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

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3
votes
1answer
379 views

What conditions can lead us to $\arctan(\tan x)=x$?

I happened to find that the graphs of $\cos(\arctan(\tan x))$ and $\cos(x)$ are different. Why? Is there something wrong with $\arctan(\tan x)=x$? Thanks!
6
votes
2answers
130 views

Is the 'arc-' notation for inverse trigonometric and hyperbolic functions discouraged?

Any books we've used throughout high school and university preferred the '^-1' notation, leading me to believe that the 'arc-' notation is archaic. It feels like we're taught it, but discouraged from ...
2
votes
1answer
58 views

hyperbolic trigonometric relation

Let $F$ be a hyperbolic once-punctured torus, and $G=\pi_1(F)$. Fix a discrete, faithful representation $\rho\colon G\to\mathbb{P}SL(2,\mathbb{R})$ and an element $g\in G$ corresponding to a ...
2
votes
2answers
185 views

Trigonometric identity and roots of a polynomial.

Prove that $$(\operatorname{cosec} A–\sin A) (\sec A–\cos A) = \frac {1}{\tan A + \cot A} $$ Also help me with this question please If $\alpha$ and $\beta$ are zeroes of the polynomial ...
4
votes
1answer
234 views

Conversion from the linear-combination to the sinusoidal form of a sinusoidal function (simple problem, but I'm missing something.)

This is a standard trigonometric identity that can be easily verified: $$a\cos (x) + b\sin (x) = \sqrt{a^2+b^2}\cos (x - p),\text{ where }\tan(p)=\frac ba.$$ So for example, ...
2
votes
1answer
241 views

Trigonometry Table Problem

Evaluate the given trigonometric expression: \[ \frac{5\sin^2 30° + \cos^245° - 4\tan^2 30°}{2\sin30°\cdot\cos 30° + \tan 45°} \]
7
votes
3answers
748 views

Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $

It's my first post here and I was wondering if someone could help me with evaluating the definite integral $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in ...
2
votes
1answer
93 views

Trigonometric integral involving trig multiplication

$$\int \sin^3(3x)\cos^{-2}(3x)dx$$ Let$u=3x$; then $du=3dx$, so $dx=\dfrac{du}{3}$ $$\dfrac{1}{3}\int \sin^3(u)\cos^{-2}(u)du$$ Expand $\sin^3(u)$ to $\sin^2(u)\sin(u)$ $$\dfrac{1}{3}\int ...
0
votes
1answer
199 views

Transformations of sinusoidal functions

We have 2 sinusoidal functions: $f(x)= 0,30+2\sin (3x-\frac{1}{2}\pi$) and $g(x)=0,30+2\sin(3(x-\frac{1}{2}\pi))$. What would happen to both functions if you performed a translation of let's say ...
0
votes
1answer
87 views

Is this transformation of a sinusoidal function correct or not?

$f(x) = 2-4\cos(2x-\frac{1}{3}\pi)$ How do you get this function from as standard function? My answer: $ y = \cos x $ ↓ Multiply with the x-axis, -4 $y=-4\cos x$ ↓ Multiply with the y-axis, 0.5 ...
4
votes
2answers
384 views

Determining linear equation for intersection of $y=\sin x$, $y=\cos x$

I'm trying to solve the following problem: The graphs for $y = \sin x$ and $y = \cos x$ has two points of intersection in the interval $[-\pi, \pi]$. Determine the equation for the line that ...
0
votes
0answers
35 views

Transformations of sinusoids [duplicate]

Possible Duplicate: Transformations of sinusoidal functions Let's say we have the functions $f(x)= 0,30+2\sin (3x-\frac{1}{2}\pi$) and $g(x)=0,30+2\sin(3(x-\frac{1}{2}\pi))$. What would ...
5
votes
1answer
150 views

How to evaluate $\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$?

How can I integrate the following: $$\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$$ for $a,b>0$? Maple gives a compact result: $$\frac{1}{b} \tan^{-1}(c) - \frac{1}{ac^2} ...
4
votes
2answers
145 views

Proving the trigonometric identity

Please help me in proving the following idenity: $$8\cdot \cos 40^\circ\cdot \cos 20^\circ \cdot \cos 10^\circ = \cot 10^\circ$$
0
votes
0answers
42 views

best way to detect the trigonometric identites that shall work on a given expression so as to simplify it accordingly?

how to tell that what trigonometric identity (a.k.a. Pythagorean trigonometric identity) will work on the given equation , so then you can simplify the equation accordingly in order to apply that ...
0
votes
1answer
368 views

Trigonometry: Solve equation for $\alpha$

I have the following trigonometric equation: $$2\sin(\alpha - 45)\sin(2\alpha) = \sin(\alpha + 45)\sin(\alpha)$$ Is it possible to find $\alpha$? Please also include each step in your solution. ...
0
votes
2answers
53 views

Merging 2 formulae

We have to solve: $ \sin(\dfrac{1}{2}x + \pi) $ = $\dfrac{1}{2}\sqrt{2}$ I get these answers: $ x = -1\frac{1}{2}\pi + k2\pi\quad \lor\quad x =-\frac{1}{2}\pi + k2\pi$ Both these answers can be ...
43
votes
5answers
3k views

Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?

Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this ...
1
vote
1answer
178 views

Length bisection from circular arc

I am not sure if the following result is well known. I stumbled across it from the paper The Perimetric Bisection of Triangles by Dov Avishalom, where the result was stated without proof. I am ...
-1
votes
1answer
135 views

Function whose graph resembles the shape in this image

What is the function whose graph would resemble the shape found in the image below? I looked this up on Wikipedia, tried making my own, but I can't find an equation the Electromagnetic Spectrum. I ...
2
votes
3answers
224 views

Rigorous proof of the Taylor expansions of sin $x$ and cos $x$ revisited

I asked this question a while ago. I exchanged comments with a member(mixedmath) about the rigorous proofs that $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ and the addition formula for $\sin x$. He ...
0
votes
1answer
353 views

Sum/Difference Cosine Waves

This question has been troubling me for days, I really haven't got a clue how to handle it: $f(x) = -3+2cos(x)$ $g(x) = cos(x-\dfrac{1}{4}\pi)-2 $ Get the sum ($s(x)=f(x)+g(x)$) and difference ...
1
vote
1answer
148 views

Sum and Difference of 2 cosine functions

This question has been troubling me for days, I really haven't got a clue how to handle it: $f(x) = -3+2\cos(x)$ $g(x) = \cos(x-\dfrac{1}{4}\pi)-2 $ Get the sum ($s(x)=f(x)+g(x)$) and difference ...
4
votes
4answers
4k views

Drawing sine and cosine waves

I like mathematics and pretty much every mathematical subject, but if there is one thing I thoroughly dislike, it is drawing (functions, waves, diagrams, etc.) We have this important trig test coming ...
2
votes
1answer
135 views

Trig question I don't really understand

$4\cos^2 \left( x + \dfrac{1}{4}\pi \right)$ = 3 My final answer: $ x = \frac{11}{12}\pi+k\pi $ and $x = \frac{7}{12}\pi + k\pi $ In the correction model it is $x = \frac{7}{12}\pi + k\pi $ and ...
1
vote
1answer
81 views

How to tackle these trig questions correctly

$ \sin(2x) \cdot \cos(2x) + \sin(2x) = 0 $ In the correction model I have something I don't understand is done in the first step: $ \sin 2x(\cos 2x + 1) = 0 $ Is this step correct? And can someone ...
1
vote
1answer
356 views

How to solve these types of trig equations

Lets use an example: $$ \sin^2 \left(\dfrac{\pi}4x\right) = 1 $$ I am at this point: $$ \frac{\pi}4 x=\frac{\pi}2 + k\cdot2\pi \quad\text{or}\quad \frac{\pi}4 x=-\frac{\pi}2 + k\cdot2\pi $$ But ...
1
vote
1answer
282 views

Application of Trigonometry 2

My question is- At the foot of a mountain the elevation of its summit is 45 degrees.After ascending one kilometer towards the mountain up an incline of 30 degrees,the elevation changes to 60 ...
6
votes
5answers
958 views

how to calculate the exact value of $\tan \frac{\pi}{10}$

I have an extra homework: to calculate the exact value of $ \tan \frac{\pi}{10}$. From WolframAlpha calculator I know that it's $\sqrt{1-\frac{2}{\sqrt{5}}} $, but i have no idea how to calculate ...
1
vote
2answers
199 views

Application of Trigonometry

My question is- From an aeroplane vertically over a straight road,the angles of depression of two consecutive kilometer-stones on the same side are 45 degrees and 60 degrees.Find the height of the ...
2
votes
1answer
265 views

How to solve a set of cosine equations?

suppose I have an equations of the following with two unknowns $A$ and $\theta$ $A\sin(x+\theta)=D$ I have two points $(E,F) (G,H)$ how do I go about solving this equation analytically. I can solve ...
3
votes
5answers
1k views

Prove that $\cos(2\pi/n)+\cos(4\pi/n)+\cdots+\cos(2(k-1)\pi/n)=-1$

How do you prove that $$\cos(2\pi/n)+\cos(4\pi/n)+\cdots+\cos(2(n-1)\pi/n)=-1,$$ where $n \geq 2$?
2
votes
1answer
82 views

triangles and trigonometry

A triangle has sides $a,b,c$ and angles $\alpha,\beta,\gamma$ such that: $$ a \,\cos\beta + b \, \cos\gamma+ c \, \cos\alpha = \frac{a+b+c}{2}$$ Prove that the triangle is isosceles. I tried writing ...
0
votes
2answers
173 views

