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
16 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
24 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
0answers
22 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
24 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
1answer
31 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 ...
2
votes
1answer
25 views

Indefinite integral and a trigonometric substitution

I have this integral: $\displaystyle\int\dfrac{x^3}{({\sqrt[2]{4x^2 + 9})^3}}\,dx$. I tried to solve it with a trigonometric substitituon but I can't get any result. I would appreciate if somebody can ...
-2
votes
2answers
33 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
0answers
42 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
37 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
59 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
21 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
52 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
24 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
51 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.
2
votes
1answer
72 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
31 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
35 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
26 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
26 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
43 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
62 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
22 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
20 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 ...
3
votes
2answers
194 views

What method can i use to find the first 3 roots of y(t)=tan(t)+t?

Just by looking at the function: $$y(t) = \tan(t)+t$$ I can immediately see that there is a root at $t=0$, though after graphing it I can see many more roots and I can calculate them using computer ...
3
votes
2answers
26 views

Usage of law of sines

The vertex angle of an isosceles triangle is 35 degrees. The length of the base is 10 centimeters. How many centimeters are in the perimeter? I understand the problem as there are two sides with ...
0
votes
2answers
48 views

Show that complex numbers are vertices of equilateral triangle

1)Show if $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$ then $z_1,z_2,z_3$ are vertices of equilateral triangle inscribed in a circle of radius. I thought I can take use from roots of unity here, since ...
0
votes
1answer
12 views

Trigonometry (non right angled triangles)

The height of a vertical tower is to be found by a surveyor. The angle of elevation of the top of the tower from a point on the horizontal ground some distance away is measured as 28.7 degrees. From ...
2
votes
3answers
18 views

Show Trigonometric Identities from Complex indentity

So the exercise says to show $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ and $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$ By using the following identity: $e^{i(a+b)}=e^{ia}e^{ib}$ How do ...
0
votes
1answer
24 views

Is this solution about unit circle right?

If cos(x)=-2/pi Are these solutions for x right: X=129 and x=230 If they are not correct please correct them
1
vote
2answers
40 views

Is it possible to gain intuition into these trig compound angle formulas - and in general, final year high school math?

Does anyone have any insight into the trig sum and difference formulas? The formulas in themselves are very elegant, but I don't really like the proofs that have been given, even the geometric proofs. ...
0
votes
2answers
26 views

Trigonometric Graphs - Point of intersection with the curve and line

The diagram shows the graph of y=(a)sin(b)x +(c) . 1)write down the value of a,b and c. 2)Find the coordinate of P an Q, the points of intersection with this curve and the line y=2.
0
votes
0answers
19 views

For which value of $ r \in \mathbb N $ is satisfied that $r\sin (\frac{1}{r})\leq sin(\frac{1}{r+1})(r+1)$ [on hold]

For which value of $ r \in N $ is satisfied that, $\qquad r\sin (\frac{1}{r})\leq sin(\frac{1}{r+1})(r+1)$
1
vote
1answer
100 views

if we konw :$\sin{A}:\sin{B}:\sin{C}$ then How Find the value $\sin{(2A)}:\sin{(2B)}:\sin{(2C)}$

Question: let $x,y,z>0$ is give numbers, and the postive number $k$ such $$\dfrac{x^2}{x^2+k}+\dfrac{y^2}{y^2+k}+\dfrac{z^2}{z^2+k}=1$$ in $\Delta ABC$, ...
0
votes
0answers
13 views

Solving/simplifying a trig function

Given: $\tan\:a=\frac{5}{12}$ and $a\in QIII$ $\csc\:B=-\frac{5}{4},\:\cos\:B>0$ $\cos\:\theta =-\frac{8}{17},\:\frac{\pi }{2}<\theta <\pi $ Solve: ...
0
votes
1answer
17 views

Solving/simplifying a trig expression

My problem sheet says that $\tan a= 5/12$ and $a \in {\rm Q\,III}$ ($a$'s in quadrant III). Using this information, I am to solve/simplify the expression $\quad \quad \cos\left(\frac{1}{2}a\right)$ ...