
5answers
117 views

Trigonometry multiplication?

How would I solve the following problem? $$\cos(67.5^\circ) \cos(22.5^\circ)$$ I multiplied them using wolfram alpha and got $.353553$ but how would I find an exact value?
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vote
2answers
55 views

half-angle trig identity clarification

I am working on the following trig half angle problem. I seem to be on the the right track except that my book answer shows -1/2 and I didn't get that in my answer. Where did I go wrong? ...
3
votes
3answers
1k views

Trigonometry with multiple angle and exact value of $\tan\pi/5$

By considering the equation $\tan5\theta=0$, show that the exact value of $\tan\pi/5$ is $\sqrt{5-2\sqrt{5}}$. Do I need to evaluate the multiple angle for $\tan5\theta=0$?
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2answers
341 views

Solve $\sin 2x\sin 3x-\cos 2x\cos 3x>\sin 10x$ for $x$

I need help solving this: $$ \sin 2x\sin 3x-\cos 2x\cos 3x>\sin 10x. $$ I derived formulas for $\sin 3x$ and $\cos 3x$, but substituting them just gives me the polynomial of fifth degree on ...
5
votes
1answer
179 views

Integral representation of cosecant function

According to Wolfram website http://functions.wolfram.com/ElementaryFunctions/Csc/introductions/Csc/05/, There exists a "well-known" integral representation for the cosecant function, i.e. ...
2
votes
3answers
64 views

Two ways of solving same equation

$$\cos x - \sin x = -1$$ There are 2 methods to solve the equation: Dividing by $\sqrt{2}$ to get $\cos (\frac{\pi}{4} + x) = \cos(\frac{3 \pi}{4} \rightarrow x=(2n(\pi)+ \frac{\pi}{2} ~\text{or} ...
2
votes
1answer
578 views

Find the maximum and minimum of $\cos x\sin y\cos z$.

Given $x\geq y\geq z\geq\pi/12$, $x+y+z=\pi/2$, find the maximum and minimum of $\cos x\sin y\cos z$. I tried using turn $\sin y$ to $\cos(x+z)$, and Jensen Inequality, but filed. Please help. ...
1
vote
1answer
141 views

How to rotate base points of a scalene triangle to represent an isosceles triangle

Given the following graph: Where SP is the start point and EP is the end point of a line - how would I rotate p1 and p2 on axis SP in such a way that they would represent an isosceles triangle e.g. ...
12
votes
1answer
558 views

How does one calculate the product of $\tan 1^{\circ} … \tan 45^{\circ}?$

I have seen a question asked on yahoo asking to find the value of $\tan 1^{\circ} \cdot \tan 2^{\circ} \cdot \dots \cdot \tan 45^{\circ}$ (in degrees) I have seen various results concerning ...
1
vote
2answers
689 views

What's the integration of $\int \tan^6(x)\sec^4(x) dx$

I know I need to use $1+\tan^2(x) = \sec^2(x)$, but I don't know what to do next ? How do I get my $U$ variable so I can use $U$ substitution?
2
votes
2answers
2k views

Calculus integration problem: $\int \sin^5 (x) \cos^2 (x)\,dx$

What's the integration of $$\int \sin^5 (x) \cos^2 (x)\,dx?$$
3
votes
2answers
415 views

How to prove that $\tan x$ is strictly increasing without derivatives

How can I show that the function $f(x) = \tan x$ is strictly increasing on $[-\pi/2, \pi/2]$ without using derivatives?
0
votes
1answer
45 views

Proving the area function has an inverse

I am able to differentiate A at x using the FTC, but then I was wondering how one could show that A was one to one and prove that it has an inverse. If anybody could please help.
1
vote
2answers
50 views

Trigonometric Functions

The question is to show that $A\sin(x + B)$ can be written as $a\sin x + b\cos x$ for suitable a and b. Also, could somebody please show me how $f(x)=A\sin(x+B)$ satisfies $f + f ''=0$?
2
votes
1answer
33k views

How do I find the angles of a triangle if I only have the lengths of the sides?

Is it possible to find the angles of a triangle if I only have its sides? If so, how can I achieve this? Regarding my knowledge of triangles: Whilst I was taught trigonometry a few years ago, I ...
6
votes
0answers
244 views

Integral of a gaussian function of trigonometric functions

I need help with the analytical solution of this integral: ...
6
votes
2answers
120 views

Missing and parasite roots in the trigonometric equation.

