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

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2
votes
6answers
116 views

Finding the value of $\tan 20^\circ$.

Just a small thought popped up in my mind; and now I'm stuck on it. Any idea on how to find the value of $\tan 20^\circ$? I tried doing it by using the multiple angle formulas, but I didn't get an ...
2
votes
2answers
46 views

$\sin(2x)-\cos(2x)-\sin(x)+\cos(x)=0$

I would like to share a trigonometry question here. Wonder is there another way to solve it or not. $\sin(2x)-\cos(2x)-\sin(x)+\cos(x)=0$ $2\cos(x)\sin(x)-(1-2\sin^2(x))-\sin(x)+\cos(x)=0$ ...
2
votes
1answer
26 views

Proving $\sin ((n-1/2)\phi) + \sin(\phi/2)=\sin({n+1 \over 2}\phi)$

I am trying to show that $\sin ((n-1/2)\phi) + \sin(\phi/2)=\sin({n+1 \over 2}\phi)$ I tried to apply $\sin (x-y) = \sin x \cos y - \cos x \sin y$ to it and I got $\sin ((n-1/2)\phi) + ...
2
votes
1answer
34 views

Generalization of $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$

I'm looking for a generalization of the following statement $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$ In particular, I want to find $\sup \limits_{\theta} (a \sin ...
2
votes
2answers
100 views

Polar Plots and square roots

When I plot a polar plot of $r=\sin (3 \theta)$, and $r=\sqrt{\sin (3 \theta)}$ I get nearly identical graphs, both $3$ pedal rose type plots. In the case without the square root, it is easy to ...
2
votes
4answers
126 views

asymptotically sharp upper and lower bound for for arctan [closed]

How do I prove that $\frac{\pi}{2}-\frac{1}{x}<\arctan(x)<\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3x^3}$ for all $x>0$?
2
votes
1answer
109 views

Prove $ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $

$ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $ This is what I have so far: I know that $A + B + C = 180^\circ$, so $C = 180^\circ - (A+B)$. ...
2
votes
1answer
56 views

Find $r$ knowing that $r=\frac{60}{\sin^{-1}\frac{60}{r}}$

I'm trying to find the value of $r$ knowing that: $$r=\frac{60}{\sin^{-1}\frac{60}{r}}$$ I'm not really sure how to approach finding the solution. Can anyone help me out? I've spent well over an ...
2
votes
6answers
218 views

How to find $\lim_{x\to 0}\frac{\tan 3x}{\tan 5x}$?

I am asked to find the following limit: $$ \lim_{x \to 0} \frac{\tan 3x}{\tan 5x}$$ My problem is in simplifying the function. I followed two different approaches to solve the problem. But both ...
2
votes
0answers
44 views

From $\tan(1/A) = \tan(1/B) + \tan(1/C)$ to $A + B + C = ABC$

In this recent question, the equation $$\tan\left(\frac{1}{A}\right) = \tan\left(\frac{1}{B}\right) + \tan\left(\frac{1}{C}\right)$$ is said to imply $$A + B + C = ABC$$ without any stated ...
2
votes
8answers
162 views

What is $\tan50^\circ$

What is $\tan50^\circ$? (without using a calculator) 1 a little less than 1 a little bigger than 1 none of the above answers I think the answer is 3, but I can not explain this mathematically. The ...
2
votes
2answers
40 views

Writing answers to trigonometric equation

I wonder how to write answers to trigonometric equations in more elegant form. For instance if we have $ \displaystyle \sin x = \frac{\sqrt{2}}{2} \vee \sin x=-\frac{\sqrt{2}}{2}$ then I write four ...
2
votes
4answers
75 views

Can someone explain this?

$\sec(x/2) = \cos(x/2)$ I worked on this and got here... (Let (x/2) = u) $\cos u - \sec u = 0$ $\cos u(1 - \sec^2u) = 0$ $\cos u[ -1(-1 + \sec^2u)] = 0$ $\cos u(-\tan^2u) = 0$ So, the solutions ...
2
votes
6answers
270 views

How would you solve the inequality $\sin x \gt \cos x$?

$$\sin x \gt \cos x, \qquad (-2\pi <x <2\pi)$$ I tried an approach saying that $\tan x\gt1$ but apparently the solution, which is $\frac{\pi}{4}<x<\frac{5\pi}{4}$ is not good. It's a ...
2
votes
4answers
433 views

Proof that $\sin(x) > x/2$

I need to prove that $\sin(x) > \frac{x}{2}$ if $0<x<\pi/2$ I've started working with the derivative, but if it's possible, I'd rather something simpler than that.
2
votes
1answer
121 views

Radius of circle by knowing a cross section.

