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

learn more… | top users | synonyms (1)

1
vote
1answer
618 views

Distance between centers, given angle between tangents

Two circles, whose radii are 12 and 16 inches respectively intersect. The angle between the tangents at either of the points of intersection is 29°30`. Find the distance between the centers of the ...
1
vote
3answers
269 views

area of shaded figure herons formula

I am trying to figure out how to go about solving the following figures. Find the area to two decimal places. Finding the area of a triangle is fairly simple, but when it comes to these figures I am ...
13
votes
3answers
539 views

How can I calculate $\sin\left(10^{10^{100}} - 10\right)^\circ$?

How can I calculate the sine of a googolplex minus 10 degrees?
1
vote
1answer
59 views

Name of trigonometric identity

Is there a name of this trigonometric identity: $$\cos(a+b) \cos(a+c+b) \equiv \frac{1}{2} \left[\cos(c) + \cos(2a+2b+c) \right]$$ Bsaically we are "changing" a product of cosines into a sum of ...
1
vote
3answers
232 views

Manipulating trig limit functions

I'm having difficulty understanding how my calc teacher manipulated this problem What I dont understand is how he minipulated the second line to the third line. In other words, I don't understand ...
2
votes
4answers
779 views

Finding solutions using trigonometric identities

I have an exam tomorrow and it is highly likely that there will be a trig identity on it. To practice I tried this identity: $$2 \sin 5x\cos 4x-\sin x = \sin9x$$ We solved the identity but we had to ...
8
votes
2answers
244 views

How to find finite trigonometric products

I wonder how to prove ? $$\prod_{k=1}^{n}\left(1+2\cos\frac{2\pi 3^k}{3^n+1} \right)=1$$ give me a tip
18
votes
2answers
2k views

A hard definite integral with trigonometric functions

How could we get a closed form for this one? $$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x \right)\text{d}x}$$
4
votes
3answers
153 views

What is the solution for $\lim\limits_{m\to\infty}\left(\cos\frac xm\right)^{m}$?

Please, help me in solving of $\lim\limits_{m\to\infty}\left(\cos\frac xm\right)^{m}$.
7
votes
1answer
151 views

Triangle with $\sin^2 A +\sin^2 B =\sin C$

I want to know the triangles which satisfies the following equation : $\sin^2 A + \sin^2 B = \sin C$. Here $A$, $B$, $C$ are angles of a triangle. If we let $a$, $b$, $c$ to be the lengths of edges ...
0
votes
1answer
120 views

Find two points across diameter a circle, based on a known position and an isosceles triangle produced from these three points.

I want to find two points across diameter a circle, based on a known position and an isosceles triangle produced from three points that I'll describe as a (the known position) and b and c ( the ...
1
vote
2answers
128 views

Find the values of the positive constants $k$ and $c$ such that $-37\le k(3\sin\theta + 4\cos\theta) +c\le 43$ for all values of $\theta$

Hi how do i go about solving this? Find the values of the positive constants $k$ and $c$ such that $$-37\le k(3\sin\theta + 4\cos\theta) +c\le 43 $$for all values of $\theta$ $$\rightarrow-37\le ...
2
votes
2answers
511 views

Trigonometric Substitution - Arc Length

Good day to everyone, I'm not so good in maths as I wish. But, I'm doing my job to improve it. So, be patience and thanks in advance for any detailed clue. Description of the problem: The arc is ...
0
votes
1answer
3k views

Why do you multiply one way and divide the other way with these trig problems?

I am practicing finding a side of an angle on Khan Academy. I understand SOH CAH TOA and which sin, cos, tan to choose from. But, I don't understand why they multiply sometimes to find the side and ...
1
vote
2answers
95 views

What is the equation that fits this curve?

I have a curve that looks like this (it's cyclical): Curve I can get a partial fit by fitting a 3rd degree polynomial, but I have a feeling there must be a better fit (something that involves sin ...
2
votes
2answers
476 views

$\cot x=0$ is $x$ undefined?

I'm having trouble with finding the values of $x$ when $\cot x=0$ $$\cot x=\frac{1}{\tan x}=0$$ $$\tan x = \frac{1}{0}$$ Which is not possible. But $$\cot x=\frac{\cos x}{\sin x}=0$$ $$\cos x = 0$$ ...
1
vote
1answer
4k views

How would I find the area of a triangle given three sides and using either the sine/cosine laws?

Triangle ABC has sides $8.5m$ (a), $7.1$ (b), and $9$ (c). I have been asked to find the area of the triangle using trigonometry.
2
votes
1answer
378 views

How to determine if function increasing more rapidly than another function?