Simplification of trigonometric expression regarding planetary orbits

I am trying to solve an orbital problem concerned with analytically calculating the solstice points of an orbit. I have managed to reach a point in the problem where I need to simplify the ...
2
votes
2answers
157 views

Finding $\alpha$ for $\sin(4 \alpha + \frac{\pi}{6}) = \sin (2\alpha + \frac{\pi}{5})$

I try to find $\alpha$ for $\sin(4 \alpha + \frac{\pi}{6}) = \sin (2\alpha + \frac{\pi}{5})$. Left side: $$\sin(4 \alpha + \frac{\pi}{6}) =$$ $$= \sin4\alpha \times \cos \frac{\pi}{6} + \cos 4\alpha ...
3
votes
1answer
151 views

evaluation of the integral $\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $

Can the integral $$\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $$ be expressed in terms of elemental functions or in terms of the sine and cosine integrals ? if possible i would need a hint thanks. ...
2
votes
2answers
2k views

$\sin 4\alpha = 2\sin 2\alpha \times \cos 2\alpha $?

A trigonometry rule says that $\sin 2\alpha = 2\sin \alpha \times \cos \alpha$. Does this also apply to $\sin 2x$ when $x = n \times \alpha$? For example: $$\sin 4\alpha = 2\sin 2\alpha \times \cos ...
5
votes
3answers
639 views

Verifing $\int_0^{\pi}x\ln(\sin x)\,dx=-\ln(2){\pi}^2/2$

I used all I know to show that $$\int_0^\pi x\ln(\sin x)dx=-\ln(2) \pi^2/2$$ This is my homework but don't know where to start. I appreciate your help.
2
votes
1answer
206 views

calculated reflected point within circle

The problem to solve is this. Imagine a circle. We know two points on the circumference, anchor A and anchor B, they could be anywhere on the circumference of the circle. Draw a line between these ...
2
votes
2answers
69 views

Calculate $\cos(v+\frac{\pi}{6})$ when $\cos v = -\frac{2}{3}$

I am to calculate $\cos(v+\frac{\pi}{6})$ when $\cos v = -\frac{2}{3}$ and $0 \lt v \lt \pi$. I know I can change $\cos(v+\frac{\pi}{6})$ into $$\cos v \times \cos \frac{\pi}{6} - \sin v \times \sin ...
0
votes
1answer
90 views

A simple question about angles on a circumference

Given two points on a circumference of radius $R$, $P_0$ and $P_1$ subtended by an angle $\theta$ at the center of the circumference, what is the angle at which a generic point $P_m$ inside the circle ...
1
vote
1answer
468 views

Auxiliary trigonometric identities [Identidades Trigonométricas Auxiliares]

I need to show two auxiliary trigonometric identities: 1) $\sec^2x = \tan ^2x + 1 (\cos x \neq 0)$ 2) $\csc^2x = \cot^2x +1 (\sin x \neq 0)$ How could I do it? [Original Portuguese] Identidades ...
4
votes
3answers
530 views

solution to equation $a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$

Does the equation $$ a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$$ have a closed-form solution for $\theta$? What about the case where $a^2 + b^2 = 1$?
1
vote
2answers
326 views

Starting point of a Sine wave

We learned this at school, the function: $$y= a + b\sin (c(x-d))$$ has a starting point of $(d,a)$. But when I had to draw this function: $$g(x) = -2-\cos(x-1/2π)$$ I thought the starting point ...
1
vote
1answer
508 views

Triangle two angles and one length [closed]

In a triangle ABC, the angle at A is 41 degrees, the angle at B is 71 degrees, and the length of side AB is 8. To 2 decimal places, what is the length of side BC? I got this answer 5.66. is that ...
1
vote
1answer
78 views

Triangle one angle and two lengths [closed]

In a triangle ABC, the angle at B is 108 degrees, the length of side BC is 16, and the length of side AB is 12. To 2 decimal places, what is the length of side AC? So i worked out and out and got ...
2
votes
2answers
245 views

calculating limit without using direct formulae

In case of $\delta$-$\epsilon$ definition of limit, books usually show that some $L$ is the limit of some function and then they prove it by reducing $L$ from the function $f(x)- L$ and showing ...
3
votes
1answer
405 views

How to find the limit of $\sin(f(x))$, given the graph of $f(x)$

The full question is uploaded here: http://imgur.com/EZekb Basically, given the graph shown in the image above, I thought that the limit of $\sin(f(x))$ would be $\sin(2)$, since the limit of just ...
2
votes
3answers
445 views

Expressing in the form $A \sin(x + c)$

Express in the form $A\sin(x+c)$ a) $\sin x+\sqrt3\cos x$; b) $\sin x-\cos x$ sol: a) $A=\sqrt{1+3}=2$, $\tan c=\frac{\sqrt 3}1$, $c=\frac\pi3$. So $\sin x+\sqrt3\cos x=2\sin(x+\frac\pi3)$ ...