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

learn more… | top users | synonyms (1)

0
votes
0answers
10 views

Angle of elevation problem

the angle of elveation on the top of a tower at a point A on the ground is $30$ degrees. On walking $20$ m toward the tower,the angle of elevetion becomes $60$ degrees.Find the height of the tower and ...
1
vote
3answers
44 views

Sum of $\sin$ and $\cos$

We are given a trigonometric equation to solve: $$a\sin x+b\cos x=c$$ with $a,b,c$ nonzero real numbers. We are also given that $$a\sin x+b\cos x=R\sin(x+\varphi)$$ with $R^2=a^2+b^2$ and ...
0
votes
2answers
10 views

Verify the trig formula with complex representation

Verify the trigonometric formula for sin(a+b)=sinacosb+sinbcosa and cos(a+b)=cosacosb-sinasinb by using complex representation. I tried to use Euler's formula to start but I am unsure how to use ...
0
votes
3answers
30 views

Verify Euler's formula

Verify Euler's formula for $e^{ix}$ by considering $\frac{dz}{dx}$ where $z=r(\cos x+i\sin x)$ I tried taking the derivative of z but could not get to Euler's from there.
0
votes
1answer
33 views

Express $\sinh x$ in terms of the exponential function

I found that $\cosh x=\frac{e^x + e^{-x}}{2}$ but I am unsure how to find $\sinh x$ in terms of the exponential function by using Euler's formula.
1
vote
3answers
20 views

Finding Lipschitz for trigonometric functions

how would i find the Lipshitz constant for $$\sin(x)\times \cos(x)$$ or other trigonometric functions? How would I get my $\operatorname{abs}{x_1 - x_2}$
2
votes
2answers
39 views

trouble solving the integral of $\cos(x^2)$

No, I really mean the integral of $\cos(x^2)$, not $[\cos(x)]^2$. Can the chain rule be applied here?
0
votes
1answer
21 views

Complex Trigonometry

Find the moduli and argument of $(1+i)e^{\pi/6}$. I converted $(1+i)$ to polar form, that is $\sqrt2e^{i\pi/4}$ and multiplied by $e^{i\pi/6}$ to obtain $\sqrt2e^{i5\pi/12}$. But am unsure how to ...
1
vote
2answers
38 views

Equation with tangent and powers

I need to solve this equation for x: $$2000 \sigma = 1 - \frac{20x}{\pi^2x^2 + 100} - \frac{2 \arctan(\frac{\pi x}{10})}{\pi} $$ $\sigma$ is a known value. I need to solve this for $ \sigma = ...
0
votes
2answers
23 views

Intersection points of two trignometric equations

I am studying for a SAT II Math2C and I came across this question in Barron's book. Solve $2 \sin(x) + \cos2(x) = 2 \sin^2(x) - 1$ [0<= x <= 2pi] The solution says put the equations in a ...
0
votes
1answer
10 views

Converting to Spherical Coordinates that have a Large Azimuth?

I've run into a problem converting Cartesian coordinates to spherical coordinates. Say I've got a vector/point $p=(-1,5,7\frac{2}{3})$. Obviously, finding the polar angle/inclination isn't going to ...
1
vote
2answers
33 views

Finding the fourth roots of $\,5(\cos(3)+i\sin(3)).$

Find the four fourth roots of $\,5(\cos(3)+i\sin(3)).$ I tried to convert to polar form so I could set up an equation like $\,x^4=5e^{i3},\,$ but I am unsure to continue.
78
votes
3answers
5k views

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
0
votes
1answer
18 views

Half Angle Trig and Complex Numbers

By making use of the half-angle formulae, or otherwise, prove that $$\frac{1+\cos x+i\sin x}{1-\cos x+i\sin x}=\cot{\frac x2} e^{i(x-\frac\pi2)}$$
1
vote
1answer
38 views

how to solve asinx+bcosx

Let's solve: $\sqrt{3}\sin x - \cos x=2$ The left hand side may be expressed as $R\sin(x+ \phi)$ We know that $R=\sqrt{3+1}=2$ We also know that $\tan \phi= \frac{-1}{\sqrt{3}}$ The solution to ...
1
vote
2answers
17 views

Trig height problem using elevation and shadow length? [closed]

The angle of elevation of the sun is 28 degrees 10 minutes, and the shadow of a flagpole on horizontal ground is 97.3 ft. long. How tall is the flagpole?
1
vote
2answers
29 views

How can we prove that $\cot(A/2)-3\cot(3A/2)=4\frac{\sin A}{1+2\cos A}$ [closed]

Prove that $\cot(A/2)-3\cot(3A/2)=4\frac{\sin A}{1+2\cos A}$
0
votes
2answers
21 views

Why does this trig addition equal what it does?

