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

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527 views

Calculate Radius for Apparent Diameter

Background Animate approaching planets, as per a journey though the Solar System. At a distance of 2 Blender units, a sphere fills the entire height of the camera's view port: This generates the ...
1
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2answers
184 views

Why the dramatic difference in the arc tangent?

Why are the two following calculations so dramatically different in results? Given ATAN(y2 - y1 / x2 - x1) ...
3
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2answers
2k views

Solving system of equations which contain sin and cos

I require some help to push me in the right direction to solve these equations. $$t_1 = P_1\sin(A)\sin(B) + P_2\cos(A)\cos(B)$$ $$t_2 = P_3\cos(A)\sin(B) + P_4\sin(A)\cos(B)$$ where $t_1, t_2, P_1, ...
2
votes
2answers
572 views

superposition of trigonometric functions with a probability consideration

This is a completely practical problem in application. Considering $$ A \sin(\phi + \phi_{0}) = \sum_{k=1}^{N} A_k \sin(\phi + \phi_k), \forall \phi \in \mathbb{R} $$ where $\phi$ is a deterministic ...
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2answers
747 views

Angle between 2 faces of tetrahedron

Two faces ABC and DBC of a tetrahedron ABCD are right-angled triangles with $\angle ACB = \angle DCB = 90^{\circ}$. Given that the edge DA is perpendicular to the face $ABC$, $\angle CBD = ...
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votes
5answers
4k views

Calculating point on a circle, given an offset?

I have what seemed like a very simple issue, but I just cannot figure it out. I have the following circles around a common point: The Green and Blue circles represent circles that orbit the center ...
9
votes
4answers
780 views

Find $\cos(x+y)$ if $\sin(x)+\sin(y)= a$ and $\cos(x)+\cos(y)= b$

Find $\cos(x+y)$ if $\sin(x)+\sin(y)= a$ and $\cos(x)+\cos(y)= b$.
6
votes
4answers
974 views

Trigonometry: Solve $(1-\cos\alpha)^2 + \sin^2\alpha = d^2$ for $\alpha$

My next step in implementing my algorithm in Java is following. It is quite difficult to explain, but I know what I need. I have this equation: Given: d Asked: $\alpha$ $$(1-\cos\alpha)^2 + ...
3
votes
2answers
400 views

Calculate measurements for a diagonal fence beam

Given the width W and the height H of a rectangle, and the thickness T of a beam extending exactly from the upper left corner to the lower right corner as shown, how do I solve for length X and angle ...
10
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1answer
572 views

Prove that minimum of $\lambda \sin \theta + (1 - \lambda) \cos \theta \le -\dfrac{1}{\sqrt 2}$

I need a little nudge to the finish for the last bit of this problem. Express $\lambda \sin \theta + (1 - \lambda) \cos \theta$ in the form $R \sin (\theta + \phi)$, where $R(R>0)$ and $\tan ...
7
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7answers
462 views

Trigonometric equality: $\frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = \tan \frac{A}{2}$

Can you guys give me a hint on how to proceed with proving this trigonometric equality? I have a feeling I need to use the half angle identity for $\tan \frac{\theta}{2}$. The stuff I have tried so ...
4
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4answers
314 views

Integrate $\frac{1}{\sqrt{1 - x^2}}$

I have to calculate $\int \frac{1}{\sqrt{1 - x^2}} \operatorname{d}x$ forwards, using known rules like partial integration or substitution. What I'm not allowed to do is simply show that ...
4
votes
1answer
154 views

If $|z| \leq \pi/2$ and $|\sin z| \leq 1/4$, then $|z| \leq (4 \sin(1/4))^{-1} |\sin z|$

I came across the following assertion and am having trouble justifying it: If $z$ is a nonzero complex number with $|z| \leq \pi/2$ and $|\sin z| \leq 1/4$, then $$ \left| \frac{z}{\sin z} \right| ...
2
votes
2answers
173 views

Derivative, sensitivity and implicit function

I have this implicit function $$y=f(x) \iff \sin(x+y)=k \sin(x), \quad$$ where $k>1$ is a constant. I would like to know how a small variation in $x$ propagates on $y$. I think I need to do an ...
7
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4answers
293 views
2
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2answers
1k views

Find $\sin \theta$ and $\cos \theta$ given $\tan 2\theta$

Can you guys help with verifying my work for this problem. My answers don't match the given answers. Given $\tan 2\theta = -\dfrac{-24}{7}$, where $\theta$ is an acute angle, find $\sin \theta$ ...
5
votes
1answer
587 views

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...
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3answers
7k views

How to find triangle height?

