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

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2
votes
1answer
18 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
votes
3answers
61 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
votes
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
votes
4answers
34 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 ...
4
votes
4answers
396 views

Solve the following trigonometric integral

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
0
votes
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
votes
3answers
35 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
votes
2answers
24 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
37 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
votes
0answers
14 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
votes
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
votes
1answer
32 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
votes
1answer
55 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 $$ -\sqrt{ab}\cdot \sin(p \cdot (a-b))+\sqrt{ac}\cdot \sin(p \cdot (a-c)) -\sqrt{bc}\cdot \sin(p \cdot (b-c)) =0 $$ ...
-6
votes
0answers
53 views

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

As title said, How to find angles $a$, $b$ and $c$? Thanks in advance!
7
votes
2answers
53 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 ...
0
votes
0answers
11 views

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
votes
1answer
43 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
votes
2answers
71 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
votes
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
votes
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
votes
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
votes
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
votes
3answers
30 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
votes
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
vote
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
vote
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
votes
2answers
68 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
votes
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 ...
6
votes
4answers
892 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 ...
0
votes
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 ...
0
votes
1answer
33 views

Plane rotation: range of angles to produce all posible x'y' planes

Given an $(x, y, z)$ system I create a new system $(x', y', z')$ by applying two rotations $\theta$ and $\phi$. In the new system the $(x',y')$ plane, i.e.: the $z'=0$ plane, can be written as: $$ ...
0
votes
1answer
24 views

Rational solutions for $\sin(n)$ in radians

This is completely for my own curiosity. Does $y = \sin(n)$ have rational solutions for $n$, an integer number of radians. I know that this is strange because usually integers are only used in ...
2
votes
3answers
63 views

$\sin2(x) - \tan(x) = 0$ , solve for $-180\le x\le 180$

I have been unable to solve the following question, If $$\sin(2x) - \tan(x) = 0$$ Find $x$ , $-\pi\le x\le \pi$ So far my workings have been Use following identity: $$\sin(2x) = ...
1
vote
1answer
49 views

Find the minimum value of $\frac {\sin \alpha \sin \beta}{\sin^2 \frac {\gamma}{2}}+…$

Find the minimum value of the following expression $$ \frac {\sin \alpha \sin \beta}{\sin^2 \frac {\gamma}{2}}+ \frac {\sin \beta \sin \gamma}{\sin^2 \frac {\alpha}{2}}+ \frac {\sin \gamma \sin ...
1
vote
0answers
36 views

Proving cosines product identity [duplicate]

Prove that $\cos\left({\pi \over 11}\right)\cdot\cos\left({2\pi \over 11}\right)\cdot\cos\left({3\pi \over 11}\right)\cdot\cos\left({4\pi \over 11}\right)\cdot\cos\left({5\pi \over 11}\right)={1 ...
2
votes
1answer
54 views

Joining the Midpoints of the Sides of a Quadrilateral

$ABCD$ is a quadrilateral. $P$, $Q$ and $R$ are the midpoints of $AB$, $BC$ and $CD$ respectively. If $PQ = 3$, $QR = 4$ and $PR = 5$; find the area of $ABCD$. Since, $5^2 = 3^2+4^2$, So, ...
1
vote
0answers
29 views

Find the expected value of the matrix

$\require{cancel}$ I want to see if I have solved this problem appropriately or not. If we have ...
1
vote
1answer
42 views

How to calculate $\lim_{x \rightarrow 0} \frac{\int_0^{G(x)} \arctan(s+2s^2) ds}{x^2}$ based on the following assumption?

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. To start this question, we need to integrate $\arctan(s+2s^2)$ but how do you do that? Then, what do we do ...
1
vote
0answers
28 views

How to prove that there a constant $C$ such that $\arcsin \frac{1-x}{1+x}+2\arctan\sqrt{x}=C$? [duplicate]

How to prove that there a constant $C$ such that $\arcsin \frac{1-x}{1+x}+2\arctan\sqrt{x}=C$? I have no idea using which theorem to prove. Could someone show me how to start the problem?
0
votes
0answers
21 views

Find the rotation angles of a 2-D rotation matrix between two vectors

I am trying to solve the following to find $\theta$. I was given two vectors $\begin{bmatrix}-4.95 \\ -.7\end{bmatrix}$ and $\begin{bmatrix}3 \\ 4 \end{bmatrix}$ and asked to compute the rotation ...
1
vote
2answers
21 views

Integral of $-4\sin(2t - (pi/2)) $ weird behavior on wolfram alpha

I'm confused by what Wolfram Alpha is doing with my function: $$-4\sin{(2t - (\pi/2))}$$ on why the it gets replaced by $$4\cos{(2t)}$$. Is it equal? Link: See behavior here
4
votes
1answer
118 views

Why is $s$ used for arc length?

Why is $s$ used for arc length? I looked around online, but I can't find a definite answer. Thank you!
1
vote
2answers
35 views

Problem integrating (substitution)

can you help me identify the mistake I'm making while integrating? Question: $$\int{\frac{2dx}{x\sqrt{4x^2-1}}}, x>\frac{1}{2}$$ my solution ...
0
votes
1answer
31 views

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the ...
0
votes
0answers
21 views

Find the third angle

Three planes are orthogonal to each other. I have found the rotation about the 2 axis (x and y). Is there a way to find the third angle around z provided the angles around x(70 degrees) and y(-1 ...
0
votes
2answers
39 views

Elementary integral for square roots of trig functions?

What's an easy way to calculate something like $\int \sqrt{1+\cos x} \text{ d}x$?
4
votes
1answer
63 views

Quaternions: Why is the angle $\frac{\theta}{2}$? [duplicate]

The equation for creating a quaternion from an axis-angle representation is $$x'= x \sin\left(\frac \theta 2\right)$$ $$y' = y \sin\left(\frac \theta 2\right)$$ $$z' = z \sin\left(\frac \theta ...
1
vote
1answer
50 views

Express this expression in terms of $x_1$ and $x_2$?

We have the following definition: $$ x_1=A \cos(\omega t_1 +\phi) \\ x_2= A \cos(\omega t_2 +\phi) $$ The expression we want to simplify is: $$ S=A^2\omega \left[\sin 2(\omega t_2+\phi)-\sin 2(\omega ...
0
votes
0answers
26 views

Expand a $\arctan(x)$ function [duplicate]

I want to expand a function $\arctan(x)$ as a polynomial form. I know that I can use Taylor expansion in the case of x <1. But in my case, the x can be pretty large. Is there any way to expand or ...