I have a curve on an ellipse where I know the length of a cross section and need to find out it's radius (vertically and horizontally) and calculate the angle of the curve. In the following diagram ...
2
votes
1answer
70 views

Equation $2\cos(x)-3\tan(x)=0$

I solved this equation $2\cos(x)-3\tan(x)=0$ and I got, $\frac{1}{2}=\sin(x)$ and $-2=\sin(x)$. For the first solution I got $\arcsin(1/2)=x, 30°=x$, but second is invalid because the domain of ...
2
votes
4answers
158 views

Finding $\lim_{x \to 0} \frac {a\sin bx -b\sin ax}{x^2 \sin ax}$ witouth L'Hopital, what is my mistake?

I was working on this question. $\lim_{x \to 0} \dfrac {a\sin bx -b\sin ax}{x^2 \sin ax}$ $\lim_{x \to 0} \dfrac {1}{x^2} \cdot \lim_{x \to 0} \dfrac { \frac {1}{abx}}{\frac {1}{abx}} \cdot \dfrac ...
2
votes
2answers
75 views

How to prove: $-\frac{1}{\sec2x}=\frac{\cos^3x-\sin^3x}{\cos x +\sin x}+\frac{\cos2x}{(\cos x +\sin x)^2}$

How do you do it? I'm really stuck on this proof. Can someone please explain? Thanks
2
votes
0answers
420 views

Reduction formula for a trigonometric integral

I have come upon the following trigonometric integral: $$\int (\alpha + \sin x)^n \cos^2 x\,\mathrm{d}x,$$ where $\alpha \in \mathbb{R}$ is an arbitrary real constant and $n \in \mathbb{N}$ is a ...
2
votes
2answers
110 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 ...
2
votes
2answers
101 views

Finding $\cos(A+B)$ from $\sin(A+B)$

The question states to start with the identity: $$ \sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$$ It then asks to use the above to prove that: $$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$$ All I can think ...
2
votes
1answer
343 views

Graph the following function: $y = \sin \frac{1}{2}\left(x-\frac{\pi}{6}\right)$

I am working my way through Trigonometry by Gelfand and Saul. I am trying to work out the following question on p 183: Graph the following function: $$y = \sin ...
2
votes
2answers
90 views

How to find $P_1$ in $(x,y)$ form

From following diagram, $A_1$ is center of circle of radius $r$. All distances are in coordinate system $(x,y)$. Distance from $A_1P_2$ is known. Distance $A_1,A_2,A_3$ is also known from origin. I ...
2
votes
3answers
101 views

If $p =\frac{4\sin\theta \cos\theta}{\sin\theta +\cos\theta}$ Find the value of $\frac{p+2\sin\theta}{p-2\sin\theta}$

Problem : If $\displaystyle p =\frac{4\sin\theta\cos\theta}{\sin\theta +\cos\theta}$, find the value of $\displaystyle \frac{p+2\sin\theta}{p-2\sin\theta} + \frac{p+2\cos\theta}{p-2\cos\theta}$. ...
2
votes
1answer
84 views

find parameter for maximize area

suppose that we have Cartesian coordinate system.and suppose that we have three point which depend on parameter $t$,where t belongs to $(0,1)$;points are $A(cos(3-t),sin(3-t))$ $B(cos(t),sin(t))$ ...
2
votes
1answer
149 views

Trigonometry and algebra question

Given: The total length of ad + dc The lengths of each ab, bc and ...
2
votes
1answer
219 views

A cosine function has maximum value of 14 and a minmum value of 4, a period of 7, and a phase shift of 12.

A cosine function has a maximum value of 14 and a minimum value of 4, a period of 7, and a phase shift of 12. Write an equation representing this cosine function... Could someone tell me if I'am ...
2
votes
1answer
312 views

A Trigonometric Sum Related to Gauss Sums

This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it. Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} ...
2
votes
1answer
215 views

Given that $\sec \theta = k$, $|k| \geq 1$, and that $\theta$ is obtuse, express in terms of $k$: $\cos \theta$, and $\csc \theta$

Given that $\sec \theta = k$, $|k| \geq 1$, and that $\theta$ is obtuse, express in terms of $k$: $\cos \theta$, and $\csc \theta$ For $\cos \theta$ I get: $\frac{-1}{k}, (as \theta$ is obtuse, ...
2
votes
2answers
2k views

Calculus integration problem: $\int \sin^5 (x) \cos^2 (x)\,dx$

What's the integration of $$\int \sin^5 (x) \cos^2 (x)\,dx?$$
2
votes
1answer
42k views

How do I find the angles of a triangle if I only have the lengths of the sides?

Is it possible to find the angles of a triangle if I only have its sides? If so, how can I achieve this? Regarding my knowledge of triangles: Whilst I was taught trigonometry a few years ago, I ...
2
votes
1answer
124 views

Inverse trig question?

