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

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0
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2answers
362 views

Solve $\cos x \cos 3x -\sin x\sin 3x =0 \,\,\, \forall x \in [0,\pi]$

We had this problem $$\cos x \cos 3x -\sin x\sin 3x =0 \,\,\, \forall x \in [0,\pi]$$ in an assignment and I wasn't able to solve without graphing the function first. I tried using $\cos (x+y)$ but ...
1
vote
5answers
1k views

Proving the identity: $\sin3x + \sin x = 2\sin2x\cos x$

I need some help proving this identity: $$\sin3x + \sin x = 2\sin2x\cos x$$ I don't know where to start. I thought about expanding $\sin 3x$ into $\sin (2x + x)$ but I don't think that does me any ...
0
votes
1answer
86 views

Using a matrix vector product to show a specific example

I am suppose to use a matrix vector product to show that if $\theta$ is 180 degrees then $A_\theta v = -v$ for all v in $R^2$ I have no idea what this means and it is really confusing, as far as I ...
2
votes
2answers
2k views

How the arc length of sine wave is calculated?

According to this site the arc length of the curve $y = f(x)$ from $x=a$ to $x=b$ is given by: $$\operatorname{length}_{ab} = {\large\int}_a^b \!\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx.$$ So, we ...
2
votes
2answers
145 views

Proving $\sqrt{\tan(\alpha)\tan(\beta)+5}+\sqrt{\tan(\alpha)\tan(\gamma)+5}+\sqrt{\tan(\beta)\tan(\gamma)+5}\le4\sqrt{3}$ [closed]

Prove that if $\alpha+\beta+\gamma=90^{\circ}$, then we have following inequality: $$\sqrt{\tan(\alpha)\tan(\beta)+5}+\sqrt{\tan(\alpha)\tan(\gamma)+5}+\sqrt{\tan(\beta)\tan(\gamma)+5}\le4\sqrt{3}$$
3
votes
2answers
86 views

trig equation $(\sqrt{\sqrt{2}+1})^{\sin(x)}+(\sqrt{\sqrt{2}-1})^{\sin(x)}=2$

Please help me to solve this trig equation. $$(\sqrt{\sqrt{2}+1})^{\sin(x)}+(\sqrt{\sqrt{2}-1})^{\sin(x)}=2$$
0
votes
2answers
115 views

Solving $\cot^2(x) = \{\sin^2(x)\} + [\cos^2(x)]$ [closed]

Solve the following trigonometric equation: $$\cot^2(x)=\{\sin^2(x)\}+[\cos^2(x)],$$ where $\{x\}$ and $[x$] are fractional and integer part accordingly.
0
votes
1answer
161 views

An alternative form for $\frac{\sin x}{1+\cos x}$

$\frac{\sin x}{1+\cos x}=$(choose one option from the followings) a) $\frac{\sin x}{\cos x}$ b) $\frac{\cos x-1}{\sin x}$ c) $\frac{1-\cos x}{\sin x}$ d) $\frac{\sin x+1}{\cos ...
-2
votes
1answer
144 views

Is this function periodic? [closed]

Is the following function periodic? $$f(x)=\cos(x)*\cos(x\sqrt5)$$ A function $f$ is said to be periodic with period $P$ ($P$ being a nonzero constant) if we have $$f(x+P) = f(x)$$ for all ...
1
vote
3answers
646 views

Find $x$-coordinates of all horizontal tangents on the graph of $f(x) = \sin^2x + \cos x$

The function is $f(x) = \sin^2x + \cos x$. I found the derivative which was $f'(x) = 2x\cos^2x - \sin x$. I think what you do next is make $f'(x) = 0$ so it becomes: $0 = 2x\cos^2x - \sin x$. ...
0
votes
3answers
1k views

How to prove $\sin 3A+\cos 3A=(\cos A-\sin A)(1+2\sin 2A)$

How to prove the following equation? \begin{eqnarray} \sin 3A+\cos 3A&=&\left(\cos A-\sin A\right)\left(1+2\sin 2A\right)\\ \end{eqnarray} Let's start with the left hand side. ...
1
vote
1answer
74 views

Calculate the inverse for $\arctan(x^2+1), x≥0$

I have no idea how to solve this problem. Calculate the inverse for the function: $$f(x) = \arctan(x^2+1),\quad x≥0 .$$ Also specify $D_{f^{-1}}$ and $V_{f^{-1}}$. I would really ...
3
votes
2answers
123 views

Prove $2\cos^2(x)=1+\cos(2x)$

I need help to prove that: $2\cos^2(x)=1+\cos(2x)$. I know that $\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$, but I don't know how to get to this step without memorizing ...
1
vote
0answers
65 views

Can this trigonometric equation be solved exactly or approximately?

