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 (2)

-2
votes
1answer
44 views

Prove using the given condition… [on hold]

If $\displaystyle\frac{\sin^4{x}}{a} + \frac{\cos^4{x}}{b}= \frac{1}{a+b}$, then prove that $\displaystyle\frac{\sin^6{x}}{a^2} + \frac{\cos^6{x}}{b^2}= \frac{1}{(a+b)^2}$.
0
votes
1answer
36 views

In triangle $ABC$, the Euler line is parallel to $BC$. Prove that $\tan B \tan C = 3$

I am trying to finish the solution to this exercise from Geometry Revisited by Coxeter and Greitzer: In triangle $ABC$, the Euler line is parallel to $BC$. Prove that $\tan B \tan C = 3$. This ...
2
votes
1answer
35 views

Integrating $ \frac{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| |\sin(x)| dx }{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| dx } $

I'm trying to integrate $ \frac{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| |\sin(x)| dx }{\int_{-\pi}^{\pi} \cos^2(x)|\cos(x)| dx } $ I understand that $|\cos(x)|$ and $|\sin(x)|$ when integrated over $- ...
1
vote
2answers
41 views

The Golden Ratio in a Circle and Equilateral Triangle. Geomertic/Trigonometric Proof?

Geogebra gave me 1.61 for the following Golden Ratio construction shown below. Firstoff, has anyone seen anything similar to this construction? Basically begin with an equilateral triangle. ...
2
votes
1answer
30 views

Distance between incentre and orthocentre.

I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C} $$here $r$ is inradius and $R$ is circumradius. I considered $\triangle API$ ($P$ is ...
44
votes
12answers
5k views

Are there theoretical applications of trigonometry?

I am a high school student currently taking pre-calculus. We have just finished a unit on analytic trig. I am curious to know if there are any purely theoretical uses for trigonometry. More ...
1
vote
3answers
60 views

Solve the equation on the interval $\; 0 \le \theta \lt 2\pi $

Hey I have two questions for math exchange! Let me list them first and show you what I have tried. By the way one can not use a calculator on the test review! Solve the equations on the interval ...
0
votes
1answer
20 views

Find the perimeter of the given trapezoid

Find the perimeter of the given trapezoid (The diagram is not drawn to scale) I thought I could use the pythagorean theorem, but I have two unknow sides. What do I do now?? Thank you
-4
votes
0answers
38 views

The Golden Ratio in a Circle, Triangle, and Square: simple geometry/trigonometry construction.

I believe that I have found the golden ratio in the below figure. Geogebra is saying it is very close. Might anyone have a geometric/trigonometric proof? An equilateral triangle ABC is inscribed in a ...
0
votes
3answers
35 views

If $y=9\sec^2x+16\csc^2x$,find the minimum value of $y$

If $y=9\sec^2x+16\csc^2x$,find the minimum value of $y,\hspace{1cm}\forall x\in R$ $y=9\sec^2x+16\csc^2x$ $\frac{dy}{dx}=18\sec^2x\tan x-32\csc^2 x\cot x$ Put $\frac{dy}{dx}=0$ $18\sec^2x\tan ...
0
votes
2answers
33 views

Find the minimum value of $3 \sin 2x + 5$ and the value of x where this occurs in the interval $0\le x \le\pi$

Find the minimum value of $3 \sin 2x + 5$ and the value of x where this occurs in the interval $0\le x \le\pi$ Turning points occur where $\frac{dy}{dx} = 0$ $f'(x) = 3\cos 2x$ $3\cos 2x = 0$ ...
0
votes
2answers
22 views

Solve $2\cos x + 1 = 0$ for x, where $\pi \le x \le \frac{3\pi}{2}$

Solve $2\cos x + 1$ for x, where $\pi \le x \le \frac{3\pi}{2}$ A) $5\frac{\pi}6$ B) $7\frac{\pi}6$ C) $5\frac{\pi}4$ D) $4\frac{\pi}3$ $2\cos x + 1 = 0$ $2\cos^2 x -1 + 1 = ...
0
votes
3answers
56 views

