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

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6
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1answer
140 views

Help with trigonometry problem

Hey guys! I'm preparing for my college entry test and I ran into this problem in my book: $$\tan\alpha = \frac{(1+\tan 1^{\circ})\cdot (1+\tan 2^{\circ})-2}{(1-\tan 1^{\circ})\cdot(1-\tan ...
3
votes
1answer
622 views

Trig substitution for a triple integral

This problem involves calculating the triple integral of the following fraction, first with respect to $p$: $$ \int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^{2} ...
0
votes
1answer
718 views

Get the relation between X and Y axes in triangle based on the degree between

I have a given degree (0 - 360), and based on it, I'd like to be able to calculate the length of X and Y axis of a triangle built on that angle , if the third side of that triangle is equal to 1. I ...
4
votes
2answers
139 views

Simple Trig Question

Hi I am not sure I am solving this trig question correctly: $\tan\left(\sin^{-1}\left(\dfrac{2\sqrt{x}}{1+x}\right)\right) = ?$ I drew a right triangle and set an angle equal to ...
6
votes
0answers
613 views

Nasty Integral - Closed form solution?

Any suggestions on how to integrate this beast?: $$\int_0^{\omega_t}\int_{\omega_t}^f\sin^2(\omega_{12}/2)\sin^2(\omega_{23}/2)d\omega_{23}d\omega_{12}$$ where: $f = 2\pi+2\tan^{-1}(y,x)$ $y = ...
0
votes
0answers
251 views

Trig identities for $A\sin^2(x)+\cos^2(x)$

Does anyone know of any useful trig identities for manipulating $A\sin^2(x)+\cos^2(x)$? The only thing I come up with is: $A\sin^2(x)+\cos^2(x)=\frac{1}{2}(1+A)+\frac{1}{2}(1-A)\cos(2x)$ I'm trying ...
1
vote
1answer
172 views

Simplifying and finding PDF: $ \frac{\arcsin{(x \sin{\theta})} - \theta}{\sin{\theta}} $

\begin{align*} D = \frac{\arcsin{(X \sin{\theta})} - \theta}{\sin{\theta}} \end{align*} and \begin{align*} X \sim \text{Uniform}[-1, 1], \hspace{0.5in} \theta \sim \text{Uniform}[0, 2 \pi] ...
0
votes
1answer
376 views

Shooting a projectile in a 2d-space

Okay, I struggled a bit choosing wheter to put this on StackOverflow or here, Math. I concluded that it was actual math my problem and not "exactly" programming. You see, I am making a 2d game, ...
2
votes
2answers
178 views

Simplifying a Trigonometric Expression

I need to provide a simplified version of this expression for a homework: $$ \frac{\cos^{3}x - 2\cdot\cos x + \sec x}{\cos x \cdot \sin^{2}x} $$ Basically, there aren't restrictions. The simpler ...
1
vote
2answers
260 views

How to calculate the new intersection on the x-axis after rotation of a rectangle?

I've been trying to calculate the new intersection on the x-axis after rotation of any given rectangle. The rectangle's center is the point (0,0). What do I know: length of B ( that is half of the ...
1
vote
1answer
115 views

How to prove $a\cos(\frac{2n\pi}{N}) \neq \cos(\frac{(n-1)\pi}{N}))$

How to prove $a\cos\left(\frac{2n\pi}{N}\right) \neq \cos\left(\frac{(n-1)\pi}{N}\right)$ for $n = \{0,1,...,N-1\}$ where $a$ is a scalar and $N \geq 3$. I proceeded like this ...
4
votes
3answers
436 views

Prove that $ax = \cos(\pi\cdot x)$ has exactly one solution

Prove that $ax = \cos(\pi\cdot x)$ has exactly one solution when $0 \le x \le 1$. a is any positive real number. I can solve this question fine by drawing $\cos(\pi\cdot x)$ out but it's considered ...
1
vote
0answers
148 views

Formula for finding the distance between two lat/long coordinates? [duplicate]

Possible Duplicate: How do I measure distance on a globe? Say I have the coordinates of two locations. What formula could tell me the distance between the two? I'm not a math person at all, ...
10
votes
5answers
5k views

Calculate the area of the crescent

I found this problem on a thread on Stack overflow where it was posted as "job interview question". Unfortunately I cannot find the question. But I saved the picture and just cannot figure it out. ...
1
vote
2answers
624 views

Find radians reflection angle?

