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

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24 views

Why does $\DeclareMathOperator{arccot}{arccot}\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$ diverge?

Why does $$\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$$ diverge? In my textbook it says that from the positive side it's zero, and from the negative side it's $\pi$. However, when entering ...
0
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0answers
16 views

Trigonometry markup

Imagine we have the following problem; $$\cos(x) = \cos(a) \Rightarrow x=a+k\times 2\pi\\ or \\x=-a+k\times 2\pi$$ And we have the following answers.. : $$a=\frac{\pi}{3} \\or \\a=-\frac{\pi}{3}$$ ...
0
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1answer
16 views

How do you calculate the change in thickness of a cylinder, if you shave off a flat section?

I have a piece of steel, cylindrical (hollow), 200mm outside diameter with 160mm inside diameter (...
0
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2answers
31 views

Solving $\sec(3\alpha+30^\circ)=\csc(7\alpha-40^\circ)$

Can you solve for $α$ in degrees/radians and tell me exactly how to do so? $$\sec(3\alpha+30^\circ)=\csc(7\alpha-40^\circ)$$
0
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0answers
28 views

How can I change a summation of cosines to a product of cosines for higher degree functions?

I was wondering if I can get an alternate form of a sum over cosines $$\sum_{n=1}^m\cos{f(n)}$$ and I found that I can. We must however make a modification to the upper limit with $m\rightarrow2^m$. ...
2
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3answers
24 views

New coordinates after clockwise rotation of triangle?

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X-Y$ plane about the vertex $P$ by angle ...
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1answer
34 views

Trigonometric solution for $\int_{0}^{2 \pi} \sin^n (x) \cos^m (x) dx $?

At home I came across the exercise and had to compute: $\int_{0}^{2\pi} \sin^n (x) \cos^m (x) dx $ with $m$, $n \in \mathbb{N} $ My current set of tools for solving problems of that kind is rather ...
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1answer
27 views

Conditional Proof in Trigonometry

If $\sin\theta + \sin\alpha=m$ and $\cos\theta + \cos\alpha=n$, prove that: $$\frac{\sec(\theta+\alpha)}{2}=\frac{\sqrt{m^2+n^2}}{2}$$ My attempt\ given: $$\sin\theta+\sin\alpha=m$$ $$2 ...
0
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1answer
15 views

Trigonometry Proving

If $\sin\theta + \sin\alpha=x$ and $\cos\theta + \cos\alpha=y$, prove that ; $$\frac{\tan(\theta - \alpha)}{2} = \pm\sqrt{\frac{4-x^2-y^2}{x^2+y^2}}$$ Attempts: Here $\sin\theta + \sin\alpha=x$ ...
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2answers
41 views

How do I show that $n=2$ is the only integer satisfy :$\cos^n\theta+ \sin^n\theta=1$ for all $\theta$ real or complex?

It is well known that :$\cos²\theta+ \sin²\theta=1$ for all $\theta$ real or complex ,I would like to ask about the general equality :$\cos^n\theta+ \sin^n\theta=1$ if there is others values of the ...
2
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4answers
83 views

Show if $A^TA = I$ and $\det A = 1$ , then $A$ is a rotational matrix

Show if $A^TA = I$ and $\det A = 1$ where $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $, then $A =\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & ...
2
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0answers
31 views

Find elevator height given rope length?

This question is deceptively difficult. I feel like it's probably some classic example somewhere, but I'm not sure how to describe it in enough detail to get valid results in searching online. ...
2
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4answers
41 views

Prove that $16\cos^5A-20\cos^3A+5\cos A=\cos5A$

Prove that $$16\cos^5A-20\cos^3A+5\cos A=\cos5A$$ My solution begins here; $$ \begin{align} \text{RHS} & =\cos5A \\ & =\cos(A+4A) \\ & =\cos A\cos4A-\sin A\sin4A \\ & =\cos ...
7
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5answers
437 views

Solve the following trigonometric integral [on hold]

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
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0answers
13 views

Prove Bernoulli Function is Constant on Streamline

I have an incompressible, inviscid fluid, under the influence of gravity, with a velocity potential: $$ \mathbf{u} = (-\cos(x)\sin(y), \sin(x)\cos(y), 0) $$ Using Euler's equations, $$ \mathbf{u} ...
0
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3answers
36 views

Why dividing by trigonometric functions gives wrong answer when solving trigonometric equations?

Hello I have a problem with solving Trigonometric equations. Why this is not true for $0\le\theta\le360$ $$2\sin\theta\cos\theta=\sin\theta$$ $$2\cos\theta=1$$ Set of solutions $\theta=60,360$ and ...
2
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2answers
27 views

Find Length of line which has rotating object.

