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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1
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2answers
16 views

Reverse of an iterative function

Here, i have a function for an iterative series. Next value = x + sin(x). converging on a value I want to make it so that i can find the current value, when i know the convergence value, The only ...
3
votes
1answer
96 views

Evaluation of the integral $\int \sqrt{t^4-t^2 + 1}\,dt$

My friend took his Calculus $2/3$ test yesterday. One of the questions he had trouble with was this integral: $$\int \sqrt{t^4-t^2 + 1}dt$$ My attempt It seems rather clear that the only approach ...
4
votes
2answers
56 views

A Characterization of the Tangent Function?

The tangent function has the amazing property that if $\alpha+\beta+\gamma=\pi$ with $\alpha,\beta,\gamma\in (0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi)$ then ...
-3
votes
3answers
118 views

find the value of sin(90-a)

I was wondering if anyone here knows how to find the value of sin (90 - a) using a right-angled triangle. I can find the value using ...
2
votes
3answers
58 views

Sine Subtraction Law - a Geomtrical Proof

I can't solve the following problem (From the book: The Forgotten Art of Spherical Trigonometry): I can't manage to prov it geometrically. I could get that I have to find AB and the values ...
2
votes
5answers
104 views

What kind of curve is made of half circles?

I have this curve. It's definitely no sine or cosine. It consists of half circles. How do you call it and how do you describe it mathematically?
2
votes
2answers
81 views

Find $t$ in $i = 50\sin\left(120\pi t -\frac{3\pi}{25}\right)$ where $i = 25$

An alternating current generator produces a current given by the equation $$i = 50\sin\left(120\pi t - \frac{3\pi}{25}\right)$$ where $t$ is the $\text{time}$ in $\text{seconds}$. (Q) Find ...
0
votes
0answers
30 views

Critical Points of a Complex Sine Function within Bounds

I need a method to find the critical points of the function below. f(x) = 3.8*sin(2.4*x + 1) - 2.3*sin(7.2*x - 2) + 3.2*sin(8.1*x - 3) Bounds [-10, 10] I ...
0
votes
1answer
17 views

Writing a trigonometric function in terms of another for theta in a given quadrant

$\tan ^{ }{ \theta }$ in terms of $\cos ^{ }{ \theta } $ in quadrant III I suppose that using the second pythagorean identity: $\tan ^{ 2 }{ \theta } +1=\sec ^{ 2 }{ \theta } $ I have a ...
1
vote
3answers
20 views

Trig simplification rules

Can some one please explain to me how $(1+\sin x)(\cos x)-(\sin x)(\cos x)$ simplifies to $(\cos x)$. I'm having a hard time finding the trig rules. I'm ok at calc but horrible at trig!
0
votes
1answer
13 views

Graphing in Polar Coordinates

I´m currently using polar coordinates to calculate some double and triple integrals. However, I have an small doubt; when you are want to express, lets say, a circle of radius $a$ centered in $(a,0)$ ...
2
votes
3answers
47 views

$\cos(nx)=Q_n(\cos(x))$ for polynomial $Q_n$ of degree $n$ [duplicate]

Are there any proofs of this equality online? I'm just looking for something very simply that I can self-verify. My textbook uses the result without a proof, and I want to see what a proof would look ...
2
votes
2answers
45 views

Prove that $\cos(nx)=2\cos((n-1)x)\cos(x)-\cos((n-2)x)$

I'm trying to prove this with the following identities: $\cos(a)+\cos(b)=2\cos(\frac{a+b}{2})\cos(\frac{a-b}{2})$ $\cos(a)-\cos(b)=-2\cos(\frac{a+b}{2})\cos(\frac{a-b}{2})$ Whenever I try to reduce ...
0
votes
2answers
42 views

calculate for any natural number n $\cos(2\pi/(2n+1))+\cos(4\pi/(2n+1))+\cdots+\cos(2n\pi/(2n+1))$ [closed]

How to calculate for any natural number? $$\cos\bigg(\frac{2\pi}{2n+1}\bigg)+\cos\bigg(\frac{4\pi}{2n+1}\bigg)+\cdots+\cos\bigg(\frac{2n\pi}{2n+1}\bigg)$$
0
votes
1answer
17 views

How do I solve this derivative given limited information?