I need to know the height (h) of a triangle with two unknown angles (alpha and beta) and the known length of two sides AB and BC. Is it possible to have that value of h (height)?
3
votes
1answer
196 views

$ \cos(\hat{A})BC+ A\cos(\hat{B})C+ AB\cos(\hat{C})=\frac {A^2 + B^2 + C^2}{2} $

What more can be said about the identity derived from law of cosines (motivation below)$$ \cos(\widehat{A})BC+ A\cos(\widehat{B})C+ AB\cos(\widehat{C})+=\frac {A^2 + B^2 + C^2}{2} \tag{IV}$$ RHS ...
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votes
2answers
209 views

Simplifying trigonometric expressions, is there a unified theory?

$\frac{1}{3}\cos^3 x \cos(2x)+\frac{1}{12}\sin(2x)(\sin(3x)+3\sin x)=\frac{1}{3} \cos x$ I got this as the result of a differential equation that I solved. The answer in the book is (1/3) cos(x), but ...
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2answers
620 views

Rotating a rectangle

Lets say we have a rectangle with height h and width w. If we rotate it by d degrees, what ...
4
votes
3answers
725 views

How quickly we forget - basic trig. Calculate the area of a polygon

I think the easiest way to do this is with trigonometry, but I've forgotten most of the maths I learnt in school. I'm writing a program (for demonstrative purposes) that defines a Shape, and ...
11
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5answers
732 views

Deriving the rest of trigonometric identities from the formulas for $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, and $\cos (A-B)$

I am trying to study for a test and the teacher suggest we memorize $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, $\cos (A-B)$, and then be able to derive the rest out of those. I have no idea how to get ...
5
votes
1answer
443 views

Solving for the center of mass of a Semi Circle (without integration) [duplicate]

Possible Duplicate: Solving $2x - \sin 2x = \pi/2$ for $0 < x < \pi/2$ For fun, I was trying to solve this problem without doing calculus. After dinking around with it for a while, I ...
4
votes
2answers
299 views

Find angle subtended by overlapping circles [duplicate]

Possible Duplicate: Solving $2x - \sin 2x = \pi/2$ for $0 < x < \pi/2$ I need a little nudge for this problem. I have figured out everything but the last bit(albeit the most ...
-2
votes
1answer
246 views

Using the law of cosines [closed]

I am supposed to find the value of $\theta$ in a triangle. I am given a triangle with 3 lengths and no angles. The bottom left is $\theta$, bottom line is 5, right is 7 and left is 3. I have no idea ...
9
votes
6answers
468 views

Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$

Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$ (Hint: set $x = \sin y$ and use implicit differentiation) So, I tried to use the hint and I got: $x = \sin y$ ...
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3answers
363 views

Finding $\cos(s+t)$

I am attempting to find $\cos(s+t)$ and $\cos(s-t)$ I am given that they are in quadrant II and that $\cos s = -1/5$ and $\sin t = 3/5$. I have no idea what the relation between these numbers $s$ or ...
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votes
3answers
1k views

Euler formula and $\sin^3$

Using the formula: $$e^{i\omega t} = \cos {\omega t} + i\sin{\omega t}$$ I would like to prove that: $$\sin^3\;x = -\frac{\sin{3x} - 3\sin{x}}{4} $$ However I haven't found any approach ...
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votes
3answers
673 views

Finding $\csc$ with $\cot$

I know that $\cot\theta = 4/3$ how do I find $\csc\theta$? I tried to do $\csc^2\theta - \cot^2\theta = 1$ This gives me $\csc^2\theta = 1 + \cot^2\theta$ this gives $csc^2\theta = 9/9 + 16/9 = ...
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5answers
373 views

Trig test review [closed]

Well I just failed a trig test. Any help on why I did things wrong or what went wrong in my thought process? I triple checked all my answers and was positive I had 100 percent on this test, instead I ...
2
votes
1answer
45 views

determing x,y increase by angle?

If I had a position of 0,0 and I had an angle of 45 degrees (or any number) and my velocity was 1 what would be the x,y increase? For example if I had a 90 degree angle, and I had a velocity of 1. ...
5
votes
1answer
210 views

Why are these two functions equal?