I need to find a solution for $\alpha$ that satisfies the following equation for $i=1,2,...,N$ $$\frac{2i}{K} = \sin(2 \alpha i) \left(\frac{\frac{\cos(k-1)\alpha}{2}-\frac{\cos(k+1)\alpha}{2}}{1- ...
2
votes
2answers
354 views

How to prove that $\cos\theta$ is even without using unit circle?

The proofs I have come across on showing that $\cos \theta$ is even is something like this: In a unit circle, $\cos\theta$ gives you the $x$ coordinate after traveling $\theta$ radians ...
1
vote
1answer
503 views

If inside a big circle , exactly n $(n \geq 3)$ small circles, each of radius r,can be drawn in such a way that each small circle touches t…

Problem : If inside a big circle , exactly n $(n \geq 3)$ small circles, each of radius r,can be drawn in such a way that each small circle touches the big circle and also touches its adjacent small ...
2
votes
0answers
81 views

Relating $x$ to $\frac{dx}{dt}$ in a right triangle.

I have a right triangle with sides of $x$ and $y$. I know $y$ is a constant (500) and that $\frac{d \theta}{dt}$ (where $\theta$ is the angle opposite from side $x$) is also constant ($8\pi$ rad/s). I ...
2
votes
4answers
174 views

Trigonometric substitution

I am having troubles understanding when I should use trigonometric substitution to find an indefinite integral. Is there a general rule for when to apply this technique?
0
votes
3answers
168 views

What is the theta in this cartesian to polar coordinates problem?

If x=1 and y=1, what is the theta and why? I know for a fact that the answer is pi/4 but I do not get why.
1
vote
1answer
245 views

How to prove $\tan x+\tan \left(x+\frac{\pi }{3}\right)+\tan \left(x+\frac{2\pi }{3}\right)=3\tan 3x$?

The following is the equation. \begin{eqnarray} \tan x+\tan \left(x+\frac{\pi }{3}\right)+\tan \left(x+\frac{2\pi }{3}\right)&=&3\tan 3x\\ \tan x+\frac{\tan x+\tan \frac{\pi }{3}}{1-\tan ...
4
votes
2answers
194 views

Solve the following definite integral: $\int_{0}^{\infty}\frac{x^2dx}{({1-x^2})^2}$

Solve the following integral: $$\int_{0}^{∞}\frac{x^2dx}{({1-x^2})^2}$$ I know that substituting some trigonometric functions may help. But I was not able to solve. Can you give me some ...
3
votes
1answer
87 views

Let P be a moving point such that if $PA$ and $PB$ are two tangents drawn from $P$ to the circle $x^2+y^2=1 ( $ A ,B being the points of contact) ,…

Problem : Let $P$ be a moving point such that if $PA$ and $PB$ are two tangents drawn from $P$ to the circle $x^2+y^2=1$ ( $A$, $B$ being the points of contact) , then $\angle AOB = 60^{\circ}$, ...
2
votes
2answers
114 views

If $\frac{\cos x}{\cos y}=\frac{a}{b}$ then $a\tan x +b\tan y$ equals

If $\frac{\cos x}{\cos y}=\frac{a}{b}$ then $a \tan x +b \tan y$ equals ( options below ) (a) $(a+b) \cot\frac{x+y}{2}$ (b) $(a+b)\tan\frac{x+y}{2}$ (c) $(a+b)(\tan\frac{x}{2} ...
1
vote
3answers
181 views

Prove Trigonometric Identity $\frac{\sin x}{1-\cos x} = \frac{1+\cos x}{\sin x}$

I'm doing some math exercises but I got stuck in this problem. In the book of Bogoslavov Vene it says to prove that: $$\frac{\sin x}{1-\cos x} = \frac{1+\cos x}{\sin x}.$$ It is easy if we do it ...
0
votes
4answers
1k views

How to prove $\cot ^2x+\sec ^2x=\tan ^2x+\csc ^2x$?

