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

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66 views

A trigonometric integral identity from Krylov's “Approximate Calculation of Integrals”

In the theory of Fourier series the following expansion is known $$ \operatorname{sign}\left(\sin\left((n + 1) x\right)\right) = \frac{4}{\pi} \sum_{k = 0}^\infty \frac{\sin\left((2k + 1) (n + 1) ...
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2answers
29 views

Proving a trignometric equality

I am having difficulty in proving the following trigonometric equality. It represents the conservation of energy in my physics context. The two variable $\theta_i$ and $\theta_t$ depends on each other ...
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2answers
40 views

(surface) Area of an ellipse by integrating

Given is an ellipse with $x=a\cos(t),~~y=b\sin(t)$ I do this by using $S=|\int_c^d x(t)y'(t) dt|$, so calculating the area regarding the vertical axis. Since $t$ runs from $0$ to $2\pi$ I figured I ...
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3answers
42 views

Norm of the sum of two vectors

This problem has two parts. Part a): $x$ and $y$ are vectors. If $||x|| = 7, ||y|| = 11$, what is the smallest value possible for $||x+y||$? (Note: the || || denotes the norm of a vector). This is ...
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1answer
34 views

Simplifying trigonometric equations

Here is the question: simplify the expression $$\frac{\sin(f+g)+\sin(f-g)}{\cos(f+g)+\cos(f-g)}.$$ For this questions, are all of the addition and subtraction identities of sin and cos required? I ...
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35 views

Solving Trigonometric Identities - Thinking questions

The question I have is a thinking question: If $\sin(x+y)=0.9$ and $\sin(x-y)=0.6$, determine $\sin x \cos y$. I am really not sure how to go about it. Could I use the addition formula of sin and the ...
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0answers
33 views

Combinations of Chebyshev polynomials and sin functions

By chance, I see this formula $\int_0^1 T_{2n+1}(x)\sin(ax) { dx \over \sqrt{1-x^2}}=(-1)^n\frac{\pi}{2}J_{2n+1}(a)$ but what is the closed form if we have $\int_0^1 T_{2n}(x)\sin(ax) { dx \over ...
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1answer
22 views

Limits of Integration Trig, Mag Field Infinite Length Wire

I don't understand how the limits of integration should be defined when doing basic integrals of trig functions. It seems like it's an arbitrary decision, I don't understand it. Here's the set up: ...
2
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1answer
42 views

Finding $\cos(\pi/8)$ with half angle identities

I did $$\cos\left(\frac{45^\circ}{2}\right) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$$ and ended by getting $\sqrt{\frac{2 + \sqrt{2}}{4}}$. But the answer in the book is $\frac{\sqrt{2 + ...
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1answer
29 views

Law of Cosines, Trigonometric Angle Addition Theorems, and Dot Product Relations

Just as the derivative, slope, and gradient are essentially the same thing I've realized that the Law of Cosines, trigonometric angle addition, and dot product are saying the same thing. My question ...
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56 views

Trigonometry (sec x)

How $\sec (x)=\tan (x)+\frac{\cos (x)}{1+\sin (x)}$? I'm doing some integral of sec x. Can I know how to derive it? Thanks in advance
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3answers
23 views

Inverse of trigonometric functions

I understand the intuition behind the inverses of trigonometric functions. But on the other hand formally there's no way that the trigonometric functions have an inverse because they are neither ...
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1answer
23 views

Finding the measurement of an angle.

My question is the following:Is there any way we can calculate ( by hand ) the inverse trigonometric function of a trigonometric function so that we get the measurement of the angle to which the ...
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3answers
93 views

what is the period of cos (sin nx)? [closed]

Could anyone tell me what the period of $$f(x) = \cos ( \sin {nx}); \qquad n \in \mathbb{N} $$ is.
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0answers
33 views

Values of $\arctan(x)$ by hand

How can I solve arctan problems by hand? Is this a matter of hoping for good forms and memorizing a long list of them? My question had as part of it: $$z=1-\sqrt3i$$ and I can immediately see that ...
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0answers
42 views

Why is the argument of complex number determined up to integer multiple of $2 \pi$?