I have 3 Images. A, B and C. if I place it on graph, its look something like this. Now main image is A and I place B and C on that image's (A) center point. For easy understanding, let's consider ...
2
votes
1answer
38 views

Are there complex numbers whose sines are zero?

I recently learned that $\sin(z)$ has an extension into the complex plane, namely: $$\frac{e^{iz}-e^{-iz}}{2i}$$ Is there any complex number $z=a+bi$, with $b≠0$ such that $\sin(z)=0$ ? I am ...
0
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0answers
16 views

Translate Pitch and Roll Angles of Object to those at different Yaw

I have been trying to find a method to translate the pitch and roll angles of one object to those of another connected object at a different yaw - i.e I have an IMU mounted on a quadcopter frame and a ...
0
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2answers
35 views

Finding the solutions of $\sin\left( x - \frac{\pi}{4} \right) = \sin\left( 3x + \frac{\pi}{4} \right)$

Find all the solutions for $x$ in the following equality. $$ \sin\left( x - \frac{\pi}{4} \right) = \sin\left( 3x + \frac{\pi}{4} \right) $$ I tried using the following formulas for both ...
0
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1answer
39 views

Finding if there is a maximum or minimum on a curve?

My apologies for being very brief with this question, the reason for this is because I don't know where to start. The question is as follows: A curve has the equation $\lambda \cosh(x) + \sinh(x)$, ...
2
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1answer
62 views

Proof that there are no solutions this equation. (3 variables, Square root and Sine) [on hold]

Hypothesis: There do not exist three different positive integers $a,b,c$ such that $$ -\sqrt{ab}\cdot \sin(p \cdot (a-b))+\sqrt{ac}\cdot \sin(p \cdot (a-c)) -\sqrt{bc}\cdot \sin(p \cdot (b-c)) =0 $$ ...
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0answers
54 views

Find $a$, $b$ and $c$. [on hold]

As title said, How to find angles $a$, $b$ and $c$? Thanks in advance!
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2answers
55 views

Is there no formula for $\cos(x^2)$?

I was wondering if there was a "formula" or an "identity" for $\cos(x^2)$, as there is for $\cos(2x)$. My question is closely related to this one, which was only asking for $\cos(ab)$. For instance ...
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0answers
20 views
+100

Trigonometric position function and intersection

I have the following position function for a point: $x(t) := C_x - (S_x-C_x) \cdot \cos(t\cdot\theta) + (S_y-C_y) \cdot \sin(t\cdot\theta) + t \cdot v_x$ $y(t) := C_y - (S_x-C_x) \cdot ...
0
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1answer
44 views

Using derivatives to get some trigonometric identities

Is there a way of using derivatives to get some trigonometric identities in a straight-forward fashion? I use to forget them, so that would help me a lot... For example, since when we get the ...
6
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3answers
75 views

What is the integral value of $\frac{\tan 20^\circ+\tan40^\circ+\tan80^\circ-\tan60^\circ}{\sin40^\circ}$?

I have tried possibly all approaches. I first expressed $80$ as $60+20$ and $40$ as $60-20$ and then used trig identities.I later used conditional identities expressing $\tan ...
0
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3answers
53 views

how **(1)** $(2n-1)\pi/2 + (-1)^n\pi/3$ and **(2)** $2n\pi±\pi/6$ indicates the same angle?

I'm learning Trigonometry right now with myself and at current about General solution. I have a question in my book which I don't understand how to proof. The question is Show that the two angles are ...
-3
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2answers
51 views

Equetion of difference sinus and cosinus functions

Hi i have question i have something like this: $2 \sin 5t - 5 \cos 5t = A \sin(5t + \varphi)$ The main question is : Is there any formulas or something to change left side of "$=$" to look similar ...
0
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0answers
13 views

what functions/formula gives me linear progression from 90 -> 180/-180 -> -90

in my programming code the angles of a camera goes from TOP 90 to MIDDLE 180 to -180 to BOTTOM -90 what formula would give me a linear progression between those values? i need to move the camera up ...
2
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3answers
68 views

Trigonometric Equation Simplification

$$3\sin x + 4\cos x = 2$$ To solve an equation like the one above, we were taught to use the double angle identity formula to get two equations in the form of $R\cos\alpha = y$ where $R$ is a ...
0
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3answers
31 views

Find the ratios of the sides of a triangle

If the perimeter of a the right-angle triangle is six times its smallest side, find the ratios of the three sides. I tried solving it by using the normal area and volume.
0
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1answer
30 views

What is the third positive number starting from zero that will satisfy $\cos 6x = \cos x$? [on hold]

What is the third positive number starting from zero that will satisfy $\cos 6x = \cos x$? I need help on this one. I don't know how to approach this problem.
1
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1answer
22 views

Compound angle formula

I understand how to use the compound angle formula when solving $\sin(\pi/12)$. However I dont understand how I can use a compound angle formula to show $$\arctan(3)-\arctan(1/2)=\pi/4$$ Thankyou Any ...
1
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0answers
31 views

Mentally approximating an inverse sine?