Hello everyone I have a question about trig. How would I solve the following. $$\tan\left(2\arcsin(4/5)+\arccos(12/13)\right)=\frac{253}{204}$$ Please help.
2
votes
3answers
341 views

Range of a trignonmetric function

I came across this in an Engineering entrance book, What is the range of this: $a^2 \sin^2 x + b \sin x \cos x + c \cos^2 x$ What is the method to find it? I tried the graph approach but didn't know ...
2
votes
3answers
762 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 $\sin x+\sqrt3\cos x=2\sin(x+\frac\pi3)$ ...
2
votes
1answer
146 views

A pseudo Fejér-Jackson inequality problem

$x\in (0,\pi)$ ,Prove that: \begin{align} \sum_{k=1}^{n}\frac{\sin{kx}}{k}>x\left(1-\frac{x}{\pi}\right)^3 \end{align} the inequality holds for all integer $n$ I tried Fourier, or Dirichlet ...
2
votes
5answers
988 views

Notation for powers of trigonometric functions [duplicate]

Possible Duplicate: $\arcsin$ written as $\sin^{-1}(x)$ When I learnt the trig identity $\sin^2\theta + \cos^2\theta \equiv 1$, I learnt that $\sin^2\theta = (\sin\theta)^2$. So why isn't ...
2
votes
4answers
1k views

Finding square roots of $\sqrt 3 +3i$

I was reading an example, where it is calculating the square roots of $\sqrt 3 +3i$. $w=\sqrt 3 +3i=2\sqrt 3\left(\frac{1}{2}+\frac{1}{2}\sqrt3i\right)\\=2\sqrt ...
2
votes
2answers
807 views

amplitude of sine wave with multiple frequencies

I'm having some troubles determining the amplitude/magnitude of the following equation. $$ A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3) $$ Since each part is at a ...
2
votes
2answers
860 views

Calculated rotated point coordinate: is my solution correct

Is my calculation correct for this rotation around a point? A point a(-19.94,392.11) is rotated -49.45 degrees, what is the new coordinates of point a? My solution: ...
2
votes
1answer
95 views

I need to factor this function so it is entirely dependent on x- semicircle displacement.

In the attached image are three functions. The first is a displacement function which takes angle $t$, and returns a radius. The third one is a semicircle and the second one is a semicircle with the ...
2
votes
1answer
3k views

Horizontal and vertical asymptotes of polar curve $r = \theta/(\pi - \theta) \, , \, \in[0,\pi]$

I as supposed to find the vertical and horizontal asymptotes to the polar curve $$ r = \frac{\theta}{\pi - \theta} \quad \theta \in [0,\pi]$$ The usual method here is to multiply by $\cos$ and ...
2
votes
2answers
697 views

Evaluating $\int\sin^3t \, dt$

I have this integral: $$\int\sin^3t \, dt$$ I have tried partial integration with $\sin t \cdot \sin^2t$, but then I get another integral to evaluate which needs partial integration: $$\dots \int ...
2
votes
1answer
359 views

How does $\int_1 ^x \cos(2\pi/t) dt$ have complex values for real values of $x$?

This question is closely related to one I just asked here. I believe that it is just different enough to warrant another question; please let me know if it does not. In the question mentioned above, ...
2
votes
1answer
505 views

How to calculate the coordinates of the middle point of a given arc? [duplicate]

Possible Duplicate: How to calculate the coordinates of the middle point of a given arc? I am trying to calculate the green sides of this triangle: I know/have: the arc length, the ...
2
votes
1answer
230 views

Finding Angle of Elevation to hit X, Y

My ultimate goal is to find the angle of elevation necessary to launch a projectile from the origin to (x,y) with initial velocity V and under gravitational acceleration g. Wind resistance is ignored. ...
2
votes
3answers
558 views

Simplifying the expressions for the magnitude and phase of a Fourier transform

$$h[n] = 2( \delta[n-2]-\delta[n-1]-\delta[n-3])$$ i computed my frequency response and i have this now: $$H[e^{j \omega}] = 2[ e^{-2 j \omega} - e^{-j \omega}-e^{-3 j \omega}]$$ $$H[e^{j \omega}] = ...
2
votes
1answer
307 views

How to rearrange formulas to calculate orbit from tangent and apoapsis

I need some help rearranging some orbital mechanics formulas. All images have been borrowed from http://www.braeunig.us/space/orbmech.htm which has a through treatment of orbital equations, but ...
2
votes
2answers
350 views

Angle for pointing at a certain point in 2d space

Recently, I have been programming a simple game. Very simple: There is a tank, and the cannon will aim at whatever position the mouse is at. Now lets talk about the cannon graphic. The cannon graphic ...
2
votes
2answers
844 views

Find the value of $\sin 10^\circ + \sin 20^\circ + \sin 30^\circ - \sin 360^\circ $

How to solve this manually ? EDIT: In my module the answer is given as $0$ but when I used mathematica ...