Why does $\frac{3\pi}2+\frac \pi 2 = 2 \pi$? Where does the $3$ go?
0
votes
1answer
19 views

How to find the tangent line of arctany=arcsinx

I am really confused on how to find the derivative/tangent line of arctan(y) = arcsin(x) when x = sin(1). I subbed x in and got arctan(y) = 1 but I am unsure where to go from here.
0
votes
1answer
36 views

Derivative of all real x

Find the derivative of the function for all real $x$. $f(x)= (\sin(x^\frac 13)^3$) It also gives a hint saying extra attention needs to be placed on $x = 0$. Getting the basic derivative isn't the ...
0
votes
1answer
40 views

Finding the limit using $\lim_{x\to 0} (\sin x)/x = 1$ identity

I was having trouble with a question that asked me to utilize the $\lim_{x\to 0} \frac{\sin x}{x} = 1$ identity. It looks like $\lim_{x\to \frac{\pi}{2} }$ $\frac{\tan 2x }{x-\frac{\pi}{2}}$. I 'm ...
4
votes
1answer
59 views

Given $\alpha$, can we always find $\beta$ such that both $\sin(\alpha+\beta)$ and $\sin(\alpha-\beta)$ are rational?

Given $\alpha$, can we always find $\beta$ such that both $\sin(\alpha+\beta)$ and $\sin(\alpha-\beta)$ are rational?
1
vote
1answer
34 views

Find the maximum perpendicular height between a chord and an arc.

I am doing a maths modelling project, and I am stuck on a part. I have a (arc length) and L (chord length), but I want to find H, the maximum perpendicular distance between them! Any help would be ...
0
votes
3answers
37 views

convergente of the sum of sines of the terms of the alternating harmonic series

I want to know about the convergence or divergence of the following series: $$\sum \sin (a_n) $$ where $$a_n=\frac{(-1)^n}{n}$$ The tests that I tried were inconclusive. Is it possible to know? ...
2
votes
0answers
25 views

Solve $x/(4x^2+1) = \tan(6x)$ for $x$

$$ \frac{x}{4x^2+1} = \tan(6x) $$ Can this equation be solved for $x$ algebraically and can I get exact answer for this question? Or do I have to approximate it?
1
vote
0answers
8 views

Calculate angle of view from 2D image

I want to calculate the angle of view (or the field of view) from a photograph, without knowing anything about the camera, as to use that information in a 3D environment. I have to use trigonometry ...
1
vote
2answers
55 views

Derivative of $\sin x$ for small $x$.

I have a question which requires me to prove from first principles that the derivative of $\sin x$ is $\cos x$. You may use $$\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$$ without proof. I managed ...
4
votes
4answers
28 views

If $x=t^2\sin3t$ and $y=t^2\cos3t$, find $\frac{dy}{dx}$ in terms of $t$

If $x=t^2\sin3t$ and $y=t^2\cos3t$, find $\frac{dy}{dx}$ in terms of $t$. This is how I tried solving it: $$ \frac{dx}{dt} = 2t\sin3t + 3t^2\cos3t \\ \frac{dy}{dt} = 2t\cos3t - 3t^2\sin3t \\ ...
2
votes
2answers
41 views

Express $\sin4\theta$ in terms of powers of $\sin\theta$ and $\cos\theta$

As far as I know $\sin4\theta$ = $4\sin\theta \cos\theta$, but I don't know if that's correct or what to do from there?
0
votes
2answers
37 views

Gelfand trigonometry question

If we start with a lemma that states that when $ a^2+b^2=1$ there exists an angle $ \theta $ such that $ a=\cos\theta $ and $ b=\sin\theta$ Suppose that $\alpha$ is some angle if ...
1
vote
3answers
57 views

Calc I limit question involing trig functions

Find the limit $$\lim_{x\to2}\frac{\sin^2(x^2-4)\sec^2(3x-6)}{(x^3-8)\tan(2x-4)}$$ i have been having trouble finding this limit, i have tried using having trig identities and making all terms sin ...
0
votes
1answer
18 views

Finding the initial direction of a parametric curve?

With the parameters: $$x(t)=1-\sin^2t$$ $$y(t)=2+\cos^2t$$ It starts at (1,3) and when t=pi/2 it's at (0,2), so I'm tempted to say it's going down to the right; is this correct? In general, is there ...
1
vote
0answers
51 views

How to easily prove Euler's theorem, $OI^2=R(R-2r)$?

If $R$ is the circumradius and $r$ is the inradius of some triangle $ABC$, with its circumcenter being $O$ and incenter being $I$, then how to prove: $$OI^2=R(R-2r)$$ I have seen many mentions of ...
0
votes
1answer
21 views

What is the cartesian equation of $r = 4 + \frac{\sin(\theta)}{2}?$

This is extremely similar to this question, but as there is no r next to the constant 1, when I multiply everything by r I'm going to end up with: $r^2 = 4r + r\frac{\sin(\theta)}{2}$ And I don't ...
0
votes
3answers
34 views

Trigonometry proof- finding an angle

I don't even know where to start with this problem. Suppose $\alpha$ is some angle less than $45^\circ$. If $a=\cos^2\alpha - \sin^2\alpha$ and $b = 2\sin\alpha\cos\alpha$, show that there is an ...
1
vote
1answer
32 views

Simplify this fraction with square roots; application to arctangent equation.

