Tagged Questions

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
votes
2answers
31 views

Simplify $\def\Arctan{\operatorname{Arctan}}f(x) = \Arctan(2x) + \Arctan(3x)$

$\def\Arctan{\operatorname{Arctan}}$ Simplify $f(x) = \Arctan(2x) + \Arctan(3x)$ I had a go at it and this is what I got to : We have: $-π<\Arctan(2x)+\Arctan(3x)<π$ Let $a=\Arctan(2x)$ and ...
6
votes
5answers
287 views

How do calculators evaluate inverse trig functions?

I know for simple inputs, you can just memorize the answer, but what if I wanted to find arcsin(0.554). My calculator instantly tells me that the answer is 0.5752 radians. How can I do that by hand, ...
2
votes
1answer
60 views

circular reasoning in proving $\frac{\sin x}x\to1,x\to0$

The classic proof for $\frac{\sin x}x\to1,x\to0$ is using a squeezing theorem based on arguments about areas of circles. But as far as I know, all deduction of formula of circles' area is based on ...
0
votes
2answers
12 views

Find the values of $a$ and $b$ ~ Trigonometry

The function $f$, where $f(x) = a \sin x+b$, is defined for the domain $0 \leq x \leq 2\pi$. Given that $f(\frac{1}{2}\pi)=2$ and that $f(\frac{3}{2}\pi)=-8$, find the values of $a$ and $b$. I know ...
1
vote
2answers
41 views

Rewrite the expression in the form $A \sin(x+C)$

Rewrite the following expression in the form $A \sin(x+C)$ $$4 \sin x + 4\sqrt{3} \cos x$$ This is what I have so far, and I'm not even sure it's the right approach. I just dont understand this ...
-12
votes
1answer
49 views

Find all real numbers in the interval $[0,2\pi]$ that satisfy equation. Write exact answer ($\pi$ as needed) Show Work. Please Help!!!!!!!!!! [on hold]

Find all real numbers in the interval $[0,2\pi]$ that satisfy the equation. Write an exact answer using pi as necessary. Show work $$2 \sin^2 x = \sin x$$
0
votes
1answer
23 views

How does a complex exponential turn into the sinc function?

Suppose I have a complex variable $j$ such that we have $f(u) = \frac{1}{ju}[e^{\frac{ju}{2}} - e^{\frac{-ju}{2}}]$. Could somebody please explain how this turns into a sinc function ? I know I ...
1
vote
1answer
36 views

How to calculate this $\sin\frac{\pi}{9}\sin\frac{2\pi}{9}\sin\frac{4\pi}{9}$?

I'm stuck with the expression $$\sin\frac{\pi}{9}\sin\frac{2\pi}{9}\sin\frac{4\pi}{9}.$$ I have no idea how to begin, please give me a hint! (The answer should be $\sqrt3/8$.)
4
votes
3answers
30 views

Limit with Arctan

Here's a hard limit I've been trying to answer for a while : $$\lim_{x\rightarrow 1} \dfrac{-2x\arctan{x} + \dfrac{\pi}{2}}{x-1}$$ I've tried all the tricks that the teacher has taught us and still ...
0
votes
1answer
28 views

Trigonometric identity involving half angles

Ok.here is the problem in the picture below. How do I get these results? Given that d equals
1
vote
1answer
54 views

How to prove SinA/A+sinB/B+SinC/C<(9*(3)^.5)/2pi

Only for an acute angle triangle. $A$,$B$,$C$ are angles of a triangle. This isnt sine rule form. Ive tried Cauchy Schwarz theorem , A.M, G.M form but am unable to get the above result. Could someone ...
1
vote
1answer
20 views

Simplify a LHS of the trigonometric equation to obtain RHS

Is this equality correct? If so, how to simplify the following expression on the LHS to get RHS: $$\frac{\sin(x+\frac{nh}{2}) \sin(\frac{(n+1)h}{2})}{\sin\frac{h}{2}} ...
0
votes
0answers
16 views

Finding an angle of a line segment [on hold]