If I have two functions $$ f_0(x) = \sin(\pi a_0 x ) $$ and $$ f_1(x) = \sin(\pi a_1 x ) $$ How do I determine which one is increasing more rapidly with respect to itself over a closed interval (say ...
2
votes
2answers
1k views

Derivative of $\cos nx$

How to calculate derivative of $ \cos ax$? Do I need any formula for $ \cos ax$? The answer in my exercise book says it is $-a \sin ax$. But I don't know how to come to this result. Could you maybe ...
3
votes
0answers
2k views

Proof of the sine rule

So I made my first attempt at a proof. I think it turned out well. Maybe not. But I was wondering if someone could take a look at it and tell me what they think. I'd be glad to hear some criticism on ...
0
votes
1answer
56 views

Trigonometry $1+\tan{x}=\frac{\sin{(\frac{\pi}{4}+x)}}{\cos{\frac{\pi}{4}}\cos{x}}$

How does one get the following equalities ? $1+\tan{x}=\frac{\sin{(\frac{\pi}{4}+x)}}{\cos{\frac{\pi}{4}}\cos{x}}=\sqrt{2}\frac{\cos{(\frac{\pi}{4}-x)}}{\cos{x}}$
1
vote
1answer
304 views

Determining camera orientation (possibly using calibration images).

I need to generate a camera calibration pattern. Cameras are expected to be placed at an average height of 15 to 30 feet above ground pointing downwards at roughly 30 degrees. These cameras are ...
0
votes
1answer
231 views

Solving trigonometry equations from graphs when transformations (e.g. $3\tan^2$) are involved?

Preface; I'm not asking for answers (that won't help me learn Maths at all!), but certainly some guidance would be very much appreciated. I'm trying to solve some trigonometry equations from graphs, ...
1
vote
1answer
117 views

trigonometry (oblique triangle) [closed]

A surveyor wants to find the distance from point A on one side of a small lake from a vantage point C he measures two distances AC=325m and BC=235m. He also measures the angle between AC and BC to be ...
1
vote
1answer
191 views

Simplifying trigonometric summation: $ x[n] = \sum_{k = 1}^{\infty} (-1)^{k} \frac{\sin \left( 2 \pi k M \frac{a}{b} n \right)}{\pi k}$

I was reading an engineering publication and attempting to follow the math and got stuck at this "easy to show but somewhat lengthy" step. The author starts with $$ x[n] = \sum_{k = 1}^{\infty} ...
8
votes
1answer
254 views

Circle on sphere

Foreword This question was inspired by initial mistakes in this question. I wanted to explore the strange circle with $A>\pi r^2$ and got lost into geometrical jungle. A spherical cap is usually ...
0
votes
2answers
5k views

$\tan^2x - \sec^2x$ express in terms of sin/cos

I am trying to express this problem in terms of sin/cos and simplify. I couldn't figure out where to go, I tried as best I could. I know the answer is -1 but I am more interested to know how to do ...
1
vote
1answer
387 views

How to expand trigonometric equation into non quadratic terms in MAPLE

I'm working with partial differential equations at the moment and at a point I get a trigonometric equation like this: $$\sin^{2}(\pi x)\cos(2 \pi x)$$ Now I would like to know how I can expand ...
3
votes
4answers
417 views

What is limit of: $\displaystyle\lim_{x\to 0}$$\tan x - \sin x\over x$

I want to search limit of this trigonometric function: $$\displaystyle\lim_{x\to 0}\frac{\tan x - \sin x}{x^n}$$ Note: $n \geq 1$
0
votes
1answer
60 views

Is there an analytic solution to the following equation

I have the following general equation in $x$ $$a\cos(b - cx) - d\cos(e - fx) = 0$$ with constants $a,b,c,d,e,f$. Is there an algerbraic solution to this or only a numeric one?
17
votes
1answer
272 views

Some trigo identities

I aacidently found the following: $$\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}-\sin\frac{6\pi}{7}=\frac{\sqrt{7}}{2}$$ ...
1
vote
3answers
1k views

Return an array of evenly distributed points on a sphere give Radius and Origin. [duplicate]

Given a sphere of radius $r$, and origin $x,y,z$ what is the simplest way I can generate an evenly distributed array of points on the sphere $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots(x_n,y_n,z_n)$. Note I ...
4
votes
4answers
868 views

Getting the equation of an ellipse using the constant and the foci

Find the equation of the ellipse with the foci at (0,3) and (0, -3) for which the constant referred to in the definition is $6\sqrt{3}$ So I'm quite confused with this one, I know the answer is ...
4
votes
4answers
183 views

Proof of tangent half identity

Prove the following: $$\tan \left(\frac{x}{2}\right) = \frac{1 + \sin (x) - \cos (x)}{1 + \sin (x) + \cos (x)}$$ I was unable to find any proofs of the above formula online. Thanks!
2
votes
1answer
272 views