These functions are equal. But I don't understand why. $$a \leftrightarrow f(x) =|\cos(2\pi x)|^2$$ $$b \leftrightarrow f(x) = \dfrac{\cos(4\pi x)}{2} + 0.5$$ Which results both in this plot:
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2answers
169 views

solving a problem using degrees OR radians

hey so i'm programming something that finds an angle of a line between 0 and 180 degrees based on two points.... the equation to find the answer is ...
6
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3answers
2k views

Definite integral: $\displaystyle\int^{4}_0 (16-x^2)^{\frac{3}{2}} dx$

The following integral can be computed using the substitution $x = 4\sin\theta~$ and then proceeding with $dx = 4\cos\theta~ d\theta~$, and evaluating the integral of $\cos^4\theta~$: ...
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1answer
66 views

find point of object in the future given direction and velocity

I have a point A, and another point B. I know the distance from point A to B. I also know the velocity with which B is moving and its direction. I also know the angle of B with respect to A. I would ...
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2answers
200 views

Trig identities

I need to prove that: $$1+\tan x \tan 2x = \sec 2x.$$ I started this by making sec 1/cos and using the double angle identity for that and it didn't work at all in any way ever. Not sure why I can't ...
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1answer
133 views

Trig identities

Finally got to double angles. Anyways I need to show that these are identities. $$\sin(4x) = 4 \sin(x) \cos(x) \cos(2x)$$ The book does some magic and gets $$2(2\sin(x)\cos(x))\cos(2x)$$ This makes ...
6
votes
3answers
427 views

$\int \cos^2 x$ - Where did I go wrong?

So when looking on the question: $$\int_{0}^{\pi} \cos^2 x \ \text{d}x$$ I would just subtract $\cos^2(0)$ from $\cos^2(\pi)$, but doing so would get me 1 - 1 = 0. When the answer is $\pi/2$. Where ...
0
votes
1answer
162 views

Calculus question - Trig Identities

Alright so I've got the question: $\int2\sin^2(x)\cos^2(x)dx$ And in class I learned: $\sin^2(x) = ((1-\cos(2x))/2)$ $\cos^2(x) = ((1+\cos(2x))/2)$ So when I substitute I get: ...
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1answer
81 views

Check if a rectangle bisects another rectangle

I've seen many examples of checking for a rectangle (A) intersecting another rectangle (B), but I'm developing something where I need to check if A is bisecting (crossing outside of) B. My intersect ...
3
votes
5answers
341 views

Finding $\sin(4a)$ if we know $\cos a$

I need to show that $\sin 4a = 0$ if $\cos a = 0$. I am not sure how to do this really. I know I can take $\sin^2 x + \cos^2 x = 1$ but I don't think that helps. I was also suppose to find the ...
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0answers
206 views

Maximum size of a rotated-then-cropped rectangle

With regard to topic/question New size of a rotated-then-cropped rectangle: The answer by Isaac, the maximum area is $b^2\csc\alpha\sec\alpha$ when $x=0.5b\csc\alpha = 0.5b/\sin\alpha$ seems to ...
2
votes
1answer
163 views

Solving trigonometric equation involving summation

For $ 0 <\theta<\frac{\pi}{2}$ find the solution of $$\sum\limits_{m=1}^{6}\csc\left(\theta+\frac{(m-1)\pi}{4}\right)\cdot\csc\left(\theta+\frac{m\pi}{4}\right)=4\sqrt{2}$$ I thought of ...
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1answer
314 views

Is there a way to reverse the effect of $a\tan2$?

I have a specific question about reversing $a\tan2$. (I am programmer, sorry for the jargon). I also use the $a\tan2$ function in my example, but I think everybody knows what it means. radial = ...
8
votes
2answers
566 views

Reduction formula for $I_{n}=\int {\cos{nx} \over \cos{x}}\rm{d}x$

What would be a simple method to compute a reduction formula for the following? $\displaystyle I_{n}=\int {\cos{nx} \over \cos{x}} \rm{d}x~$ where $n$ is a positive integer I understand that it ...
3
votes
3answers
209 views

$\frac{\pi}{2} =\tan^{-1}(\infty)$

Using the result, $\tan^2{\alpha} - A \tan{\alpha} + 1 = 0~$, where A is a constant, prove that the two solutions to this equation (such that $0 \leq \alpha \leq \frac{\pi}{2}$) are complementary ...
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vote
3answers
376 views

Factoring trig expressions

This should be simple but I am horrible at math. Anyways I forgot basic math properties and when I try to work it out in more simple terms I can't make sense of anything. Anyways I have to factor ...
2
votes
3answers
122 views

Trig identities

I need to perform the indicated operation and simplify $(1+\sin t)^{2} + \cos^{2} t$ The book is telling me that it turns into $1 + 2\sin^2t + \cos^2t$, how is is possible? Basic math tells me that ...
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2answers
3k views

How would one calculate the cosine of an obtuse angle?

How would you calculate the cosine of an obtuse triangle's largest angle? Cos = adj/hyp. But which side is the adjacent side?