I am trying to rotate an icon of a marker on google map. For that i need to know the angle of rotation. I have two lat-long point, and i need to know the angle of the line between those two points. I ...
1
vote
1answer
28 views

Solving trancendental with variable argument. $20 = ax\sin(ax)$

Approaching transcendental equations is in general new to me. My experience with numerical methods is limited, and this equation seems to require such a method. But there's a catch - it contains an ...
1
vote
1answer
24 views

Trigonometric Functions on a unit circle

I have to find all solutions for $\theta$ in the given range: \begin{equation} tan (\theta) = \frac {-1}{\sqrt3}, -\pi \le \theta \lt 2\pi \end{equation} I said that if $(x,y)$ is on the unit circle ...
1
vote
1answer
25 views

Confused about the answer to the inverse of a cosine function

$$\arccos { (\cos { (\frac { 17\pi }{ 6 } ) } } )$$ No matter how I try and look at this problem, I end up with $\frac { 5\pi }{ 6 } $ I counted $\frac { \pi }{ 6 } $ 17 times counter clockwise ...
0
votes
1answer
26 views

Area of a triangle inside a larger triangle

It's been a while since I've done any geometry so I'm a bit confused by this question. We have a triangle $\triangle PQR$ whose total area is $90 \mathrm{cm}^2$. Another triangle $\triangle PTU$ is ...
3
votes
2answers
43 views

Maximum of $\sin A\sin B\cos C+\sin B\sin C\cos A+\sin C\sin A\cos B$ in triangle

What is the maximum value of $$\sin A\sin B\cos C+\sin B\sin C\cos A+\sin C\sin A\cos B,$$ where $A,B,C$ are angles in a triangle? We can rewrite as $$-\sin A\sin B\sin(A+B)+\sin B\sin(A+B)\cos ...
3
votes
3answers
44 views

Evaluating indefinite integral using a trigonometric substitution

I have this integral: $$\int\frac{x^3}{\left(\sqrt{4x^2+9}\right)^3}\,dx$$ I tried to solve it with a trigonometric substitituon but I can't get any result. I would appreciate if somebody could help ...
-2
votes
2answers
34 views

how to get $\theta$ and $\phi$ in these two equations

I would like to solve the below equations but I'm not really sure how to get $\theta$ and $\phi$ . \begin{align*} &a \sin \theta = h \\ &b \cos\phi \cos\theta = r \end{align*}
1
vote
1answer
48 views

Show that f solves the so called wave equation

Task $\text{Let } \; c \in \mathbb{R} \; \text{ be a given parameter, with } \; c > 0$ $\text{ Show that } \; f: (\mathbb{R}^3 \setminus \{ \vec{0} \}) \times \mathbb{R} \to \mathbb{R} \; ...
1
vote
1answer
45 views

Finding circle with two points on it and a tangent from one of the points

Two points P1(x1,y1) and P2(x2,y2) are known. In addition, a line slope passing through P1 is known. The aim is to construct a circle (or circular arc) that it passes through both P1 and P2 and it is ...
2
votes
3answers
63 views

Finding the roots of $\sec^2(x)=0$

I need to find the roots of $\sec^2(x)=0$ in my works. I know there are no real roots of this equation; are there complex roots?
0
votes
1answer
23 views

Why are trigonometric substitutions valid?

Within an integral, when you make a trigonometric substitution like $x = \sin(\theta)$ for $x$, aren't you changing the possible range of values for $x$? Aren't you limiting the possible values of $x$ ...
2
votes
1answer
55 views

Constructively solving a trig equation

Solve the equation $$\frac{\sin(18°+x)}{\sin(x)}=\frac{\sin48°}{\sin18°}$$ If we use a computer we quickly note that $x=12°$, which can be easily proven: ...
0
votes
1answer
18 views

Find the matrix of the transformation with respect to the basis $\cos(t),\sin(t)$

Let $V$ be the space spanned by the two functions $\cos(t),\sin(t)$. Find the matrix of the given transformation $T$ with respect to the basis $\cos(t),\sin(t)$, and determine whether $T$ is an ...
0
votes
3answers
25 views

Determine the value of $x$ in $3 \sin x = 2$ for the range $0<x<2\pi$ [on hold]

Determine the value of $x$ in $3 \sin x = 2$ for the range $0<x<2\pi$
1
vote
1answer
11 views

Help finding the solution of the following equation system.