I'm coding a simple Flash game, just to learn flash and improve my maths, but I'm getting very confused with Radians as there are new to me. What I've done so far is using your mouse you (click & ...
0
votes
1answer
135 views

Trigonometry. Sec

$\sec(\theta)=x/5$. What does $\sin(\theta)$ equal? What does $\tan(\theta)$ equal?
2
votes
2answers
99 views

Identity for solving trig equation

I have the following type of equation which I wish to solve for $t$: $$\frac{x}{\cos(t)} - \frac{y}{\sin(t)} = z$$ I have used $c^2 + s^2 = 1$ to get it into the following form: ...
2
votes
1answer
147 views

Half angle formulas

Say I have the trig identity $$ \tan \frac{\theta}{2} = \frac{ 1 - \cos \theta }{ \sin \theta } $$ And you have the 3,4,5 triangle: taking the angle $\theta$ as marked, why can't we just say ...
1
vote
0answers
2k views

Compute Altitude and Azimuth using either Quaternions or Rotation Matrix or Roll, Pitch and Yaw component

I am struck with a mathematical problem. I want to convert the iPhone device's attitude information which is available in one of the following forms: Quaternion Rotation Matrix Roll, Pitch and Yaw ...
2
votes
1answer
373 views

3d axis rotation

I have a vector V= and several line segments Seg1, Seg2, Seg3, Seg4. I want to know how to rotate each of the line segments so that the X axis is parallel to my given vector. How can I do this? ...
33
votes
2answers
1k views

Is there an interpretation for this trigonometric identity?

A while ago I came across the following identity in an online math forum (of which I don't remember the name): $$\tan\left(\frac{\pi}{11}\right)+4\sin\left(\frac{3\pi}{11}\right)=\sqrt{11}.$$ It is ...
4
votes
1answer
453 views

Find the minimum for a trigonometric function

Find the local minimum of the following function: $$\tan\left(x+\frac{2\pi}{3}\right)-\tan\left(x+\frac{\pi}{6}\right)+\cos\left(x+\frac{\pi}{6}\right)$$ I am wondering how can I simply this ...
2
votes
3answers
517 views

Simplifying the expressions for the magnitude and phase of a Fourier transform

$$h[n] = 2( \delta[n-2]-\delta[n-1]-\delta[n-3])$$ i computed my frequency response and i have this now: $$H[e^{j \omega}] = 2[ e^{-2 j \omega} - e^{-j \omega}-e^{-3 j \omega}]$$ $$H[e^{j \omega}] = ...
5
votes
5answers
277 views

Calculate $x$, if $y = a \cdot \sin{[b(x-c)]}+d$

I am not an expert when it comes to trigonometric functions. I need to calculate value of $x$ for a program. If $y = a \cdot \sin{[b(x-c)]}+d$ then what will be the formula to calculate $x$? Please ...
4
votes
2answers
134 views

Solving Trigonometric Equation

I'm trying to solve the following equation for $t$ in the first cycle $0.8=-1.2\sin(2t)+0.8\cos(t)$ I've got it down to $0.8=[\cos(t)](0.8-2.4\sin(t))$ Is there any algebraic way to continue this ...
4
votes
2answers
252 views

Help finding solution for trigonometric equation

I have a flat mirror and a target. Given the sun's light angle of incidence, I must calculate how much to spin my mirror to hit the target with the light. The problem is shown in the following figure. ...
3
votes
1answer
5k views

Find the angle of depression

I am having trouble solving this word problem: A cellular tower that is 150ft is placed on top of a mountain that is 1200 feet above sea level. what is the angle of depression from the top of the ...
3
votes
1answer
216 views

Geometrical interpretation of trigonometric antiderivative

I know about geometrical explanation of [defininte] integral as an area under the curve, and I wonder if there are some ideas, which may give similar insight in taking antiderivatives [indefinite ...
0
votes
3answers
166 views

Trig identities solve for $\frac1{2} \tan(x)$

How to prove $$\frac{\sin^2(x)}{1+\cos(2x)} = \frac1{2} \tan^2(x)$$
4
votes
0answers
97 views

how understand if a segment is inside a lissajous curve

i am a programmer and not a math guru, but i like geometry. so if i'm not accurate in math terminology or i have folly question please sorry me. i'm drawing with a programming language the lissajous ...
3
votes
1answer
102 views

Trigonometric re-write I don't understand

$$\int \sin^3{x}\,\cos^5{x}\,dx = \int \sin{x}\,(\cos^5{x}-\cos^7{x})\,dx$$ My ignorance amuses me hehe. Even if I multiply it out I still don't get it.
1
vote
1answer
90 views

How can this be re-written with the following identity?

Can this: $$\frac{\cos x}{4 + \sin^2 x}$$ Be re-written using the fact that: $$\cot(t) = \frac{\cos (t)}{\sin (t)} = \frac{1}{\tan (t)}$$ I'm not good with algebra, but I'm getting there. I'm ...
4
votes
1answer
275 views

Confusing question: try and prove that $x -\tan(x) = (2k+1)\frac{\pi}{2}$ has no solution in $[\frac{3\pi}{4},\frac{5\pi}{4}]$

I am trying to show that $x - \tan(x) = (2k+1)\frac{\pi}{2}$ has no solution in $[\frac{3\pi}{4},\frac{5\pi}{4}]$. However, I seem to be stuck as I don't know where to begin. The only sort of idea ...
2
votes
1answer
203 views

How to find $x$ for $1 + \sin(x/2) = \cos x$?