How can I prove the following equation? \begin{eqnarray} \cot ^2x+\sec ^2x &=& \tan ^2x+\csc ^2x\\ {{1}\over{\tan^2x}}+{{1}\over{\cos^2x}} &=& ...
4
votes
2answers
1k views

If $\tan(\pi \cos\theta) =\cot(\pi \sin\theta)$, then what is the value of $\cos(\theta -\frac{\pi}{4})$?

Problem : If $\tan(\pi \cos\theta) =\cot(\pi \sin\theta)$, then what is the value of $\cos(\theta -\frac{\pi}{4})$? My approach : Solution: $\tan(\pi \cos\theta) =\cot(\pi \sin\theta)$ ...
0
votes
1answer
27 views

Need help understanding this result

I am trying to solve a question, where the following result occurs in the solution - $\sin y \sin(2x+y)=0 \ and \sin x \sin(x+2y)=0$ Then it implies that 1) Either $x=0=y$ 2) or $\tan x=\tan y ...
0
votes
1answer
37 views

Half and double angle

when I know that $tan(\frac{x}{2})$ is, then what is $tan(x)$ ? I recently have seen a formula for this, but I can't fit it and I do not remember how to get it :) Thank you.
-2
votes
3answers
96 views

Trigo problem : Find the value of $\tan \left(\dfrac{\pi}{4}\sin^2x\right), x \in \mathbb{R}$

Trigo problem : Find the value of $\tan \left(\dfrac{\pi}{4}\sin^2x\right), x \in \mathbb{R}$ Let $f(x) = \tan \left(\dfrac{\pi}{4}\sin^2x \right), x \in \mathbb{R}$ therefore, $-\infty < f(x) ...
3
votes
2answers
291 views

Find the value of $\sin25^{\circ} \sin35^{\circ} \sin85^{\circ}$

Trigo problem : Find the value of $\sin25^\circ \sin35^\circ \sin85^\circ$ My approach : Using $2\sin A\sin B = \cos(A-B) -\cos(A+B)$ $$ \begin{align} & \phantom{={}}\cos10^{\circ} ...
3
votes
2answers
166 views

Integer multiples of $2\pi$ on $\cos$ function

Suppose you know that the smallest positive value $t$ such that $\cos t=0$ is $t=\dfrac{\pi}{2}$, but you don't know other values of $\cos$ and $\sin$. You also know that $u$ is a real number such ...
7
votes
1answer
104 views

Prove $\frac {1}{\cos 0^\circ \cdot \cos 1^\circ} + \ldots +\frac {1}{\cos 88^\circ \cdot \cos 89^\circ}= \frac{\cos 1^\circ}{\sin 1^\circ}$

Prove the following identity: $$\frac {1}{\cos 0^{\circ} \cdot \cos 1^{\circ}} + \ldots +\frac {1}{\cos 88^{\circ} \cdot \cos 89^{\circ}} = \frac{\cos 1^{\circ}}{\sin 1^{\circ}}$$ After hours of ...
1
vote
3answers
619 views

Equation of a Circle from parametric functions of sin and cos

Given: x = 2 cos (t/2) y = 2 sin (t/2) How do we find the equation of the circle? I know that x^2 + y^2 = 1, where x = cos(t) y = sin(t) so x^2 = (2 cos (t/2))^2 y^2 = (2 sin (t/2))^2 How do ...
3
votes
4answers
146 views

solving trigonometric equation $3^{\sin^2(x)}+3^{\cos^2(x)}=4$

Please help me to solve this trigonometric equation. $$3^{\sin^2(x)}+3^{\cos^2(x)}=4.$$
1
vote
1answer
90 views

Trigonometric Identity $\arcsin(\alpha+\beta)=\arcsin(\alpha|\sec(\beta)|)+\arcsin(\beta|\sec(\alpha)|)$

Is this trigonometric identity is true? $$\arcsin(\alpha+\beta)=\arcsin(\alpha|\sec(\beta)|)+\arcsin(\beta|\sec(\alpha)|).$$
2
votes
4answers
524 views

How to prove the identity $3\sin^4x-2\sin^6x=1-3\cos^4x+2\cos^6x$?