I have the following equation with some trigonometric functions: ( where: $u_{x}, u_{y}, a, b$ are known. To be found are: $\theta,\phi$) $$ \begin{cases} u_{x} = a\sin(\theta)\cos(\phi) + ...
0
votes
2answers
53 views

Differentiate $y =\sin(1+x^2)^{1/2}$

I've tried differentiating $y= \sin(1+x^2)^{1/2}$ using the chain rule, but I keep getting the wrong answer. Can anyone give me a step by step so I can see what I'm doing wrong? Thanks.
4
votes
0answers
54 views

Finding an angle between side and a segment from specified point inside an equilateral triangle

Here is the question: $\overset{\Delta}{ABC}$ is an equilateral triangle. D is a point inside triangle. $m(\widehat{BAD})=12^\circ$ $m(\widehat{DBA})=6^\circ$ $m(\widehat{ACD})=x=?$ I managed to ...
2
votes
1answer
75 views

How to show $\frac {\cos(x)+\sin(x)}{\cos(x)-\sin(x)}=\frac{1+\tan(x)}{1-\tan(x)}$

A step in trig expression simplification, from a textbook: $$\frac {\cos(x)+\sin(x)}{\cos(x)-\sin(x)}\to\frac{1+\tan(x)}{1-\tan(x)}$$ Please give a hint on how was this transformation achieved.
4
votes
2answers
36 views

Solve equation for $0^\circ < x < 360^\circ$

Solve the following equation for $0^\circ < x < 360^\circ$ $$\cos(2x - 15^\circ) = -0.145$$ By finding out the cos inverse, I get $81.7^\circ$. Because $-0.145$ is negative, it lies on the ...
4
votes
1answer
49 views

prove this properties of triangles trigonometric question

The triangle $DEF$ circumscribes the three escribed circles of triangle $ABC$. Prove that $$\frac{EF}{a\,\cos A} = \frac{FD}{b\,\cos B} = \frac{DE}{c\,\cos C}$$
3
votes
4answers
45 views

Differentiate y=Cot²(sinx)

$$ y = \cot^2(\sin x) $$ How do I differentiate that? I tried using chain rule but I don't understand how to differentiate $\cot^2(\sin x)$.
1
vote
2answers
29 views

$\sin u\geq \frac{2u}{\pi}$ whenever $0\leq u\leq \pi/2$

How is it true that $\sin u\geq \frac{2u}{\pi}$ whenever $0\leq u\leq \pi/2$? I just simply cannot see how this is true though it looks very simple. Hope someone could give an answer thanks
0
votes
1answer
19 views

Can it be possible to write arccsc in one equation by using 2arctan?

I have proved the following two inverse trigonometric identities \begin{align} \text{arccsc}(x)&=2\arctan\frac{1}{x+\sqrt{x^2-1}}, \qquad x\geq 1,\tag{1}\\ ...
0
votes
2answers
27 views

integrate sine at denominator

This integral: $\int_{\pi/2}^0\frac{d\theta}{1-\gamma\sin 2\theta}$. I tried $e^{i\theta}=\cos\theta+i\sin\theta$ and $\sin^2\theta+\cos^2\theta=1$, but didn't succeed. Is there any one can help me? ...
4
votes
0answers
27 views

Write $\sum_{k=1}^nk\sin(kx)^2$ in closed form

$\underline{Given:}$ Write in closed form $$\sum_{k=1}^nk\sin(kx)^2$$ using the fact that $$\sum_{k=1}^nku^k=\frac u{(1-u)^2}[(n)u^{n+1}(n+1)u^n+1]$$ $\underline{My\ Work:}$ I substituted ...
0
votes
1answer
24 views

