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

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42 views

Highschool Geometry: Finding Common Tangent

I can't seem to identify what the arrows are indicating in this question, obviously the two lines are parallel but what does it mean? I don't know where to begin. Any suggestions?
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0answers
40 views

Confused about integration over zeroes.

Does for example $\int_{-\pi}^{\pi} \sin(x) \, dx$ cancel out to zero (following WolframAlpha/normal integration technique), or do we have to take the absolute value of all the areas between bounds ...
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2answers
40 views

An identity involving the Chebyshev polynomials

Let $n \in \{0, 1, 2, \dots\}$ and let $T_n$ denote the Chebyshev polynomial of degree $n$: $T_n(x) = \cos\left(n \arccos(x)\right)$. Let $t_0, t_1, \dots, t_n$ be $T_{n + 1}$'s roots: $t_i = \cos\...
4
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3answers
545 views

Finding the exact value of arctan function then adding it?

The question is $x = \arctan\frac 23 + \arctan\frac 12$. What is $\tan(x)$? I'm having trouble figuring out how to calculate the arctan values without a calculator, or do I not even need to find ...
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1answer
64 views

$y=3^{\cos(x)}$ how to graph this trigonometric function

Please help me with graphing this function $y=3^{\cos(x)}$ without graphing software. Thanks in advance for all your procedures.
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0answers
42 views

Can you always cover a circle in a finite number of steps with this “radar” algorithm?

Suppose you have a disc $C$ of radius $V$ with center $c$ and you randomly place a point $p$ in it. $p$ Behaves as follows: at every time-step, $p$ calculates its angle to $c$, and moves a distance of ...
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0answers
34 views

Trying to Find Bounds on a Trig Function

I have the function $|\sin(\frac{N+1}{2}x)\sin\frac{Nx}{2}|$ and I want to use inequalities to get it to the form $c\sin\frac{Nx}{2}$ for some constant $c$. For a little perspective, I am going to ...
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1answer
162 views

Find triangle angle knowing side length change and angle change

There is triangle A with angle $\alpha=x$ and adjacent size $a$, and triangle B with angle $\beta=x-20$ and adjacent size $b=2a$, so \begin{align*} cos(x)/cos(x-20)=1/2 \\ \end{align*} How do I find $...
2
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0answers
79 views

Is this integral is right or wrong?

We did this exercise in class in a way, but at home I tried to solve it in a different way and I do not know if it is right or wrong. May you help me please? $\mathbf{\int tan^{5}x \, \, \, sec^{4}x ...
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3answers
55 views

If $\triangle ABC\;,\frac{a}{b}=2+\sqrt{3}$ and $\angle C=60^0,$ Then find ordered pairs $\left(\angle A,\angle B\right)$

In a $\displaystyle \triangle ABC\;,\frac{a}{b}=2+\sqrt{3}$ and $\angle C=60^0,$ Then find the ordered pairs $\left(\angle A,\angle B\right)=$ $\bf{Options}::$ $(i)\; \left(45^0,75^0\right)\;\;\;\...
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3answers
117 views

Establish the identity: tan u(csc u - sin u) = cos u

I'm struggling to establish the identity below: $$\tan\,u(\csc\,u - \sin\,u) = \cos\,u$$ I've ended up with: $${1 - 2\sin\,u \over \cos\,u}$$ I don't know if this is correct so far, and if it is, ...
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3answers
127 views

Evaluate the indicated trigonometric function

Question states: Assume that theta is an acute angle in a right triangle satisfying the given conditions. Evaluate the indicated trigonometric function $\tan \theta = 1/5;$ What is $ \csc \theta$ ...
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3answers
58 views

Prove the following trigonometric identity

$$\frac{\tan{(\frac{\pi}{4}+x)}-\tan{(\frac{\pi}{4}-x)}}{\tan{(\frac{\pi}{4}+x)}+\tan{(\frac{\pi}{4}-x)}} = 2\sin{x}\cos{x}$$ ============== On L.H.S, I've tried to write it using the sum and ...
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1answer
55 views

Find the position of a circle tangent to two other circles

Say there are 3 circles, A, centered at point a, B centered at point b, and C, centered at point c. Each has a known radius independent of the others, Ar, Br, and Cr. The positions of a and b are ...
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1answer
20 views

Trigonometry / Sum of two angles (α + β) if sinα = 8/17 and sinβ = 15/17

Find the sum of two angles α and β if sinα = 8/17 and sinβ = 15/17 if they are A) acute B) obtuse How do you approuch this problem? I'm stuck at the begging. Please help.
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1answer
23 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
25 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 explain ...
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2answers
66 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 $\sin{...
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2answers
41 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 = \...
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1answer
30 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: $$\int_0^{\frac{\pi^2}{4}}{\frac{1}{\sqrt{x}}\,\sin{(4\pi+\sqrt{x})}}\,...
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1answer
22 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
152 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
134 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
72 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
68 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
70 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 ...
0
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1answer
31 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
40 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 $$u=2\sin{x},\,\,\...
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3answers
38 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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1answer
125 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
84 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
49 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 ?
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1answer
39 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 cos$x$+sin$x=...
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4answers
77 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
65 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
93 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
414 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
28 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 $0$...
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2answers
62 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.
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1answer
65 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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1answer
26 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
38 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
67 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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2answers
908 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 ...
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1answer
53 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 ...
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3answers
3k 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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1answer
14 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 ...
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7answers
78 views
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2answers
57 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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4answers
52 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? $$ \...