Given $y=\cos^{-1}(t^{-1})-\sec^{-1}(t)$, I am to find $y'$. I am also given the following definitions: $\frac{d}{dx}\cos^{-1}(x)=\frac{-1}{\sqrt{1-x^2}}$ ...
0
votes
1answer
39 views

Help with trigonometry

In a game I'm developing I have a ball which is connected to a rope and it moves in a pendulum motion. At some point this ball should move down, taking into account it's current position and an ...
3
votes
2answers
45 views

Finding the maximum area of polygon?

Firstly , I divided the polygon to three triangles and I used the Heron's formula to find the area of triangles which formed the polygon $$A=\sqrt{p(p-a)(p-b)(p-c)}$$.I couldn't find easily the ...
1
vote
2answers
34 views

I need Arctan but only Arctan2 is supplied

I'm a new programmer and I'm programming the projectile of a missile using the equation $\theta = \arctan(v^2\pm\sqrt{v^4-g(gx^2+2yv^2)}/gx)$ where $g$: the gravitational acceleration—usually taken ...
2
votes
2answers
59 views

Solve for x in tanx-2x=0

I know homework questions are generally frowned upon here, but I've run into the following equation, which I've tried to solve and am having a genuinely hard time with: $$\tan(x)-2x=0,x\in(-\pi/2, ...
0
votes
3answers
28 views

Show that $\frac{1-\cos2 \theta}{\sin2 \theta} = \tan \theta$

I have to show that the left equation simplifies to $\tan\theta$: Show that: $$\frac{1-\cos2 \theta}{\sin2 \theta} = \tan \theta$$ I do have prior knowledge that: $$\tan \theta = \frac{\cos ...
1
vote
3answers
29 views

$\arcsin $ differentiation

I am hitting my head against a wall trying to understand how to differentiate this. Can someone please hold my hand through this? I understand that $\arcsin(2x) = \sin^{-1}(2x)$. Is this implicit ...
1
vote
2answers
60 views

“Release” the variable on the left side of equation

I have a particular formula (source) for solar azimuth angle, which I would like to derive in a way, that on the left side I have teta only(that's the solar azimuth angle) and on the left side ...
-1
votes
1answer
37 views

Drawing shapes and stuff with trigonometry? [closed]

I was wondering if there is any notes or links or ideas on how to draw different stuff with trigonometry. I know how to draw a circle with trig.
0
votes
1answer
19 views

Are the two legitimate ways to factor out $\alpha$ in the equation $v_0cos(\alpha)t=x$?

The first one would be $\alpha=\arccos(\frac{x}{tv_0})$ and the second one $\alpha=\frac{1}{v_0}\arccos\frac{x}{t}$ . Ok, I figured out, why it makes sense. Whether you write $cos(\frac{\pi}{2})$ or ...
0
votes
1answer
21 views

Solving trigonometric system of equations

What are the solutions for this system of equations when $\alpha \in \mathbb{R}$ is considered a constant and $0 \leq x < 2\pi$. $$ I) \ (y - \cos x)\sin x + (\alpha - \sin x) (-\cos x) = 0$$ $$ ...
1
vote
2answers
44 views

How would I calculate the area of a rectangle on a sphere using vertical and horizontal angles?

Imagine a sphere being one's eyeball and the rectangular area being the picture of one's view. Like putting a name tag sticker on a balloon. How can I find the area of the rectangle on the sphere?
1
vote
2answers
41 views

Integrating $\frac{x\cos x}{\sin^3x}$.