I have this equation: $$\boxed{\cos(2x) - \cos(8x) + \cos(6x) = 1}$$ RIGHT And its right solution from the textbook is: $$ \begin{align} \cos(2x)+\cos(6x)&=1+\cos(8x)\\\\ ...
0
votes
0answers
42 views

Solving for $R$ given $\tan p=\frac{18H}{243-H^2}$ and $R(243-H^2)\cos p+18HR\sin p=1$

Ok so we start with $$\tan p=\frac{18H}{243-H^2}$$ And use this in the equation $$R(243-H^2)\cos p+18HR\sin p=1$$ To find $R$ in terms of $H$ without trig functions I have the answer by the way, ...
42
votes
3answers
950 views

Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$

Prove that $$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
2
votes
1answer
95 views

Asymptotic behavior of $\cos(\sqrt{4n+1}x)-\cos(\sqrt{4n+\alpha}x)$

While reading a paper in physics i came across asymptotic behavior of $\cos(\sqrt{4n+1}x)-\cos(\sqrt{4n+\alpha}x)$ and it was written this is equal to $O(n^{-1/4})$ for any real $\alpha$. I tried to ...
1
vote
0answers
160 views

What would be a close equation representation of this repeating line pattern?

A quick observation might conclude that this is just a sin function, but the thing I'm looking to find the answer to is the straightness between each maximum, and brief dip before and after the ...
2
votes
1answer
171 views

Find the solutions to ${ 450 }^{ { (\sin x) }^{ 3 } }+{ 273 }^{ { (\cos x) }^{ 5 } }=2$

Find solutions to, $${ 450 }^{ { (\sin x) }^{ 3 } }+{ 273 }^{ { (\cos x) }^{ 5 } }=2$$ where $0≤x≤8π$ Since $\sin x$ and $\cos x$ are in powers hence $450$ and $273$ will never be zero for any $x$. ...
0
votes
1answer
79 views

$\tan(x)$ with phase shift and period

I'm trying to transform $$y=A \tan(Bx-C)+D$$ so that there are two consecutive vertical asymptotes at $x=17$ and $x=19$. I want the period to equal $2$ and so I set $B=\pi/2$. I want a vertical ...
0
votes
2answers
117 views

How do I solve this trigonometry?

L and H are known values, how do I get (solve) $RH$: $RH * (1 – \cos ( \arctan(L/2 / RH ))) = H$ RH (not R*H) is circle radius H is difference in beween radius and line crossing to circle. L is ...
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0answers
84 views

Trigonometry Question

In trigonometry to measure the height or distance of objects we consider the distance between the observer and object to be straight. But the surface of the earth is curved. Assuming the line to be ...
2
votes
1answer
343 views

Definite double integral of a trigonometric function

I am in trouble in the calculation of the following double integral: $$\int_0^a d\rho\int_0^{2\pi}d\phi\exp(-ik\rho(\sin(\theta_0)\cos(\phi_0-\phi)+\sin(\theta_1)\cos(\phi_1-\phi)))\rho$$ Many thanks
14
votes
14answers
3k views

Funny thing. Multiplying both the sides by 0?

Alright this maybe really funny but I want to know why is this wrong. We often come across identities which we prove by multiplying both the sides of the identity by a certain entity but why don't we ...
4
votes
1answer
223 views

Does $\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(…\cos x))}}$ exist? [duplicate]

Possible Duplicate: Explaining $\cos^\infty$ Does the following limit exist? $$\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$$ If yes, find the limit. If no, please ...
6
votes
2answers
764 views

Evaluate integral with trigonometric functions in denominator

$$ \int \frac {1}{\sin x + \cos x} dx$$ How would I go about solving this?
0
votes
1answer
82 views

Find $m,n$ such that $1+m\cos(x+a)+n\cos2(x+b)+\cos3x \ge 0$ for all $x\in \mathbb{R}$

Find all $m,n$ such that $$1+m\cos(x+a)+n\cos2(x+b)+\cos3x \ge0 \textrm{ for all } x\in \mathbb{R}$$ Please give me some hints. I don't have any idea. Thanks.
7
votes
3answers
366 views

Ratio of areas of two triangles

Based on the figure below, what is the ratio of the area of triangle $CGI$ to the area of triangle $ABC$, in terms of $\theta$?
3
votes
1answer
100 views

Evaluating integral, first year calculus

We started integrals not too long ago, I understand it for the most part but I always have a problem figuring out how to solve ones involving trig identities. Like this: $$\int ...