I have just started learning about complex number and came across to this argument of complex number Let's say we have a complex number $z$. Then the argument of $z$ can be represented by this: $arg ...
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4answers
49 views

Proving trigonometric identity $1+\cot x\tan y=\frac{\sin(x+y)}{\sin x\cos y}$

$$1+\cot x\tan y=\frac{\sin(x+y)}{\sin x\cos y}$$ I have worked through most of this question, and I believe I am so close to finding the answer, but I have run into some issues where I am not sure ...
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3answers
31 views

Prove the trigonometric Identity involving secant

The question I am currently working on is: $\sec^2x-2\sec x\ \cos x+\cos^2x=\tan^2x-\sin^2x$. Okay, judging by the expression here, I am going to need to work with the left side of the equation ...
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2answers
45 views

Verify this identity

I need to verify this identity but I have no clue how to solve it. I have tried many different ways but haven't been able to figure it out. $$\frac{\cos^2(t)+\tan^2(t)-1}{\sin^2(t)}=\tan^2(t).$$
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2answers
19 views

$\sin x \sin(\pi/2-x)=\sin x \cos x $?

During a question involving proving a trigonometric identity, I was given help in which one of the lines showed that $\sin x \sin(\pi/2-x)$ equals $\sin x \cos x$? Could anyone please explain to ...
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1answer
31 views

How do I add roll to Pitch and Yaw?

How to get the effect of a roll axis as a sum of yaw and pitch axes? I have two axes, and I need to add 'roll' to them in a way that would change the yaw/pitch but not actually add a roll third axis. ...
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2answers
44 views

Proving Trig. Identities [duplicate]

$\cot x=\sin x \sin(\pi/2 -x) + \cos^2x \cot x$ I'm having difficulty with figuring out how to prove trigonometric identities. I know that in order to do these you need to use the trig ratios ...
0
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2answers
32 views

Rewriting an expression in the form of $A \sin(x + C)$

The problem asks to rewrite $$\sin(x) - \cos(x)$$ in the form of $A\sin(x + C)$, using the reduction formula. The answer is supposed to be $\sqrt{2}\sin(x - \pi/4)$, or $\sqrt{2}\sin(x - 45)$ using ...
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0answers
30 views

How can you quickly ease a sine wave's amplitude?

I'm currently drawing a sine wave using the following function in java: Math.sin(x)*currentAmplitude The problem I'm experiencing is that, although this works ...
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1answer
39 views

Proving trigonometric identities

Prove: $\cot x=\sin x\sin(\pi/2-x)+\cos^2x\cot x$. Hi there! So this problem asks to prove this trigonometric identity. I am not sure how to approach these problems other than needing to know the ...
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2answers
24 views

Simplify the expression using trigonometric identities

Simplify: $$\frac{\sin(x)\cos(x)}{1-\sin^2(x)}$$ This looks to be similar to the Pythagorean identities: $\cos^2(x)=1-\sin^2(x)$. However, I am not certain about how to approach this. I'm thinking ...
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0answers
39 views

How can I derive Polarization Operator algebraically

Can anyone describe for me the algebraic steps needed to create what Prof. Leonard Susskind calls, in one of his early Quantum Mechanics courses, the "Polarization Operator Matrix"? It is a 2 by 2 ...
0
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1answer
29 views

Finding Centre of a Trapezoid for rotation

I'm trying to find the centre of a trapezoid, and am attempting to then rotate it around its centre. Right now, i'm finding the centre, translating it to the origin, rotating it around the origin, ...
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2answers
55 views

Integral of $\frac{\sin x}{1+\sin^2x}$ from 0 to $\pi/2$

I am trying yo find $\int_0^{\pi/2}\frac{\sin x}{1+\sin^2x}dx$. So far I have tried using the substitution $\tan u=\sin x$ which led me to $$\int_{u=0}^{u=\pi/4}\frac{\sin x}{\cos x}du$$ ...
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2answers
39 views

Simplifying trig expression

I have $$\frac{\tan{15^\circ}}{1-\tan{15^\circ}^2}$$ and need to simplify it. The only equation I have that is even close to a match for it is $2\frac{\tan{15^\circ}}{1-\tan{15^\circ}^2}$. But the ...
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108 views

Has the age at which we teach Mathematics changed over the last two centuries?

My experience of learning Advanced Trigonometry and Calculus is that it was done to 17 and 18 year olds (School Curriculum in Australia). I assumed that it was similar in the UK, US and Europe. In ...
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1answer
47 views

Interesting trig integration pattern?

I was recently playing around with some easy volume by revolution problems that I just randomly make up for fun, and I found a weird and interesting pattern that I can't easily (or otherwise) explain. ...
0
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1answer
26 views

System of equations after reducing vectors

Part a): Find the real number k such that the equation has no solutions in a and b. This was originally a vector problem, but I reduced this down to a system of equations: $b – 2 = 2a$ $-2b -2 = ...
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0answers
30 views

How to calculate Arc Cosine, without a unit circle.