This has had me going mad for more than an hour now. I'm integrating $\frac{x\cos x}{\sin^3x}~dx$. First, I change it to $x\cot(x)\csc^2(x)~dx.$ Then, using substitution ($u=-\cot(x), ...
2
votes
2answers
88 views

Idea for primality testing based on a trigonometric product

This is an idea that I had about 3 months ago. I tried some college professors, they didn't care. I tried to solve, but with no luck. I ask for help to find the closed form of the following product ...
2
votes
3answers
51 views

Limit at 0 involving e, sine and cosine.

I'm a newbie, please pardon my ignorance. I've been trying to solve this limit, to no avail. $$\lim_{x\to0} \frac{e^{-x} + \sin x - \cos x}{x} = 0$$ I figured I'd break the fraction into 3 parts, ...
0
votes
0answers
44 views

Prove that these two definitions are equivalent

While answering this question I have used that \begin{equation}\sin x=\displaystyle\sum_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}}{(2n+1)!}\end{equation} Nwe my question is that how can it be shown that the ...
1
vote
4answers
56 views

How to simplify $\cos(x)\cos(x)-\sin(x)\sin(x)$ to $\cos(2x)$

I am trying to work on calculus, finding extrema, convacity, critical numbers and so forth but when It comes to trig problems I am struggling with problems where it is not as simple as trig$(x) = n$. ...
0
votes
1answer
23 views

find the height of the tree and its distance from the point of observation

From a point 5 meter above the water surface the angle of elevation of the top of a certain tree is 40 degree 10 minutes while the angle of depression of its image is 63 degree 20 minutes. Find the ...
1
vote
3answers
51 views

If $\sin A+\sin B =a,\cos A+\cos B=b$, find $\cos(A+B),\cos(A-B),\sin(A+B)$

If $\sin A+\sin B =a,\cos A+\cos B=b$, find $\cos(A+B),\cos(A-B),\sin(A+B)$ Prove that $\tan A+\tan B= 8ab/((a^2+b^2)^2-4a^2)$
3
votes
1answer
46 views

The length of a line segment related to a parallelogram

In parallelogram $ABCD$, angle $A$ is acute, point $E$ is on the $AD$ such that $BE$ is perpendicular to $AD$ and point $F$ is on line $CD$ such that $BF$ is perpendicular to $CD$. If $AB=BF=13$ ...
0
votes
0answers
30 views

What is the sine when the opposite is the hypotenuse?

This is such a dumb question, I'm sure. But what is the sine when the opposite is the hypotenuse? Wouldn't that always be 1? And if so, since arcsin(1) = pi/2, ...
0
votes
4answers
50 views

Determine max and min for the function: f(x,y)=x^2+y^2-2x-4y

My problem is defined to determining the max and min for the function: $$f(x,y)=x^2+y^2-2x-4y $$ within the domain: $$ x^2+y^2\leq9$$ And I've been able to locate the minimum which is found within ...
0
votes
3answers
35 views

If $y=-e^x\cos2x$, show that $\frac{d^2y}{dx^2}=5e^x\sin(2x+\tan^{-1}(\frac{3}{4}))$

If $y=-e^x\cos2x$, show that $\frac{d^2y}{dx^2}=5e^x\sin(2x+\alpha)$ where $\alpha=\tan^{-1}(\frac{3}{4})$. I've managed to figure out that $$ \frac{d^2y}{dx^2}=e^x(4\sin2x+3\cos2x) $$ But I'm ...
0
votes
2answers
23 views

Function + Differentiation

Given the function $f(x) = ax^3+bx^2+cx+d$. Determine the value of $a$, $b$, $c$, and $d$ knowing that the curves passes through points (-1,2), (2,3) and that the tangents at the points on the curve ...
3
votes
2answers
37 views

Tricky trigonometric sum evaluation

Prove that the sum $$\sum_{k=1}^{n-1} (n-k)\cdot\cos\left(\frac{2k\pi}{n}\right) $$ Is an integer for any $n\geq 3$. I found this in my textbook but am unable to evaluate this sum. Any help would be ...
2
votes
1answer
32 views

Why is $2^{n/2}(5/3)^n(\cos(n\pi/4)+\sin(n\pi/4))$ an alternate form of the complex number $(5/3)^n(1+i)^n$?

