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

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15 views

Rearrangement of harmonic oscillation formulae

Can anyone show me why the following identity is true? $$A\cos(\omega t) + B\sin(\omega t) = \sqrt{A^2+B^2}\cos\left(\omega t + \arctan\left({ \frac BA}\right)\right) $$ I ask this is relation to ...
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1answer
13 views

Inverse Trigonometry Plots ArcT(T(x))-Clarification

enter image description here The graphs are plots of functional forms $T^{-1}(T(x))$ where T is a trigonometric function:sine,cosine,tangent,cosecant,cotangent,and secant Can someone please ...
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2answers
37 views

How should trigonometric expressions be simplified?

I have been learning trigonometric identities and I am having trouble understanding how they should be applied to simplify expressions. For example, the expression $2\sin{x} \cos{x}$ is equal to ...
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2answers
36 views

Solving the indefinite integral of a trig function

I'd like to ask for some feedback on my calculation. Please let me know if you spot any mistakes in my technique: $$\int{\frac{1}{\sqrt{x}}\sin^{-1}{\sqrt{x}}}\,\,dx$$ Using substitution: $$u = ...
0
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1answer
21 views

Solving the definite integral of trig function

I'd like to ask for some feedback on my calculation. Please let me know if you think it's correct, or if I messed up somewhere: ...
0
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1answer
18 views

Terminals and co-terminals for angles

I'm trying to understand how my teacher converted these angles. I'm not sure if my title is correct but I'm assuming that's what he was doing. For a unit circle he had, \begin{align*} u & = ...
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0answers
37 views

Derivatives of hyperbolic functions and Osborne's rule.

I am slightly confused when it comes to Osborne's rule when you take derivatives of hyperbolic functions. For example. The derivative of cotx is -cosec^2x, so there is a product of sines. So should ...
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4answers
48 views

Prove a trigonometric identity [closed]

$$\frac{1}{\csc A-\cot A}-\frac{1}{\sin A} = \frac{1}{\sin A} - \frac{1}{\csc A+\cot A}$$
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4answers
49 views

Solving an equation with the sum of inverse cosine and inverse tangent

I have the below question and have to find value of $x$. $$ \cos^{-1}\left(\frac{x^2-1}{x^2+1}\right)+\tan^{-1}\left(\frac{2x}{x^2-1}\right) = \frac{2\pi}{3}$$ I took $x$ as $\tan y$ but it isn't ...
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0answers
18 views

How to find the Laplace Transform of two (independent) functions multiplied together?

How does one find the laplace transform for an equation consisting of two trig functions multiplied together, when it is not possible to use any trig identities? For example, take a function say; ...
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4answers
58 views

Characterization of the $x$ such that $\sin(x)$ is rational?

For $x \in [0,\pi/2]$, $\sin(x)$ ranges over $[0,1]$. So every rational number in $[0,1]$ is the sine of some $x \in [0,\pi/2]$. Q. Is there any characterization of the $x$ for which $\sin(x)$ is ...
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1answer
27 views

Principle of superposition

Here $\phi$ is the solution to a linear pde so the principle of superposition applies. $\theta$ is the phase. I've tried using trig identities and different linear combinations in order to try and ...
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1answer
19 views

Integral of trigonometric function using substitution

I'd like to get some feedback on the following calculation: $$\int{\frac{(\cos{\frac{1}{x}})^2}{x^2}}\,\,dx$$ Using substitution, let $$u = \frac{1}{x},\,\,\frac{du}{dx}=-\frac{1}{x^2},\,\,du = ...
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0answers
33 views

Integral of trig fraction using substitution

I'm getting to grips with the process of integration by substitution, and would like to ask for feedback on my calculation: $$\int{\frac{\cos{x}}{\sqrt{1+2\sin{x}}}}\,\,dx$$ Let ...
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3answers
36 views

Evaluating indefinite integral using substitution

I have the following integral to evaluate. I'm not sure whether to use the reverse chain rule or integration by parts, as my calculation hits a bit of a snag. Any suggestions would be appreciated! ...
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0answers
28 views

Manipulating multivariate trigonometric functions

I've manipulated a few trig identities but nothing seems to help me in showing that these two equations are equal. I'm pretty sure I haven't come across anything like this before, how do I ...
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1answer
23 views

Derivative of inverse cosecant?