About the sign of $\cos(x)-\sin(x)$ on $[0, \pi]$

Everything's in the title. I have to find the sign of $\cos(x)-\sin(x)$ this expression on $[0, \pi]$, and maybe that's a stupid question, but I just can't figure it out. I just know that it's equal ...
0
votes
2answers
43 views

Proving a $\cos(2nx)$ identity using induction

Prove that $\cos(2nx)=∑_{k=0}^n (-1)^k \dbinom{2n}{2k} \cos^{2(n-k)}(x)\cdot \sin^{2k}(x):=p(n)$ I'd start using induction, with $n=1$ we have: $$cos(2x)=\cos^2(x)-\sin^2(x)$$ True. Now assume ...
-1
votes
1answer
32 views

How does $\sin (\frac{3\pi}{4}) = \frac{1}{\sqrt{2}}$?

I cannot see how $$\sin \left(\frac{3\pi}{4} \right) = \frac{1}{\sqrt{2}}$$ This example has been taken from https://archive.uea.ac.uk/jtm/10/dg10p1.pdf Please can someone show me how this is so ?
1
vote
2answers
26 views

Find a point on a circle given a point and height

Point on a circle Given : A point on a circle (point and angle), radius of the circle , height (Orthogonal to horizon ) I would like to find A point on a circle or , and The angle between the ...
0
votes
2answers
21 views

find the values of a, b and c from sin graph

The diagram shows part of the graph of a function whose equation is $y = a\sin(bx^{\circ}) + c$ (a) Write down the values of a, b and c. (b) Determine the exact value of P ...
1
vote
2answers
29 views

Is there a formula that finds middle between two angles

I am sorry in advance. If I am shooting a laser in front of me. Assume 90 degrees. And then I have a second laser that I shoot from that has a point 130 degrees. The midpoint between the two is not ...
1
vote
1answer
22 views

How to know the quadrant $\sec(-\pi/4)$ lies in?

It is in quadrant four and since the angle is negative it moves clockwise but how do you know it is in quadrant four? For instance, I know $11\pi/6$ is in quadrant four because I divide $11$ by $6$ ...
0
votes
1answer
34 views

Finding the general formula for the solutions of the equation $\cos(2\Theta) = \frac{\sqrt{2}}{2}$

so as you can see on the title, it says finding the General Formula. So First lets take a look on the question: Solve the equation. Give the general formula for all the solutions. $$cos(2\Theta) ...
1
vote
2answers
24 views

Given $\tan a = -7/24$ in $2$nd quadrant and $\cot b = 3/4$ in $3$rd quadrant find $\sin (a + b)$.

Say $\tan a = -7/24$ (second quadrant) and $\cot b = 3/4$ (third quadrant), how would I find $\sin (a + b)$? I figured I could solve for the $\sin/\cos$ of $a$ & $b$, and use the add/sub ...
0
votes
2answers
23 views

How to express two variables in two other variables

If: $A=R\cos x$ and $B=R\sin x$ Then how can I express $R$ and $x$ in terms of $A$ and $B$ in a rigorous way? Meaning that I take the domain and range in account? I tried: $$\cos x=\frac{A}{R}$$ ...
0
votes
3answers
55 views

How to convert arccos to arctan?

Is this true? $$\arccos\frac{A}{\sqrt{A^2+B^2}}=\arctan\frac{B}{A}$$ If so, how can one show it?
-2
votes
0answers
23 views

To find/create midpoint, is it easier to bisect a line segment, or double a line segment? With only compass and ruler / straightedge. [closed]

Suppose one wants to find the midpoint of a line segment. Is it generally easier to simply draw two lines of equal length end to end, or is it easier (does it count as less steps) to draw a line and ...
1
vote
1answer
95 views

Don't know how to approach $x \cos(x) - 2 \cos^2(x) = 2$

I don't how to solve for $x$ because of the mixture of trig and $x$ outside a trig function. Can someone give me a hint how to proceed? $$x \cos(x) - 2 \cos^2(x) = 2$$ where the interval is $[0,2\pi]$ ...
1
vote
1answer
36 views

Simple Golden Ratio Construction with Three Lines, and Interesting Implied Circle?