I'm trying to prove a trigonometric identity but I can't. I've been trying a lot but I can't prove it. The identity says like this: $$3\sin^4x-2\sin^6x=1-3\cos^4x+2\cos^6x$$ The identity would be ...
4
votes
4answers
1k views

$\lim_{x\to0} \frac{x-\sin x}{x-\tan x}$ without using L'Hopital

$$\lim_{x\to0} \frac{x-\sin x}{x-\tan x}=?$$ I tried using $\lim_{x\to0} \frac{\sin x}{x}=1$. But it doesn't work :/
2
votes
2answers
108 views

Condition for $\tan A\tan B=\tan C\tan D$

Here, it is claimed that $$\tan A\tan B=\tan C\tan D$$ if one of the four following conditions holds $$\displaystyle A\pm B=C\pm D$$ If it is true, how to prove this? $\tan(x\pm y)$ did not help ...
5
votes
3answers
284 views

Solving trigonometric equation with unknown and restricted domain

Given that $ \tan^2(\fracθ3) = 1$ and $θ\in [0, 4\pi]$ find θ. I'm not sure how to progress with the restricted domain. Here's what I've got so far: Solving for the domain $[0, 4\pi]$. $$ ...
0
votes
1answer
164 views

Using De Moivre's theorem

Hello I want to solve this $x^5=32$ using De Moivre's theorem $$z^n=r(\cos(n\theta) + i \sin(n \theta))$$ I want help to find the solution and more specifically to find the missing $\theta$. ...
21
votes
1answer
711 views

A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

I need to find a value of this definite integral: $$\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx.$$ Its numeric value is approximately $0.7875720991394284$, and lookups ...
1
vote
2answers
48 views

Trigonometric Limits function

I need to find this limit $\lim_{\theta\to \frac \pi2}$ (2 $\theta$ - $\pi$) sec $\theta$ ?
2
votes
3answers
248 views

Evaluate $ \lim_{x\rightarrow{\frac\pi2 }} (\sec(x) \tan(x))^{\cos(x)}$ without L'Hôpital's rule

I have tried changing limit to $\lim_{x\rightarrow0}$ and use some trigonometry identity ($\sin^2(x)+\cos^2(x) = 1$ and $\sin (x+\pi/2) = \cos(x)$) but doesn't work I have no idea on how to do this ...
1
vote
1answer
159 views

Angle of rotation of a wheel over time

I am modelling the motion of a wheel that should rotate at a constant speed over time. Think of a Ferris wheel or a car wheel. I am trying to determine at what angle the wheel will be at about the ...
0
votes
1answer
466 views

Why does using different units of angle affect the rate of change?

The question is A triangle has sides of length 4cm and 9cm. The angle between them is increasing at a rate of 1$^\circ$ per minute. Find the rate in cm$^2$ per minute at which the area of the ...
18
votes
2answers
551 views

On the “funny” identity $\tfrac{1}{\sin(2\pi/7)} + \tfrac{1}{\sin(3\pi/7)} = \tfrac{1}{\sin(\pi/7)}$

This equality in the title is one answer in the MSE post Funny Identities. At first, I thought it had to do with $7$ being a Mersenne prime, but a little experimentation with Mathematica's integer ...
3
votes
2answers
2k views

How to find the type of triangle when given the ratio of it's sides?

Q.The sides of a triangle are in ratio 4 : 6 : 7, then the triangle is: (A) acute angled (B) obtuse angled (C) right angled (D) impossible It's definitely not (C) right-angled ...
1
vote
3answers
216 views

Prove, that $\sin x- a^3\cos x\leq \frac 1 3 \sqrt{1+a^6}$

Let a and $x$ be natural numbers with the property that $\sin x\leq a\cos x$. Prove that $\sin x- a^3 \cos x\leq \frac 1 3 \sqrt{1+a^6}$. Again, I'm looking for a second solution. I don't know how to ...
2
votes
1answer
44 views

How to start showing that a formula is valid

This is the problem on the practice exam for preparing the exam, this question seems impossible to solve, how to I get started guys? Appreciated for the tutoring in advance! ...
7
votes
2answers
114 views

$\frac{\pi}{4}=k\arctan \frac{1}{m}+l\arctan \frac{1}{n}$ has only four solutions?

Is the following true? "$$\frac{\pi}{4}=\arctan \frac{1}{2}+\arctan \frac{1}{3}$$$$\frac{\pi}{4}=2\arctan \frac{1}{2}-\arctan \frac{1}{7}$$$$\frac{\pi}{4}=2\arctan \frac{1}{3}+\arctan ...