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)

2
votes
3answers
41 views

Triangle of maximum perimeter for a given area

What type of triangle has the maximum perimeter for a certain area? Suppose I start with a rectangle of that area (axb=Z). I can stretch one dimension of the rectangle until infinity, reducing the ...
3
votes
4answers
52 views

To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$

I used the following way but got wrong answer $$A.M. \ge G.M.$$ $$ \frac{\sin \theta + \csc \theta}{2} \ge \sqrt{\sin \theta \cdot \csc \theta}$$ Squaring both sides, \begin{equation*} (\sin\theta + ...
0
votes
2answers
29 views

In a triangle ABC , $a\cos(B-C)+b\cos(C-A)+c\cos(A-B)$ is equal to…

In a triangle ABC, prove that $a\cos(B-C)+b\cos(C-A)+c\cos(A-B)$ is equal to $\frac{abc}{R^2}$, where $a$, $b$, and $c$ are sides of the triangle and $R$ is the circumradius. My work:- By ...
12
votes
1answer
205 views

Something strange about $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ and its friends

We have the nice radical identity involving $d = 163$, $$-\sqrt{ 44- \sqrt{ 44 - \sqrt{ 44-x}}}=x,\quad\quad x = 2-2\sum_{n=1}^{27}\cos\left(\frac{2\pi\, t_1(n)}{163}\right)=-6.15824\dots$$ where ...
4
votes
3answers
43 views

Generic rotation to remove Quadratic Cross-product

Show that if $b\neq 0$, then the cross-product term can be eliminated from the quadratic $ax^2 + 2bxy + cy^2$ by rotating the coordinate axes through an angle $\theta$ that satisfies the equation $$ ...
2
votes
2answers
45 views

Equivalence of trigonometric identity

Is writing $$ \cot{2\theta}=\frac{a-c}{2b} $$ equivalent to $$ \cot{\theta}=\frac{a}{b},\tan{\theta}=\frac{c}{b} $$ becuase of the trigonometric identity $$ ...
6
votes
2answers
80 views

Evaluate this Trigonometric Expression

Evaluate $$ \sqrt[3]{\cos \frac{2\pi}{7}} + \sqrt[3]{\cos \frac{4\pi}{7}} + \sqrt[3]{\cos \frac{6\pi}{7}}$$ I found the following $\large{\cos \frac{2\pi}{7}+\cos \frac{4\pi}{7} + \cos ...
4
votes
2answers
39 views

How to solve $|\tan x| \ge 1$?

I need to solve $$|\tan x| \ge 1$$ So I separated it to cases $$-{\sin x\over \cos x}\ge1$$ And $${\sin x\over \cos x}\ge1$$ But now I've got so many cases to check it seems like I'm doing it in the ...
1
vote
1answer
34 views

Calculating XY coordinates on line

I have been working on this problem for a while now and can’t figure out the solution. Hence my post on this forum. I’m trying to figure out the position of a symbol on a line. These lines are located ...
0
votes
1answer
28 views

trigonometry issue: can i continue solving with a negative angle?

i'm on the verge of solving a basic sine equation that presented 2 sides (b,c) and 1 angle (B), i found angle C (which had two possibilities) and started solving for angle A using the two ...
1
vote
0answers
18 views

Combinatorial proof for the formula of $\tan n\theta$

Is there any combinatorial proof of the formula for $\tan n\theta$ where $n\in \mathbb{N}$? Then proofs that I know are by Induction and using de Moivre's Formula but recently one of my friend asked ...
0
votes
1answer
28 views

What is the radius of a circle tangent to two lines with a known angle between them

Given angle, $\alpha$, and distance, $d$, what is the radius, $r$, and angle, $\theta$, in the image below in terms of the known quantities?
2
votes
2answers
31 views

Suppose that two polar curves are given by: $R_1 = \cos(2\theta)$ and $R_2 = \sin(3\theta)$. Find the smallest positive solution exactly.

Suppose that two polar curves are given by: $R_1 = \cos(2\theta)$ and $R_2 = \sin(3\theta)$. Find the smallest positive solution exactly. I know that we are looking for the smallest positive value ...
0
votes
1answer
33 views

Drawing lines from tangents from two circles on both sides.

I need to draw two red lines connecting the tangents from two circles on both sides. I need an algorithm that would get them based on any angle these circles are in relation to another. I need the ...
1
vote
5answers
115 views

Why sin of supplementary angles have equal values?

Given two supplementary angles (for instance, 30 degrees and 150 degrees), why is $\sin(30^\circ) = \sin(150^\circ)$? Where can I find a proof for this? Or the derivative of such proofs?
2
votes
3answers
52 views

Can someone explain this question about trig substitution?

I am not sure what I am doing wrong! Here is the question: Evaluate the definite integral $$\int_0^{7/2}\sqrt{49-x^2} \ dx.$$ What I have gotten to is that the integral (through trig ...
3
votes
2answers
77 views

How to evaluate $\lim\limits_{n\rightarrow \infty} \sin (1/n)$?

I know that the $\lim\limits_{n\rightarrow \infty} \sin \left( \frac{1}{n} \right)= 0$ But I am not sure of the right workings. My attempt: As $n$ tends to infinity, $\frac{1}{n}$ will tend to $0$. ...
1
vote
1answer
48 views

Prove the following identity in complex analysis (trigonometry) [closed]

We know $\cos(z)^2 + \sin(z)^2 = 1, \forall z \in \mathbb C$. Prove that, on the other hand, $|\cos^2(z)| + |\sin^2(z)| > 1, \forall z \in \mathbb C$ with $\text{Im}(z) \not = 0$.
2
votes
3answers
83 views

I am working on proving or disproving $\cos^5(x)-\sin^5(x)=\cos(5x)$

True or false? $$\cos^5(x)-\sin^5(x)=\cos(5x)$$ for all real x. I have no idea how to prove or disprove this. I tried to expand $\cos(5x)$ using double angle formula but I wasn't sure how to go from ...
-1
votes
1answer
53 views

In a Right Angled Triangle.

In a triangle ABC, Let $\angle$C=$\frac{\pi}{2}$. If $r$ is the inradius and $R$ is the circumradius, then what is the value of $2r+R$. Options are a+b b+c c+a a+b+c My approach. Radius of ...
0
votes
0answers
28 views

How can you calculate the module of a gear?

How can you calculate the module of a gear, knowing only the space between the teeth, the number of teeth and the contact angle? On the website "http://woodgears.ca/gear_cutting/template.html" Can ...
6
votes
2answers
47 views

Solving the equation: $3\cos x - \sin 2x = \sqrt{3}(\cos 2x + \sin x)$

Solving the equation: $$3\cos x - \sin 2x = \sqrt{3}(\cos 2x + \sin x)\tag{1}$$ I tried to write $(1)$ becomes $$\sqrt{3}\sin \left(\frac{\pi}{3}-x\right)=\sin \left(\frac{\pi}{3}+2x \right)$$ Now, ...
0
votes
2answers
24 views

Even and odd functions

Given $f(x)= \sqrt{1-\cos x}$. Period $0<x<2 \pi$ Is it a even function or a odd function? Whether the $f(x)$ has to be converted to square root of $2$ multiplied by $\sin(x/2)$.
6
votes
2answers
48 views

If $\left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right)$

If $\displaystyle \left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right) = $ $\bf{My\; Try::}$ Given $\displaystyle ...
0
votes
1answer
25 views

Given an angle and opposite side in a triangle, find constraints on the side adjacent to the angle

In triangle ABC, we are given an angle A = 42 degrees, and its opposite side length, a = 38. i) For what values of adjacent side b such that we have one unique triangle ? ii) For what values b ...
2
votes
1answer
34 views

Find cosine of acute angles in a right triangle.

If sides of a right triangle are in Geometric Progression, then find the cosines of acute angles of the triangle. [Answer] $\frac{\sqrt{5}-1}{2}$,$\sqrt\frac{\sqrt{5}-1}{2}$ My work: Using ...
0
votes
1answer
22 views

Law of sines solving for triangles

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers ...
2
votes
0answers
42 views

How to rewrite this trigonometric formula in terms of scalar and vector products between vectors?

Given two angles $\alpha$ and $\gamma$ such that $$ \cos(\alpha) = v\cdot v' $$ and $$\cos(\gamma) = f\cdot f',$$ what is the simplified form of $\cos(\alpha + \gamma)$ in terms of the vectors ...
0
votes
1answer
18 views

Two questions regarding the angle of reflection

I have two problems regarding the calculation of angles given certain values. In the first problem I need to calculate the angle X given that both angles Y are identical In the second problem I ...
2
votes
5answers
84 views

Evaluate $\lim_{x→0}\left(\frac{1+\tan x}{1+\sin x}\right)^{1/x^2} $

I have the following limit to evaluate: $$ \displaystyle \lim_{x→0}\left(\frac{1+\tan x}{1+\sin x}\right)^{1/x^2} $$ What's the trick here?
1
vote
2answers
31 views

In triangle ABC, find the range of sinAsinC.

In $\triangle$ABC, $\angle$B=$\frac{\pi}{3}$. Find the range of sinAsinC. I used C=120-A to simplify the equation in sinC and then applied $-1<sinC<1$ but answer did not matched.
1
vote
3answers
49 views

How to show $\ln(\tan(x) + \sec(x))=\frac{\ln (1+\sin x )-\ln(1-\sin x )}{2}$ is true or false

$$\ln(\tan(x) + \sec(x))=\frac{\ln (1+\sin x )-\ln(1-\sin x )}{2}$$ Is this true or false? I thought the right side looked like a sum of an even and an odd function but I ended up with $$\ln(\tan(x) ...
0
votes
3answers
47 views

How was this equation for the hypotenuse of a triangle derived?

I've been staring at this for quite a while and simply can't understand how they got the equation for the hypotenuse. Probably has something to do with it being 5am my time! I'm confused because ...
1
vote
1answer
36 views

What is the exact value of $\cos\left(\arccos\dfrac{1}{7}+\arcsin\dfrac{1}{5}\right)$

What is the exact value of $\cos\left(\arccos\dfrac{1}{7}+\arcsin\dfrac{1}{5}\right)$? I wasn't sure if I was doing this correctly or using the fastest method, but I wrote $x=\arccos\dfrac{1}{7}, ...
6
votes
1answer
52 views

Prove that : $f(\sin x)+f(\cos x) \ge 196, \forall x\in\left(0;\frac{\pi}{2}\right)$

Given: $$f(\tan2x)=\tan^{4}x+\frac{1}{\tan^{4}x}, \forall x\in\left(0;\frac{\pi}{4}\right)$$ Prove that :$f(\sin x)+f(\cos x) \ge 196, \forall x\in\left(0;\frac{\pi}{2}\right)$ Could someone help me ...
1
vote
2answers
37 views

Ratio of sines given, find ratio of cosines?

Is it possible to find $\displaystyle{\cos x \over \cos y}$ if $\displaystyle{\sin x\over \sin y}$ is given? If so, how would one approach this problem? Thank you in advance!
0
votes
2answers
16 views

Finding the tension in two ropes.

I have a problem that says to find the tension in two ropes in the following figure. The answers are 1830kg in the right rope and ...
-1
votes
1answer
56 views

Evaluating $ \int \frac{1}{\sin x} dx $

Verify the identity $$\sin x = \frac {2 \tan\frac{x}{2}}{1 + \tan^2\frac{x}{2}}$$ Use this identity and the substitution $t = \tan\frac{x}{2}$ to evaluate the integral of $$ \int \frac{1}{\sin x} dx ...
1
vote
3answers
47 views

Finding exact value of trigonometric functions

I was wondering, how do I get the exact fraction (the value) of this trigonometric function: $$\cos\left(\sin^{-1}(12/13)+\sin^{-1}(4/5)\right)$$ Usually, I would evaluate the inverse sin in degree ...
1
vote
3answers
31 views

Computing the trigonometric sum $ \sum_{j=1}^{n} \cos(j) $

I have a task to compute such a sum: $$ \sum_{j=1}^{n} \cos(j) $$ Of course I know that the answer is $$ \frac{1}{2} (\cos(n)+\cot(\frac{1}{2}) \sin(n)-1) = \frac{\cos(n)}{2}+\frac{1}{2} ...
1
vote
2answers
33 views

Find the relationship between angles in a right-angled triangle given relative lengths of hypothenous and perpendicular

Take a right angled triangle. The angle $\angle A$ is the right angle. What is the difference between $\angle B$ and $\angle C$ if the hypothenous is four times the lenght of the perpendicular ...
0
votes
3answers
41 views

Find Circumdiameter of Δ HBC.

If H is the orthocenter of an acute angled ΔABC whose circumcircle is $x^2+y^2=16$, then circumdiameter of the triangle HBC is
2
votes
1answer
46 views

In $\triangle ABC$, find the value of $\tan A\tan C$.

In $\triangle ABC$, line joining the circumcenter(O) and orthocenter(H) is parallel to side $AC$, then show that the value of $\tan A\tan C$ is 3. Let $\perp$ from circumcenter cuts $AC$ at D and ...
0
votes
2answers
27 views

Find the value of x,y in triangle

In the below figure the hypothenuse is 13.4cm and perpendicular is 7.2cm just have to find the x and y both of the angles in the diagram
0
votes
1answer
59 views

In ΔABC prove that sides are in $AP$. [closed]

In ΔABC, if $$\tan\left(\frac{A}{2}\right)=\frac{5}{6}$$ and $$\tan\left(\frac{B}{2}\right)=\frac{20}{37}$$ prove that sides lengths $a$, $b$ and $c$, when sorted, form an arithmetic progression My ...
3
votes
3answers
282 views

How to solve $\sin\theta +\sin3\theta =0$

Solve the equation by first using a Sum-to-Product Formula. $$\sin\theta +\sin3\theta =0$$ Steps I took: $$\begin{align}0&=2\sin\frac { \theta +3\theta }{ 2 } \cos\frac { \theta ...
2
votes
3answers
45 views

Using the half/double angle formulas to solve an equation

I am completely stumped by this problem: $$\cos\theta - \sin\theta =\sqrt{2} \sin\frac{\theta}{2} $$ I know that I should start by isolating $\sin\dfrac{\theta}{2}$ and end up with $$\frac ...
2
votes
1answer
34 views

Evaluate $ \int \frac{\tan(x)}{2+\sin(x)}dx $

How do you evalute this integral? $$ \int \frac{\tan(x)}{2+\sin(x)}dx $$
1
vote
4answers
34 views

Finding all the solutions of $\cos2\theta -\cos4\theta =0$

$$\cos2\theta -\cos4\theta =0 $$ in the interval $[0,2\pi)$ So I know that I can use the Double or Half Angle formulas to simplify this and help me find the solutions. So I used the Double Angle ...
3
votes
1answer
33 views

Solving $\cos\theta \cos3\theta -\sin\theta \sin3\theta =0$ , $\theta\in[0,2\pi)$

$$\cos\theta \cos3\theta -\sin\theta \sin3\theta =0$$ I know that this uses the cosine addition formula and could be rewritten as $\cos(\theta +3\theta )=0$ and then as $\cos(4\theta )=0$ However , ...