I am slightly confused by this, because when I worked out the derivative of arccosec(x), my answer was $\frac{-1}{x\sqrt{x^2-1}}$, which agrees with the answers online. However this would imply that ...
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4answers
69 views

Simplifying inverse trigonometric functions

Given below is the question $$ \sin^{-1}\sqrt{\frac{x}{1+x}} $$ I tried to put x as tany, siny, cosy nothing seems to be working. Looking forward to some quick help , I have an exam tomorrow :/
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1answer
32 views

Solutions $3 p\sin x - (p+\sin x)(p^2-p \sin x +\ sin ^{2} x) =1$

$3 p \sin x - (p+\sin x)(p^2-p \sin x + \sin ^{2} x) =1$ has a solution for $x$. Then number of integral solutions of $p$ are ?
0
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1answer
29 views

Applying boundary conditins to a differential equation confusion [closed]

Here $k, A_1, A_2$ are constants Although $A_1$cos$kx+A_2$sin$kx \not\equiv 0$, $A_1$cos$kx+A_2$sin$kx$ can equal zero at certain values of $x$ For example if $A_1=1, A_2=1, k=1$ then ...
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4answers
73 views

If $16^{\sin ^2x}=5$, then what is $2^{\cos^2x}$?

I happened to create this problem and solved it. I used only basic algebra and trigonometry. I thought it was a fun problem, so I wanted to expose the problem to the public. Please provide an exact ...
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1answer
59 views

find a solution for Trigonometric Functions

How to find a solution for $\sin(\theta)-\sqrt{\sin( \theta )+\cos( \theta )} -\cos^2(\theta) = 0 $ where $\theta $ in $[ 0 , 2\pi ] $
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4answers
78 views

What is the general solution to $\sin\theta=\frac12$?

What is the general solution to $\sin\theta=\frac12$? I have an incorrect solution but I don't know why. \begin{align*} \sin\theta & =\frac12\\ \sin\theta & =\sin\alpha\\ \alpha & ...
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2answers
73 views

What comes after seconds?

Angles can be measured in different ways. For example, one can measure angles in degrees/minutes/seconds. So $1^\circ$ is divded into $60$ min. and $1$ min is divided into $60$ sec. That way a tenth ...
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0answers
25 views

$\int_0^\pi \cos{mx}\cos{nx}\,\mathrm{d}x$, with $m$ and $n$ integers

I can't figure out this integral at all. Integrating gave the following \begin{equation} \frac{1}{2}[\frac{\sin((m-n)x)}{m-n}+\frac{\sin((m+n)x)}{m+n}], \end{equation} but when I evaluate it from ...
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1answer
19 views

Angle between a line and a plane in a cuboid

I was no good at this kind of stuff at school, probably because of bad spacial awareness. I thought I'd revisit it to see if I've improved! I'm stuck on this question: I dropped a perpendicular ...
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2answers
54 views

Simplifying an inverse trigonometric function

How can I express the following in its simplest form? $$\sin^{-1}\left(x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^{2}}\right)$$ I tried writing $x=\cos2y$ but it didn't seem to help.
2
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1answer
59 views

What did I do wrong trying to find this limit?

In another question, a user asked to find: $$\lim_{x\to 0} \frac{\exp(x^2)-\cos(x)}{\sin(x)^2}$$ I thought I could use pure trigonometric identities to find the limit. Apparently I was mistaken, but I ...
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0answers
51 views

Prove the large triangle is equilateral

Given the smaller triangle inside is equilateral, and three shorter sides of the big triangle are equal in length. Prove the large triangle is also equilateral
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1answer
21 views

Trig general solution - getting different answer to book

For the question $$ tan(2x + 1.426) = -2.156 $$ I get the general solution $$x = n\pi/2 -1.281 $$ but the answer in the book is $$x=1.571n\pi-1.281$$ I've just started this general solution thing so ...
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2answers
25 views

Show $\\Log z_1z_2 \neq Log z_1 + Log z_2$. given $z_1 = i$ and $z_2 = -\sqrt 3 + i$.

Show by evaluating both sides that for $z_1 = i$ and $z_2 = -\sqrt 3 + i$, $\\Log z_1z_2 \neq Log z_1 + Log z_2$. Recall the definition: $\\Log z = Log |z| + iArg z$ Attempt: left side: $\\Log ...
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1answer
55 views

Solving trigonometrics functions/equations

My niece ask me to help her with a school assignment, but I can't identify what type of equation are we solving. For example: $\DeclareMathOperator{\tg}{tg}$ $$\sin\alpha=\frac{8\sqrt{11}}{9}$$ or: ...
5
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1answer
142 views

Why are logarithms of trigonometric functions useful?

I have noticed that in many trigonometric tables the logarithm of the trigonometric values are given. Why this is given and not the actual values of the trigonometric functions? For example, instead ...
1
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1answer
37 views

Simple physics simulation problem

for a engineering project i'm doing it would be easier to simulate certain things beforehand, to do that I need to do some rag doll physics. I have a leg consisting of two parts, the upper and lower ...
33
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3answers
2k views

Is there a relationship between trigonometric functions and their “co” functions?