How do I get to that point? I am aware of the formula $$z = r (\cos \alpha + i \sin \alpha)$$ and that $$z^n = r^n(\cos n\alpha+ i\sin n\alpha)$$ But I don't know how to get to what is in the ...
4
votes
4answers
53 views

$\sec\theta + \tan\theta=4$ find $\cos\theta$ given: $\theta\neq90$

I tried the following : \begin{align}\sec\theta + \tan\theta&=4\\ \frac1{\cos\theta} + \frac{\sin\theta}{\cos\theta}&=4\\ \frac{1+\sin\theta}{\cos\theta}&=4\\ ...
0
votes
2answers
46 views

Law of Cosine: Trigonometry Word Problem

The longest side of a triangular lot is 21 meters. If another side of this lot is 17 meters and if the angle formed by thus side and third side is 95 degrees, how long is the third side?
3
votes
1answer
29 views

What is the number of intersections of diagonals in a convex equilateral polygon?

Question: [See here for definitions]. Consider an arbitrary convex equilateral polygon with $n$-vertexes ($n\geq 4$) and the $n$-sequence $\langle \alpha_i~|~i<n\rangle$ of its angles which ...
0
votes
3answers
46 views

Is the numerical solution for $\cos(x)=\sqrt{1-\sin^2(x)}, x=1.1$?

I typed $\cos\theta=\sqrt{1-\sin^2\theta}$ into WolframAlpha and it gave me the numerical solution $\theta=1.1$. However, it did not provide a step-by-step solution like it normally does. Is this ...
0
votes
1answer
43 views

How do I find the horizontal asymptote of $f(x)=\frac{\sin (x) }{x}$?

I can instantly see that there will be a vertical asymptote at $x=0$, however I am finding it quite a challenge to find a horizontal asymptote. I've drawn the graph and it seems as if the amplitude of ...
2
votes
3answers
85 views

ODE satisfied by $\,f(\xi) = \int_0^1 \frac{e^{-\xi x}}{\sqrt{1 - x^2}}dx$

An exercise in a textbook asked me to find the differential equation satisfied by $$f(\xi) = \int_0^1 dx \, \frac{e^{-\xi x}}{\sqrt{1 - x^2}}$$ This seems very difficult since I don't know how to ...
2
votes
2answers
57 views

How to solve trigonometric equations with a domain involving negative values of $x$?

I don't seem to understand the concept of a negative domain when solving trigonometric equations on "another interval" For example: Solve $\cos x=-\sqrt{3}/2$ given that the domain is $-\pi \le ...
3
votes
2answers
75 views

Limits with trigonometric functions without using L'Hospital Rule.

I want to find the limits $$\lim_{x\to \pi/2} \frac{\cos x}{x-\pi/2} $$ and $$\lim_{x\to\pi/4} \frac{\cot x - 1}{x-\pi/4} $$ and $$\lim_{h\to0} \frac{\sin^2(\pi/4+h)-\frac{1}{2}}{h}$$ without ...
0
votes
3answers
38 views

solve triangle law of signs

Height of a Tree A tree on a hillside casts a shadow 215 ft down the hill. If the angle of inclination of the hillside is 22 to the horizontal and the angle of elevation of the sun is 52, find the ...
2
votes
2answers
41 views

Help finding trig values with the given information

I need help with the following problem: Find $\sin x/2$, $\cos x/2$, and $\tan x/2$ from the given information. $\cot x = 4, 180^\circ < x < 270^\circ$ I thought $\sin x/2$ might ...