Consider three equal lines (as illustrated below). A red, green, and orange line of equal length all rest upon the same horizontal line. The red line is stood upon its end in a manner perpendicular ...
5
votes
4answers
96 views

Solving $2\cos\left(2\theta\right) = \sqrt{3}$

I have a question on this test review problem (that will help us on a test), and I have no clue what it's asking. We're learning trigonometry, (Analytic Trigonometry), like about the unit circle, ...
3
votes
4answers
49 views

Golden Ratio Conjecture in three simple Geogebra shapes--circle, triangle, and square.

A circle, equilateral triangle, and square of equal heights are all placed on the same horizontal line as shown below. The circle is tangent to the triangle which is centered upon the left edge of ...
1
vote
3answers
52 views

Is there a trig identity to help solve this equation? $A \cdot \cos\theta = B+ \sin\theta$

I'm trying to solve for $\theta$ in a simple equation: $A \cdot \cos(\theta) = B+ \sin(\theta)$ ($A$ and $B$ are constants) But all the trig identities I've tried just make the equation worse. ...
2
votes
5answers
77 views

Solving $\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$

If we have $$\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$$ we have to find the set of $x$ for which this is true. I tried to solve it by putting $x = \sin a$ or $\cos a, but got no ...
-1
votes
0answers
27 views

PHI Conjecture: Fibonacci Number Line Segments & Simple Golden Ratio Puzzle?

This is a golden ratio conjecture. Does the golden ratio exist as conjectured below? The three line segments in the figure below have lengths of the Fibonacci numbers 2, 3, and 5. blue=2 green=3 ...
1
vote
0answers
17 views

How can I find the distance between two points within a triangle if I have the distance between each point and each vertex of the triangle?

Title says it all. It would be useful to extend the question to finding the distance if any of the points is outside of the triangle, but I'm trying to figure out the basic problem first.
1
vote
2answers
51 views

Solving $\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$

If we have to find the solutions of equation $$\arcsin(\sqrt{1-x^2}) +\arccos(x) = \text{arccot} \left(\frac{\sqrt{1-x^2}}{x}\right) - \arcsin( x)$$ Using a triangle I rewrite it as $$2 \arctan ...
2
votes
1answer
96 views

What's the integral of $\frac{1}{x^2}\csc^2\left(\frac{1}{x}\right)$?

It's known that $\int\csc^2(x)dx = -\cot(x) + C$, but I don't know how to integrate $\int\frac{1}{x^2}\csc^2(\frac{1}{x})dx$. Can you help? Answer to integral ...
1
vote
1answer
44 views

Solve $\cot x \csc x + \cot x = 0$

$$\cot x \csc x + \cot x = 0$$ Give exact answers in radians. I tried $$\cot x(\csc x +1)=0$$ $$\cot x =0, \csc x= -1$$ But I'm not sure if that is correct. Please show me how to do this problem. I ...
3
votes
1answer
64 views

Two Equilateral Triangles and the Golden Ratio: Simple Geomtric Proof

Two equilateral triangles rest upon the same horizontal line, as shown in the linked figure below. The red triangle is tilted so that one corner touches a side of the black triangle at the midpoint ...
0
votes
1answer
76 views

Simple yet challenging integral, can it be solved analytically, and if so, the answer.

I'm trying to find solutions to the 3 following integrals. The first 2 are of the same form, only varying by a constant in the numerator within the cosine, and yes, x is a constant in the first one. ...
0
votes
1answer
30 views

Distance from Chicago to New York

An airplane flies $520$ miles from Chicago to Virginia. Then it turns $45$ degrees to face New York and flies $630$ miles to New York. What is the distance from Chicago to New York? Given the $45$ ...
0
votes
2answers
19 views

Find unit vector that bisects two directed line segments. [closed]

I'm trying to find the 2D unit vector that bisects two directed line segments with sign relative to the orientation of the line segments (left-hand side should be positive). Here is a graphic that ...
2
votes
3answers
62 views

How to show that $f(x) = x \arctan \left(x \sin^{2} \left(\frac{1}{x}\right)\right)$ is strictly increasing for $x \geq 1$?

I am trying to prove that $f(x) = x \arctan \left(x \sin^{2} \left(\frac{1}{x}\right)\right)$ is a strictly increasing function for $x \geq 1$. I try to do this by showing that $f'(x)>0$ for all ...
2
votes
3answers
47 views

Precalculus/Trigonometric Functions of Sine, Cosine, and Tangent with given parameters?

for my precalculus class I was given an assignment for extra credit however it is some material that I have yet to cover or learn as far as sine, cosine, and tangent go. Below is the prompt that I was ...
1
vote
0answers
32 views

The angle giving minimum value

We know minimum value of $\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x)=6$ by AM-GM inequality. But I wanted to know whether manually can find out that angle $x$. Is it possible?
-1
votes
1answer
35 views

Find minimal possible value of the expression $4\cos^2\frac{n\pi}{9}+\sqrt[3]{7-12\cos^2\frac{n\pi}{9}},$ where $n\in\mathbb{Z}.$ [closed]

Find minimal possible value of the expression $4\cos^2\frac{n\pi}{9}+\sqrt[3]{7-12\cos^2\frac{n\pi}{9}},$ where $n\in\mathbb{Z}.$ Is it enough to check for $\{20^\circ,\,40^\circ,\ldots,180^\circ\}$? ...
2
votes
3answers
75 views

Prove that $\tan20^\circ\tan40^\circ\tan60^\circ\tan80^\circ=3$

Prove that $\tan20^\circ\tan40^\circ\tan60^\circ\tan80^\circ=3$ ...
4
votes
1answer
382 views

Sum over fourth power of the sine

I am considering the sum $$ A_m = \sum_{j=0}^m \sin^4\left(\frac{j}{m}\cdot\frac{\pi}{2}\right). $$ I think that for $m>1$ it holds $$ A_m = \frac{3m+4}{8}, $$ but I can't really get to it.
6
votes
1answer
105 views

Does rationality of $\cosh(nx)$ and $\cosh((n+1)x)$ imply rationality of $\cosh(x)$?

Suppose that $x\in\mathbb{R}^+$ and $n\in \mathbb{N}$. If $\cosh(nx)$ and $\cosh((n+1)x)$ are rational, can we show that $\cosh(x)$ is rational too? I guess the following equalities should be useful: ...
1
vote
2answers
26 views

most general antiderivative involving sec x

I'm stumped on how to get the most general antiderivative, $F(x)$, of $f(x)=e^x+3secx(tan x + sec x)$. First, I split the equation on addition, since $\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$ ...
0
votes
1answer
34 views

max and min sine function and all intervals

I have a calculus question: The voltage signal from a standard North American wall socket can be described by the equation V(t) = 170sin(120πt), where t is time, in seconds, and V(t) is the voltage, ...
2
votes
2answers
41 views

Evaluate the following trignometric sum

I am interested in the following sum $$\sum_{\text{even } n=-\infty}^{\infty}\left(-\cos^2x\delta_{n,0}+\cos x\left(\frac{1-\cos x}{\sin x}\right)^{|n|}\right).$$ Wolfram alpha returns answer ...
6
votes
0answers
128 views
+50

Which properties characterize $\sin, \cos$?

I know a few properties of $\sin$ and $\cos$, for example: $\sin^2+\cos^2=1$ $\sin (a+b) = \sin a\cos b+\cos a\sin b$. $\cos (a+b) = \cos a\cos b-\sin a\sin b$. $\sin (x+\delta) = \sin x$ for some ...