We all know that sine is one over cosecant, cosine is one over secant, etc. But is there any relationship between sine and cosine, secant and cosecant, and tangent and cotangent? What I am asking ...
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0answers
38 views

how to plot the graphs of $\sin^{-1}(\sin x)$,$\cos^{-1}(\cos x)$,$\tan^{-1}(\tan x)$

Can someone please explain me how to plot the graphs of $\sin^{-1}(\sin x)$,$\cos^{-1}(\cos x)$,$\tan^{-1}(\tan x)$ ? I am having a little difficulty in understanding how the nature of the graphs can ...
0
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1answer
13 views

pinpoint the position of devices

My question is I know the distances d1, d2 and d3, thats the only information I have access to, but am build a android app where I need to indicate the positions of where the devices that are ...
2
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7answers
66 views
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2answers
35 views

Find line equation using another line's equation and the angle

So, I have the problem described exactly as in the figure below. I want to find the equation for the green line given the data described in the figure. I know that $$\tan(\text{angle of elevation for ...
0
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5answers
45 views

simplyfing a trigonometric equation

I don't know how to get from the left side of the equation to the right side. $$ \cos x\; \cos2x+\sin x\; \sin2x=\cos x $$ How can I do it? which equations to use? P.S How can I simplify this? $$ ...
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3answers
58 views

How can we use an identity to solve the equation $1-\tan^2 \theta = \frac{2}{3}$?

If $1- \tan^ 2\theta = \frac{2}{3}$, then find the value of $\theta.$ I cannot understand which identity fits there. What must be used to find the value of $\theta$ ?
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0answers
37 views

Find reflected angle off an arc

Note: Sorry if this question is elementary. I am trying to find the reflected angle when a ball hits a 10 degree arc. I know I have to use trigonometry in some way, but I am having trouble grasping ...
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0answers
92 views

Squeeze theorem and $\frac{\sin x}{x}$

I've been going over old calculus books to refresh my memory and have mainly been focusing on proofs. One of the things I found interesting was the squeeze theorem, even though since basic calculus i ...
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2answers
224 views

To whom do we owe this construction of angles and trigonometry?

I've come across what is, to me, the most precise, beautiful and thorough definition of what we know of as the angle between two vectors. I say this because most literature either skims over things ...
2
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1answer
56 views

Wedge Product Formula For Sine. Analogous Formula Generalizing Cosine to Higher Dimensions?

So I was day dreaming about linear algebra today (in a class which had nothing to do with linear algebra), when I stumbled across an interesting relationship. I was thinking about how determinants are ...
2
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2answers
64 views

$\sqrt{\frac{h^{2}}{4}+d^{2}-h d \cos x}-\sqrt{\frac{h^{2}}{4}+d^{2}+h d \cos x}$ equals $h\cos x$?

Trying to simplify the expression, I observed: $y=\sqrt{\frac{h^{2}}{4}+d^{2}-h d \cos x}-\sqrt{\frac{h^{2}}{4}+d^{2}+h d \cos x}$ graphically equals $y=h\cos x$ when pluging in arbitrary values of ...
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3answers
91 views

Solve $2\sin(x)^2 -\sin(x) = 1$ by hand

I'm am trying to solve $ 2\sin^2(x)-\sin(x)=1 $ and I know that the solutions are: $ x=\frac{1}{4}(4\pi~ n+\pi)$, $x=\pi(2n-\frac{1}{6})$, and $ x=\pi(2n+\frac{7}{6})$, (for any integer $n$). ...
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1answer
26 views

The maximum value of expression $ \sqrt{\sin^2x+ 2a^2} - \sqrt{-1 -\cos^2x+ 2a^2} $

If $a,x\in\Bbb R$, what is the maximum value of the expression $ \sqrt{\sin^2x+ 2a^2} - \sqrt{-1 -\cos^2x+ 2a^2} $? I tried to differentiate but it became messy.
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0answers
37 views

$\sin(2\pi/7) + \sin(4\pi/7) + \sin(8\pi/7) = (root7)/2$

How to do problems such as these 1) $\sin(2\pi/7) + \sin(4\pi/7) + \sin(8\pi/7) = \dfrac{\sqrt{7}}{2}$ 2)$ \sin(\pi/7)\sin(2\pi/7)\sin(4\pi/7) = \dfrac{\sqrt{7}}{8}$
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0answers
40 views

Show that the substitution $t=\tan\theta$ transforms the integral ${\int}\frac{d\theta}{9\cos^2\theta+\sin^2\theta}$, into ${\int}\frac{dt}{9+t^2}$

To begin with the $d\theta$ on the top of the fraction threw me off but I'm assuming it's just another way of representing: $${\int}\frac{1}{9\cos^2\theta+\sin^2\theta}\,d\theta$$ I tried working ...