I need your help. I don't know how to simplify: $\frac{-1+\sqrt{3}+\sqrt{4+2\sqrt{3}}}{2\sqrt{3}} $ and $\frac{-1+\sqrt{3}-\sqrt{4+2\sqrt{3}}}{2\sqrt{3}}$ Thank you in advance. I found $1$ and ...
2
votes
4answers
40 views

Show that $2\cos(x)$ is equal to $2\cos(2x)\sec(x)+\sec(x)\tan(x)\sin(2x)$

This is from the derivative of $\dfrac{\sin(2x)}{\cos x}$ I tried to solve it and arrived with factoring the $\sec(x)$ but I still can't get it to $2\cos(x)$. Could you help me out, please? Thanks
3
votes
2answers
44 views

What is the minimum value of $(\tan^2)(A/2)+(\tan^2)(B/2)+(\tan^2)(C/2)$, where $A$, $B$ and $C$ are angles of a triangle

What is the minimum value of $(\tan^2)(A/2)+(\tan^2)(B/2)+(\tan^2)(C/2)$, where $A$, $B$ and $C$ are angles of a triangle? I know that the sum of the angles is $\pi$, but I am unable to find the ...
1
vote
2answers
35 views

Tanget to the curve, but point not on curve?

Question is Find the $x$-coordinate of all points on the curve $$y=22x\sin(5x)+55\sqrt{3}x^2+68,\quad \frac\pi{10}<x<\frac{3\pi}{10}$$ where the tangent line passes through the point $P(0,68)$ ...
6
votes
1answer
127 views

Two integral involving logarithm and trigonometric function [on hold]

Evaluate the following integrals $$\int\limits_0^{\frac{\pi }{2}} {{x^3}{{\ln }^2}\left( {\sin x} \right)dx} ,\int\limits_0^{\frac{\pi }{2}} {{x^3}{{\ln }^2}\left( {\cos x} \right)dx} .$$ Can you ...
3
votes
1answer
71 views

Is this true? $\forall x, y \in\mathbb{Q}: (\sin(x)=\sin(y))\Rightarrow (x=y)$

I just thought about the following expression: $\forall x, y \in\mathbb{Q}: (\sin(x)=\sin(y))\Rightarrow (x=y)$ I think it is true because values of $\sin(x)$ only repeat every $\pi\times n$th time, ...
2
votes
0answers
26 views

Launch angle required to hit coordinate (x,y) with air resistance

Finding Angle of Elevation to hit X, Y and Wikipedia Angle required to hit coordinate work, but don't calculate air resistance. Is there a way to find the launch angle of a projectile required to hit ...
1
vote
1answer
25 views

Finding max and min in $f(x) = 5\sin x + 12\sin(x+\frac{\pi}{3})$

For the function $f(x) = 5\sin x + 12\sin(x+\frac{\pi}{3})$, find the max and min value the function can be. Own thoughts I first noted that the function had no constant, and so the max = ...
3
votes
2answers
95 views

Find the time interval between oscillations of SHM.

Parts i) and ii) I can solve. But for part iii) I can't do, as I don't know which equation describes the SHM motion? Is it $y=0.5sin(1.2t)$ or $y=0.5cos(1.2t)$ or $x=0.5sin(1.2t)+2.5$? I thought ...
0
votes
1answer
52 views

For which angles $a$ is $\sin^4 a - \cos^4 a > \sin^2 a - \cos^2 a$?

My questions For which angles $a$ is $\sin^4 a - \cos^4 a > \sin^2 a - \cos^2 a$? For which angles $a$ is $\sin^4 a - \cos^4 a \ge \sin^2 a - \cos^2 a$? I understand that the two sides will be ...
0
votes
1answer
45 views

Trigonometry identity proof.

Can anybody prove the following? $$\frac{\sin^3 a- \cos^3 a}{\sin a- \cos a}= 1 + \sin a \ \cos a.$$ Thanks.
1
vote
4answers
35 views

Trigonometry identity proof

I am working my way through Gelfands trigonometry book. One of the exercises asks to prove the following identity: $$ \frac{\sin(a)}{1 + \cos(a)} = \frac{1 - \cos(a)}{\sin(a)}$$ I can reduce the ...
0
votes
0answers
19 views

Simplification of trig function composition?

I was wondering if there was any way to simplify thing like sin(cos(x)), cos(sin(x)), sin(tan(x)), etc. for non-inverse functions. Maybe even things like sin(sin(x)). This is just me wanting to know ...
0
votes
2answers
64 views

Equation of circle in terms of length of arc above $x$-axis

Say I have a circle centered at $(0,b)$ that passes through $(-5,0)$ and $(5,0)$ and has upper-half length $d.$ Now I've figured out that the equation of the circle is $$x^2 + (y-b)^2 = 5^2 + b^2$$ ...
1
vote
2answers
43 views

Complicated trig limits to infinity problem!?

$$\lim_{t\to-\infty}\frac{2-t+\sin t}{t+\cos t}$$ I don't know how to do this problem! I don't think you can use the squeeze theorem on it? Any help would be appreciated.