How to find $x$ for $1 + \sin(x/2) = \cos x$ ? From the equation, I can figure out that it is satisfied at $x = 0$ by looking. How do I find the other solutions to this equation?
5
votes
4answers
135 views

Is there a statement like $\sin\left(\frac1{n^k}\right) < \frac1{n^k} \forall n,k\in \mathbb{N}$

I'm doing some exam review and I thought I might need to use something like this. Is it true? If not, is there a similar statement, and if so, how can we prove it? $$\sin\left(\frac1{n^k}\right) ...
5
votes
2answers
316 views

Trig integral $\int{ \cos{x} + \sin{x}\cos{x} dx }$

Assume we have: $ \int{ \cos{x} + \sin{x}\cos{x} dx } $ 2 ways to do it: Use $\sin{x}\cos{x} = \frac{ \sin{2x} }{2} $ Then $ \int{ \cos{x} + \frac{\sin{2x}}{2} dx } $ $ = \sin{x} - \frac{ cos{2x} ...
13
votes
2answers
998 views

Etymology of $\arccos$, $\arcsin$ & $\arctan$?

Does anyone know the origin of the words $\arccos$, $\arcsin$ & $\arctan$? That is to say, why are they named like this? What connects "arc" with inverse? Can't seem to find out via Google. ...
5
votes
3answers
3k views

In the equation $x\cos(\theta) + y\sin(\theta) = z$ how do I solve in terms of $\theta$?

In the equation $$x\cos(\theta) + y\sin(\theta) = z,$$ how do I solve in terms of $\theta$? i.e $\theta = \dots$.
1
vote
1answer
483 views

Confused about which quadrant the solution to this trigonometry question is in

Below is the textbook solution to a question I'm confused about (areas of confusion in yellow). I got $\tan\theta = \pm \frac{7}{24}$ whereas the correct answer is $\tan\theta = -\frac{7}{24}$. I ...
19
votes
3answers
495 views

If $\alpha$ is an acute angle, show that $\displaystyle \int_0^1 \frac{dx}{x^2+2x\cos{\alpha}+1} = \frac{\alpha}{2\sin{\alpha}}.$

If $\alpha$ is an acute angle, show that $\displaystyle \int_0^1 \frac{dx}{x^2+2x\cos{\alpha}+1} = \frac{\alpha}{2\sin{\alpha}}.$ My attempt: Write $x^2+2x\cos{\alpha}+1 = ...
1
vote
4answers
2k views

evaluating compositions of trigonometric functions that do not use standard points or angles from the unit circle

I need to find the exact value (so not with a calculator) of trig functions such as the following: cos(arctan(3/4)) I know how to solve these compositions when they use standard points and values ...
2
votes
1answer
597 views

Confused about “Solve $5\cos\theta = 3\cot\theta$”

I recently got this question only half correct: "Solve for values of $\theta$ the equation $5\cos\theta = 3\cot\theta$, in the interval $0 \leq \theta \leq 360$" My solution was: $$5 \cos\theta = 3 ...
3
votes
2answers
238 views

When to write that $\sqrt{x} = \pm y$?

Here is the textbook solution to a simple trigonometry problem I just completed. The Exercise was to rewrite the LHS in terms of $\cot\theta$. ...
5
votes
2answers
302 views

Solving integral $\int \frac{\sqrt{1 - x^2} - 1}{x^2 - 1}dx$

I've been asking a lot of integral questions lately. :D This is the integral I'm trying to solve: $$\int \frac{\sqrt{1 - x^2} - 1}{x^2 - 1}dx$$ By replacing $x = \sin(u)$ (thus $dx = \cos(u)du$ and ...
4
votes
4answers
397 views

A question about calculators (are $\sin^2 49^{\circ}$ and $\cos^2 49^{\circ}$ irrational?)

Are $(\sin 49^{\circ})^2$ and $(\cos 49^{\circ})^2$ irrational numbers? When you enter, $(\sin 49^{\circ})^2$ in a calculator, it shows a long number (and if it is irrational, then clearly the ...
2
votes
2answers
101 views

Calculate x and y coord where line touch

Hi I got a 10 cm long line, and it touches point 1,1 I need to calculate where it touches x and y. If I think of it like an triangle i get the following information. One side is 10 cm. You get ...
2
votes
1answer
1k views

Finding arc length

How can you find the length of the arc formed by two points on a circle? Is there any function that draws a perfect semi-circle so you can use integrals to find the value of the arc or is there a ...
3
votes
4answers
2k views

Memorizing the unit circle?

I know a quick google brings up plenty of resources on memorization techniques for the unit circle but I thought I would get the math stack exchange's opinion. What is the best way to memorize the ...
12
votes
2answers
789 views

Proving $2 ( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} )$ is a root of$ \sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$

How can one show that the number $2 \left( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} \right)$ is a root of the equation $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$?
5
votes
2answers
6k views

Is there a name for sec(x)'s relationship with tan(x)?

In a couple of trig identities, esp to do with integrals and derivatives, you see a relationship between tan(x) and sec(x). Similarly between csc(x) and cot(x). $ \frac{d}{dx}\tan(x) = \sec^2(x) $ $ ...