How to graph trigonometric functions

I am trying to complete some homework for my physics course and I have come to realise that I do not understand how parameters inside a trigonometric function affect the function and therefore a graph ...
4
votes
2answers
90 views

How to fit a sinusoid to 2 points and their gradients

Given the sinusoidal function $$f(x) = a \cos(n x + b) + c,$$ if I know $f(x_1)$, $f(x_2)$, $f'(x_1)$ and $f'(x_2)$ is it possible to determine $a, b, c$ and $n$, with $x \in [0,\tfrac{2\pi}{n})$ ...
3
votes
2answers
100 views

Need help with this integral using trig identities

I am trying to integrate the function $$\int_{-\pi/2}^0 \sin(2x)\cos(nx) \, \mathrm{d} x.$$ My professor has an answer of $$\frac{-2\cos(\frac{n \pi}{2})+1}{n^{2}-4}.$$ When I do this problem, I ...
0
votes
1answer
270 views

Circle intersection in radial coordinates?

We have two circles in the plane described by $C_0 = (x_0, y_0, r_0)$ and $C_1 = (x_1, y_1, r_1)$ We know that they intersect but one does not completely overlap the other. That is to say their ...
2
votes
2answers
2k views

Two circles overlap?

If we have two circles in the plane described by $(x_1, y_1, r_1)$ and $(x_2, y_2, r_2)$ we can determine if they are completely disjoint by simply: $$(x_1 - x_2)^2 + (y_1 - y_2)^2 < (r_1 + ...
1
vote
2answers
604 views

calculate the volume of water in a portion of a cone

Imagine we have a cone filled with water, if we were to take the upper portion of that cone how would we calculate the volume of water present. For example: So, in this example we have a surface ...
2
votes
2answers
315 views

Solve an equation involving the sine and the inverse tangent

The equation is $$ \sin\left(\frac{x}{x-1}\right) + 2 \tan^{-1}\left(\frac{1}{x+1}\right)=\frac{\pi}{2} $$ The answer is $0$, but I do not know how they got that.
1
vote
0answers
403 views

find shortest length of a isosceles trapezoid

Im not sure if this question is easy or not, the concept of what I'm asking seems simple but I cant figure it out. Given that A0 = 100, and h = 10, how would I calculate Az. I have begun by ...
4
votes
1answer
4k views

How to find an end point of an arc given another end point, radius, and arc direction?

Given an arbitrary arc, where you know the following values: end point (x1,y1), radius (r) and arc direction (e.g. clockwise or counterclockwise from start to end), how can I calculate the other ...
3
votes
3answers
404 views

How can I to solve $ \cos x = 2x$?

I would like to get an approx. solution to the equation: $ \cos x = 2x$, I don't need an exact solution just some approx. And I need the solution without higher mathematics (without derive and things ...
5
votes
4answers
193 views

$(\tan^2(18^\circ))(\tan^2(54^\circ))$ is a rational number

Assuming $$\cos(36^\circ)=\frac{1}{4}+\frac{1}{4}\sqrt{5}$$ How to prove that $$\tan^2(18^\circ)\tan^2(54^\circ)$$ is a rational number? Thanks!
6
votes
0answers
191 views

Rational multiples of $\pi/2$ whose sines are also rational

Let $f(x)=\sin(x\frac{\pi}{2})$. Let $R$ the set of $x$ such that : $0\le x\le 1$ $x \in \mathbb Q$ $f(x) \in \mathbb Q$ Hence, $0\in R$ as $f(0)=0$. $1\in R$ as f(1)=1. And $\frac{1}{3}\in R$ as ...
1
vote
1answer
244 views

Rotating a line segment in 3D to a prescribed orientation

I have a general line segment with endpoints $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ referenced to a 3D Cartesian coordinate frame E. I wish to rotate this coordinate fram E to a new coordinate system F ...
1
vote
1answer
299 views

Calculate speed an object is moving towards another direction

The title probably sucks because this is a really hard question to word properly, so I apologize for that. Basically I have these two variables: moveSpeed and ...
2
votes
3answers
170 views

Finding integral $\int_{- \pi}^{\pi} \cos(mt) \cos(\lambda t) dt$

I am a little stuck on the following problem: Prove that: $$\int_{- \pi}^{\pi} \cos(mt) \cos(\lambda t) dt = -2 \frac{(-1)^{m} \lambda \sin(\pi \lambda)}{m^2 - \lambda ^2}$$ I have used the fact ...
1
vote
2answers
99 views

Need help to solve the trigonometric equation $\tan\frac {1+x}{2} \tan \frac {1+x}{3}=-1$

How can I solve the equation $\tan\frac {1+x}{2} \tan \frac {1+x}{3}=-1$. Please give me some hint for that. Thank you.