Use a trig substitution (Half-Angle) in $a_n=2^\frac{n+1}{2}\sqrt{2^n-\sqrt{4^n-a_{n-1}^2}} \ \ \ \forall n\gt1$

I'm given $$a_1=2\sqrt{2}$$$$a_n=2^\frac{n+1}{2}\sqrt{2^n-\sqrt{4^n-a_{n-1}^2}} \ \ \ \forall n\gt1$$ I've tried finding $a_1,a_2,a_3,....$ to try and find a pattern, but it gives no simple pattern ...
6
votes
1answer
44 views

Intuition behind a certain limit.

We want to find $\displaystyle\lim_{\theta\to\frac{\pi}{2}} b_1-a_1$, we are given $c=1$ and that $\cdot=90^{\circ}$ This is my solution; $$\begin{equation}\sin \theta=\frac{b_1}{a_1} \iff b_1=a_1 ...
10
votes
0answers
68 views

How to prove that the problem cannot be solved by the four Arithmetic Operations?

The original prolbem is as in the figure: Suppose the square has unit side length, find the area of blue region. The exact solution is: $$\begin{aligned}S=&\frac{\pi-\sqrt{7}}{4}+2 ...
2
votes
1answer
17 views

Does changing side order of quad change area?

I have a quadrilateral with side lengths $10.40$, $12.33$, $11.75$, $11.50$. I am not given any other information, no angles or anything. I do not need to find the area, since I know it is ...
1
vote
0answers
25 views

Parameterize the equation

Find a way of parameterizing the following curve: $y^2=\sin x $. I have already tried $x(t) = (\sqrt t, \sin^{-1} t) $ but this only gives part of the curve because of the nature of the sqrt function ...
0
votes
0answers
19 views

algebraic determination of the correct phase angle

Let's solve $A\sin x+B\cos x=C$. We know $A\sin x+B\cos x=R\sin(x+k)$ and we easily calculate $R = \sqrt{(A^2 +B^2)}$. We calculate angle $k$ to be the $\arctan(B/A)$. We get a result from the ...
0
votes
0answers
24 views

Solving equation in maxima not placing variable on one side

I'm trying to solve an equation but the variable ($\varphi$ PHI) will not factor out to one side. Is there any other way to do this? I'm using maxima version 5.32.1 Here's the equation in latex as ...
3
votes
1answer
46 views

Show that if A is diagonalizable, then sin^2(A) + cos^2(A) = I. Does this identity also hold for nondiagonalizable matrices?

Show that if A is diagonalizable, then $\sin^2(A)+\cos^2(A)=I$. Does this identity also hold for nondiagonalizable matrices? This is what I got so far: $$ e^{iA}= \cos A +i\sin A \\ \cos A= ...
0
votes
1answer
19 views

Trigonometric Identity Symmetry

I'm currently trying to prove the following trig identity. $\dfrac{\sin \left ( \frac{\alpha}{2} \right ) \cos \left ( \frac{\alpha}{2} \right ) + \sin \left ( \frac{\beta}{2} \right ) \cos \left ( ...
0
votes
2answers
21 views

Trigonometric ratios of compound angles

If $\alpha$ and $\beta$ be two different roots of equation $a\cos\theta+b\sin\theta=c$, prove that $\sin(\alpha+\beta)=\dfrac{2ab}{a^2+b^2}$
0
votes
1answer
32 views

how to calculate the phase angle

When we transform $a\sin x+b\cos x=c$ into $a\sin x+b\cos x=R\sin(x+k)$, we calculate the $k$ angle by $k=\tan(b/a)$. By using calculator, we get a positive or negative degree value for $k$. I know ...
0
votes
1answer
17 views

Calculating a perpendicular distance to a line, when using coordinates (latitude & longitude)

I'm trying to implement the Douglas-Peucker algorithm for simplifying a recorded GPS track (a list of coordinates). All implementations I can find assume a simple X/Y grid of squares, however ideally ...