How can I calculate the $\arccos(Y)$, neither using the trig functions on a calculator, nor using a unit circle? I would be satisfied if I got $\arcsin(Y)$ or $\arctan(Y)$ instead, but I would prefer ...
0
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1answer
26 views

Roots of trigonometric equation

In the following trigonometric equation $$1 + \alpha^2 \cos^2 (n \theta) = 0$$ The complex solutions are $$\cos (n \theta) = \pm i/\alpha$$ So I thought that the correspondant angles were $$n ...
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2answers
72 views

Find the exact value of $\cos(11\pi/12)$.

This question I look at as being similar to $\sin(7\pi/12)$. You can break it up using the special triangles into $3\pi/12 + 4\pi/12$. However with this one, I can't find one of the angles in which ...
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2answers
23 views

Finding Exact values using compound angle formulae

Find the exact value of each expression: 1) $\sin{(-\frac{\pi}{2} +\frac{\pi}{3})}$ -For this question, it would appear as though you could use the addition compound angle formula ...
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1answer
67 views

What are the distances from a line to the tangents of a circle?

I have a line given by two points, and a circle given by its origin and radius. I need to find the perpendicular distance between the line and the two tangents of the circle that are parallel to the ...
0
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1answer
10 views

Simply function F and find alpha for which F will be min

I have point coordinates like [x, y], where x and y are positive natural numbers. I need to ...
3
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1answer
58 views

Proving that $\cos(\pi-\phi)=-\cos\phi$ geometrically

I want to geometrically prove that $\cos(\pi-\phi)=-\cos\phi$ without resorting to the unit circle or trigonometric formulas, but have difficulties figuring it out. It's easy enough to do the sine, ...
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2answers
94 views

Trigonometric Integration

Would someone please explain the steps taken to arrive at the third and fourth step? Why is it $(1-u^2)$? Thank you very much.
4
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1answer
138 views

Solve for sin(x)+sin(2x)+sin(3x)+…+sin(nx)=x

The question is $\sin (x)+\sin(2x)+.....+\sin(nx)=x$ where $n$ is any natural number. I used de Moivre formula to obtain the sum of $\sin(x)+\sin(2x)+...+\sin(nx)$, and I differentiated it to get $$ ...
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0answers
16 views

Workout line segment inside expanding circle

I have what is probably a fairly basic math problem for a game I'm creating. On each frame I need to work out how much a sub segment of a line passing though a circle will expand when the circle ...
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1answer
29 views

Simple trig question.

There exists an isosceles triangle. The sides are 6, 6, and 8. I am to find what all of the angles equal. My method to do this was to split the triangle in half giving me a right triangle where side ...
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2answers
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Exact values of $x$ for $2\cos^2x=1+\sin x$ [duplicate]

This question involves finding the exact values of $x$ such that $0 \leqslant x \leqslant 2\pi$. So far I have subtracted everything to the left side of the equation and then used the pythagorean ...
0
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3answers
31 views

Determine the exact value of equations involving more two trig variables

$2\cos^2x=1+\sin x$. Determine the exact values of $x$ such that $0 \leq x \leq 2\pi$. I am experiencing problems with factoring this question. First I started by getting everything on to the ...
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2answers
131 views

Is there a way to simplify a sum of cosecants?

A problem I have been working on recently results in a sum of cosecant terms. Specifically, $f(n) = \sum_{k=1}^n \csc \frac{\pi k}{2n+1}$ $g(n) = \sum_{k=1}^n [(-1)^{k+1}(\csc \frac{\pi k}{2n+1})]$ ...
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1answer
36 views

Determining the exact value of trigonometric functions using tan

Use the special triangles to give exact solutions where possible. Find all values of x such that $0 \le x \le 2\pi$. The question I have is $\tan^2x=1$. What I have done so far (it appears that ...
3
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1answer
22 views

Special Triangles and their related acute angles

So I've been working on some questions involving having to find the exact value of trig. functions involving a particular interval. I have worked through the question but now I have something I am ...
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1answer
34 views

$ \sum_{m=1}^{6}\frac{1}{\sin \left\{\theta+\left(m-1\right)\cdot \frac{\pi}{4}\right\} \sin \left\{\theta+m\cdot \frac{\pi}{4}\right\}} = 4\sqrt{2}$

If $\displaystyle 0 < \theta < \frac{\pi}{2}$ and $\displaystyle \sum_{m=1}^{6}\frac{1}{\sin \left\{\theta+\left(m-1\right)\cdot \frac{\pi}{4}\right\}\cdot \sin \left\{\theta+m\cdot ...