Is there a method to approximate the inverse of a sine function in ones head? I know one can approximate the inverse of a cosine with the following equation: $\cos^{-1}(x) = ...
2
votes
3answers
28 views

how to find $\lim_{x\to 0}\sin^2(\frac{1}{x})\sin^2 x$

How to find $\lim_{x\to 0}\sin^2(\frac{1}{x})\sin^2 x$ ? I tried using taylor expansion: $$((x-\frac{x^3}{6}+\frac{x^5}{120}+O(x^5))(\frac{1}{x}-\frac{1}{6x^3}+\frac{1}{120x^5}+O(x^{-5})))^2$$ but ...
2
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2answers
69 views

Compute $(\sin4^\circ)^2 +(\sin8^\circ)^2+(\sin12^\circ)^2+\cdots+(\sin176^\circ)^2$

Angle of sine is in degrees, can anyone show me an easy soln to this? This was question was given to us for 1minute without calcu. I know that $\sin4^\circ=\sin176^\circ$, ...
0
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0answers
12 views

Bearing of a line

Please help me to find the bearing. I've attached the image.I've tried by drawing a North a B and C, and D but couldn't figure out the angle that give me bearing of C from D. Thanks for all your ...
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4answers
894 views

What is meant by a 'pure' wave?

What is meant by a 'pure' wave? I know it might sound like a basic question, but I've never been taught this. I saw that a sine wave is a pure wave. I tried Googling what a pure wave is, but all I ...
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4answers
63 views

Using trig identities to evaluate $\int_{0}^{\pi/2} \sqrt{1-\sin x} \, dx$

Use the identities $$\cos 2x=2\cos^2 x -1=1-2\sin^2 x$$ $$\sin x=\cos \left(\frac{\pi}{2}-x\right)$$ to help evaluate $$\int_{0}^{\pi/2} \sqrt{1-\sin x} \; dx$$ I've already done ...
6
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1answer
147 views
+50

Formula for cos(k*x)

I need to prove that: \begin{align} c_k =&\; \cos(k\!\cdot\!x)\\ c_k :=&\; c_{k-1} +d_{k-1}\\ d_k :=&\; 2d_0\!\cdot\!c_k +d_{k−1}\\ d_0 :=&\; −2\!\cdot\!\sin^2{(x/2)}\\ \end{align} ...
24
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1answer
724 views

Moriarty's calculator: some bizarre and deceptive graphical anomalies

Background: This is a problem I first came across a few years ago in a calculus textbook (a James Stewart one), where it addressed some of the pitfalls of using graphing calculators. The original ...
3
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0answers
58 views

PDF of Random Variable $\sin\alpha \cdot \cos\beta$ with $\alpha,\beta$ uniform

As part of a bigger problem, I want to compute the probability density $f_Z(z)$ of $$Z = \sin\alpha \cdot \cos\beta$$ where $\alpha, \beta$ are random variables, independently and uniformly ...
12
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1answer
205 views

Inequality with summation of cosine terms $\left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| \leq 1 + 2\sum_{j=1}^k \cos (\frac{2\pi }{q}j)$

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
122
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3answers
7k views

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
31
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2answers
649 views

Is Pythagoras the only relation to hold between $\cos$ and $\sin$?

Pythagoras says that $\cos^2 \theta + \mathrm{sin}^2\theta = 1$ for all real $\theta$. (Vague) Question. Is this the only relationship between the functions $\cos$ and $\sin$? More precisely: Let ...
24
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2answers
395 views

Disproving an “almost true” trigonometric identity

The plausible looking "identity" $$\sin(\frac{\pi}{51})+\cos(\frac{\pi}{74})=\frac{3}{2\sqrt 2}$$ is not true, but it is close indeed: $$LHS=1.0606\color{blue}{598...}$$ ...
18
votes
5answers
5k views

Do “imaginary” and “complex” angles exist?

During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting: $ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $ $ \sin ...
6
votes
6answers
584 views

Proving a trigonometric identity

How can one prove the validity of this trigonometric identity? \begin{equation} 2\arccos\sqrt{x} = \frac{\pi }{2}-\arcsin(2x-1) \end{equation}
20
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10answers
22k views

Real world uses of hyperbolic trigonometric functions

I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses ...