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)

1
vote
1answer
32 views

The coincidence orthocenters of the two triangles

Let $CH -$ height in acute-angled triangle $ABC$. Some points $K$ and $N$ are on side $AB$. Let $O_1 -$ orthocenter of triangle $ACN$ and $O_2 -$ orthocenter of triangle $BCK$. Prove $$O_1=O_2=O \...
2
votes
2answers
94 views

Find all $x$ such that lim $\sin (nx)$ exists

I know how to prove that the set of $x$ such that $\lim_{n\to\infty} \sin(nx)$ exists has measure zero. And clearly the limit exists when $x = \pi k$ for $k\in\mathbb{Z}$. I'm guessing those are the ...
0
votes
2answers
71 views

Prove that $\frac{\sin 40^\circ}{\sin 80^\circ} +\frac{\sin 80^\circ}{\sin 20^\circ} -\frac{\sin 20^\circ}{\sin 40^\circ} =3$

Prove that $$\frac{\sin 40^\circ}{\sin 80^\circ} +\frac{\sin 80^\circ}{\sin 20^\circ} -\frac{\sin 20^\circ}{\sin 40^\circ} =3.$$ I tried taking the least common multiple of the denominator and then ...
2
votes
6answers
82 views

Checking nature of angles of a triangle given the equations of the three lines that form a triangle

Suppose we have three lines $\ell_i=a_ix+b_iy=c_i$, $i=1,2,3$ and we are given that they form a triangle. I need to find which angles are acute and which are obtuse without plotting the lines ...
0
votes
0answers
52 views

How to find the system transfer function corresponding to a two dimensional matrix of optical transfer functions?

I would like to find the system transfer function corresponding to a two dimensional matrix of optical transfer function where: Each of the 3 times 5 = 15 interferometers produce 15 sets of ...
2
votes
1answer
37 views

How to determine the optimal length away from the pivot point and angle to the radius to use the least amount of force to rotate the lever?

I'm writing an essay for school in physics and my topic is torque. My topic deals with the elbow joint and the tendon that attaches the radial bone to the biceps, the force that rotates the forearm ...
0
votes
2answers
27 views

Trigonometric question without given diagram

A helicopter 750m above a point A on the ground was flying horizontally towards the east . When it was at point P , the angle of elevation from A was observed to be 60 degrees . If , after 5 seconds , ...
0
votes
2answers
53 views

Proving the following identity

Prove that $${ \left( \frac { \cos (\alpha +\beta ) }{ \cos (\alpha -\beta ) } -\frac { \cos (\alpha +\gamma ) }{ \cos (\alpha -\gamma ) } \right) }^{ 2 }+{ \left( \frac { \sin (\alpha +\beta )...
2
votes
2answers
69 views

What is the period of $f(x)=\sin x\cos x$?

Problem We need to find the period of the following: $f(x)=(\sin(x))(\cos(x))$ using basic trigonometric identities which is as follows: My steps disclaimer! I know the steps but I will pin point ...
1
vote
1answer
20 views

The implicit differentiation of trignometry equations

Use the process of implicit differentiation to find $\dfrac{dx}{dy}$ given that $\sin(x)+ \sin(3y) = 1 $ Can anyone give me hints or steps on what to do?
-2
votes
3answers
47 views

Trignometry question for calculus [closed]

I just have a quick question. It's simple but I am having trouble with it. Solve the equation $\:\:12 \cos^2\theta - 6= \sin\theta\:\:$ for $\theta$ in $(-2\pi,2\pi)$. I am unsure what to do ...
2
votes
1answer
29 views

Find perpendicular height of triangle

Consider a triangle ABC. It is given that angle ABC is 38 degrees, angle ACB is 62 degrees, the length of BC is 38 cm. Find the perpendicular height of the triangle (from A to the base BC). I can ...
2
votes
2answers
38 views

Trigonometric Ratios for angles greater than 90 degrees and the Unit Circle

I am confused about the Unit Circle explanation for the trigonometric ratios for angles greater than 90 degrees. It seems that for the first (top right) quadrant, $\sin(\theta)$ is equivalent to the ...
1
vote
1answer
49 views

Substituting cot(x) with an identity in equation

Assume: $$\cos(x) / \cot(x) = 0\tag A$$ I rewrite it as $$\cos(x)/ \left(\cos(x)/\sin(x)\right) = 0\tag B$$ and get $$\sin(x) = 0\tag C$$ This implies $x = 180^\circ\times k$ But, under (B), if $\sin(...
1
vote
1answer
83 views

Integrating $\sqrt{1-x^2}$ by substitution: Why is this wrong?

When solving the following integral by substitution, I can get two different solutions (one is wrong) depending on the substitution. I'm not sure why the second option produces the incorrect solution. ...
1
vote
1answer
85 views

How were these values of sin x found?

I'm working on Trigonometry problems that have to do with inverse functions. So I have an example problem that goes like this: $$\ 10\sin^2x = \sin x$$ and apparently the solution set is: $\sin x =...
1
vote
6answers
123 views

General solution for $\frac{\mathrm{d}^2 y}{\mathrm{d} x^2} = y$?

Start with $$\frac{\mathrm{d}^2 y}{\mathrm{d} x^2} = y$$ then $$\frac{1}{\mathrm{d} x} \, \mathrm{d} \left(\frac{\mathrm{d} y }{\mathrm{d} x}\right) = y$$ $$\frac{\mathrm{d} y}{\mathrm{d} x} \, \...
4
votes
1answer
61 views

Concept of Trigonometric identities [duplicate]

The value of the expression $$\dfrac{\sin x}{ \cos 3x} + \dfrac{\sin 3x}{ \cos 9x} + \dfrac{\sin 9x}{ \cos 27x}$$ in terms of $\tan x$ is My Approach If I take L.C.M of this as $\cos 3 \cos 9x \...
1
vote
1answer
26 views

Solution of system of discrete trigonometric equations

Given an integer $N$, I am looking to find a real number $t$ to solve the following set of equations: $$\pi n\equiv t\cos\frac{\pi n}{N}\mod 2\pi,~~~~~~~n=0,1,2,3,\ldots, 2N-1$$ Given an integer $N$, ...
0
votes
2answers
74 views

Find the value of $\dfrac{\sin(A)^8}{a^3}+\dfrac{\cos(A)^8}{a^3}$ [duplicate]

The question is: We have been given $$\frac{\sin(A)^4}a+\frac{\cos(A)^4}b=\frac1{a+b};$$ find $$\frac{\sin(A)^8}{a^3}+\frac{\cos(A)^8}{a^3}.$$ My reaction: "Hmmm, hi there, tough-looking ...
0
votes
1answer
36 views

How does one solve trig functions by hand?

First of all, I am not very intelligent, so the question I am about to ask would sound inane--but I am just curious, that's all. We were always used to being attached to our calculators when it comes ...
3
votes
4answers
73 views

Find $\sin \theta $ in the equation $8\sin\theta = 4 + \cos\theta$

Find $\sin\theta$ in the following trigonometric equation $8\sin\theta = 4 + \cos\theta$ My try -> $8\sin\theta = 4 + \cos\theta$ [Squaring Both the Sides] => $64\sin^{2}\theta = 16 + 8\cos\theta ...
0
votes
1answer
49 views

What identity was used in this Trigonometry problem?

I'm trying to prove this trigonometry identity, and I can solve it to the up until the last step, where I can't figure out which identity is being used to solve it. This is the identity. $$\tan\...
2
votes
4answers
80 views

Trigonometric identities: $ \frac{1+\cos(a)}{1-\cos(a)} + \frac{1-\cos(a)}{1+\cos(a)} = 2+4\cot^2(a)$

I don't really know how to begin, so if I'm missing some information please let me know what it is and I'll fill you guys in :). This is the question I can't solve: $$ \frac{1+\cos(a)}{1-\cos(a)} + \...
9
votes
3answers
150 views

If $x^2+\frac{1}{2x}=\cos \theta$, evaluate $x^6+\frac{1}{2x^3}$.

If $x^2+\frac{1}{2x}=\cos \theta$, then find the value of $x^6+\frac{1}{2x^3}$. If we cube both sides, then we get $x^6+\frac{1}{8x^3}+\frac{3x}{2} \cdot \cos \theta=\cos ^3 \theta$ but how can we ...
-1
votes
2answers
98 views

Trigonometry. If $A+B+C =180$ then find maximum value of $\sin^2(A) +\sin^2(B)+\sin^2(C)$ [closed]

If $A+B+C =180$ then find maximum value of $\sin^2(A) +\sin^2(B)+\sin^2(C)$
3
votes
3answers
104 views

Solving the trig inequality $|\sin{x} + \cos{x}| > 1$

$|\sin{x} + \cos{x} |> 1$ How to solve this kind of question? Is there any websites to learn trigonometry inequalities? My teacher only taught us the simple question but not the complicated one. ...
0
votes
1answer
30 views

Trigonometric Conditional Identities

Prove that if $x+y = z$ , then $\cos^2x + \cos^2y + \cos^2z - 2\cos x.\cos y.\cos z = \cos(x+y - z)$. My work: I tried to solve by converting the cos squared term into double angle form and ...
1
vote
2answers
52 views

Trigonometric proof involving several identities

Show that $$\frac{1+\sin A}{\cos A}+\frac{\cos B}{1-\sin B}=\frac{2\sin A-2\sin B}{\sin(A-B)+\cos A-\cos B}$$ I brought everything to the common denominator on the right hand side. What should I ...
2
votes
1answer
36 views

Dimensions of bounding box for arbitrary circle sector

I need to determine the dimensions of bounding box for arbitrary circle sector as shown in the diagram below. Given: φ = Start angle in the range of 0 ~ 2π θ = Sweep angle in the range of 0 ~ 2π r =...
1
vote
3answers
46 views

Mistake in integration of $\sec(2x)$

I wanted to find the integral of $\frac{\cos(x)-\sin(x)}{\cos(x)+\sin(x)}= \sec(2x)+\tan(2x)$ from $0-\frac{3\pi}{2}$. I calculated the second one as $0$ so my problem is calculating the first one. I ...
5
votes
7answers
117 views

How can a $\sin x$ come out of the equation $\frac{d^2}{dx^2}f(x)=-f(x)$ as the solution, while there's no sign of a trigonometric function in it?

This is a differential equation: $$\frac{d^2}{dx^2}f(x)=-f(x)$$ Turns out that the answer is $\sin x$. But HOW?! It is impossible to achieve a trigonometric function by integrating that equation. ...
3
votes
2answers
56 views

Triangle with $3$ unknowns

I have a situation where I am trying to calculate a leading shot for a character in a 2D top down game. The enemy character moves with a certain speed $s$, which is applied to its normalized ...
2
votes
0answers
41 views

Minimum of $f(x) = \frac{1}{a}\cos^4 \frac{\pi x}{2} + a \sin^4 \frac{\pi x}{2} +\sin\pi x (b\sin\pi x-c)$ for $x\in [0,1]$?

In my quantum-physics research, I am faced with the following single-variable trigonometric optimization problem that I would wish to solve analytically. Problem: Let $a,b,c, x \in \mathbb{R}$, and ...
3
votes
1answer
132 views

show that $\prod_{k=1}^{n-1}\left(2\cot{\frac{\pi}{n}}-\cot{\frac{k\pi}{n}}+i\right)$ is purely imaginary number

Show that $$\prod_{k=1}^{n-1}\left(2\cot{\dfrac{\pi}{n}}-\cot{\dfrac{k\pi}{n}}+i\right)$$ is purely imaginary number where $i^2=-1$ where $n=2$ it is clear $$2\cot{\dfrac{\pi}{n}}-\cot{\dfrac{k\...
0
votes
2answers
15 views

How to find a diagonal given perimeter of the rhombus and one of its angle

Supposed that the perimeter of a rhombus has been given as 4 and one of the angle is 120 degrees. It asks to find the length of one of the diagonal.
0
votes
3answers
33 views

Finding the rectangle surrounding a capsule defined by two points while minimizing trig, and square root

For the visuals of a simulation I need to solve the following problem. I know points one and two (shown in yellow) and the length of the green line. I need to find the four purple points surrounding ...
3
votes
5answers
158 views

Understanding this proof that $\lim\limits_{h\to 0}\frac{\cos(h)-1}{h}=0$

I need help understanding how this limit is proved? : Show that $$\lim_{h\to 0} \frac{\cos (h)-1}{h}=0$$ Proof: Using the half angle formula, $\cos h = 1-2 \sin^2(h/2)$ $$\lim_{h\to 0} \frac{\cos (...
1
vote
1answer
67 views

Simplifying $\frac{4\sqrt{7}}{3}\cos{\left(\frac{1}{3}\arccos{\frac{1}{\sqrt{28}}}\right)}+\frac{1}{3}$

I was finding the roots of the polynomial $y=x^3-x^2-9x+1$. And I got one of the roots of the polynomial to be $$\dfrac{4\sqrt{7}}{3}\cos{\left(\dfrac{1}{3}\arccos{\dfrac{1}{\sqrt{28}}}\right)}+\dfrac{...
0
votes
1answer
84 views

Find $\theta$ and $\phi$ that maximize $\mid -2ia\sin\theta - 2ib\sin\phi + 2c(1-\cos\theta) +2d(1-\cos\phi)\mid$

How can you find for what values of the $\theta$ and $\phi$ angles the following modulus will assume its greatest possible value? $$\mid -2ia\sin(\theta) - 2ib\sin(\phi) + 2c(1-\cos(\theta)) +2d(1-\...
1
vote
1answer
34 views

Find the interval of $k$ for given conditions

In a triangle $PQR$, $\tan(P)+\tan(Q)+\tan(R)=k$. Find the interval in which $k$ should lie so that: $(A)$ there exists only one isosceles triangle $PQR$. $(B)$ there exists exactly two isosceles ...
0
votes
1answer
30 views

Give the smallest positive value of $x$ for which $\tan x$ is undefined. [duplicate]

Give the smallest positive value of $x$ for which $\tan x$ is undefined. An answer and explanation on how to solve would be great.
2
votes
3answers
62 views

How to compare $\left(\sin \left(x\right)\right)^{\cos \left(x\right)}$ and $ \left(\cos \left(x\right)\right)^{\sin \left(x\right)}$

I am new here ,can anybody help to solve this problem: How to compare $\left(\sin \left(x\right)\right)^{\cos \left(x\right)}$ and $ \left(\cos \left(x\right)\right)^{\sin \left(x\right)}$ in the ...
0
votes
1answer
49 views

How can I show that $\left\lvert\sin z\right\rvert^2= \left\lvert\sin x\right\rvert^2 + \left\lvert\sinh y\right\rvert^2$ for $z= x+iy$

I want to show that $\left\lvert\sin z\right\rvert^2= \left\lvert\sin x\right\rvert^2 + \left\lvert\sinh y\right\rvert^2$ for $z= x+iy$ We have that \begin{align} \left\lvert\sin z\right\rvert^2 &...
2
votes
3answers
93 views

Exact value of $\cos^2(\frac{\pi}{8})+\sin^2(\frac{15\pi}{8})?$

I tried separating it into $\cos^2(\frac{\pi}{8})+\sin^2(\frac{\pi}{8}+\frac{14\pi}{8})?$ and using the angle sum identity but it didn't help.
0
votes
2answers
30 views

Proof verification: $\sin(Arccos(x))$ is always positive, regardless of $x$.

$\sin(Arccos(x)) \implies \sin(Arcsin(\frac{\pi}{2}-x))$ and since $-1 \leq x \leq 1$ then we can have as the maximal and minimal values in the $Arcsin,$ $\frac{\pi}{2}-(-1)\approx2.57$ and $\frac{\pi}...
0
votes
2answers
41 views

Is $\text{arccosec}(x) = \arcsin\left(\frac{1}{x}\right)$ for all $x \in ℝ?$

Is $\text{arccosec}(x) = \arcsin\left(\frac{1}{x}\right)$ for all $x \in ℝ?$ I'm still really new to trigonometric inverses, so if the above was cleared up I'd be grateful. Thanks.
-1
votes
2answers
30 views

Trigonometric calculation help

How do I go about solving: 1.) cos 18$^\circ$ $\cdot$ (tg 36$^\circ$ + ctg 36$^\circ$) 2.) cos 10$^\circ$ + $cos^2 20$$^\circ$ + 4 cos 15$^\circ$ cos 75$^\circ$ + $cos^2 70$$^\circ$ + cos 170$^\...
1
vote
4answers
93 views

Solving $\sin \frac{\theta}{2} + \cos \frac{\theta}{2} = \sqrt{2}$

Does anyone have some tips for me how to go about the problem in the image? $$\sin \frac{\theta}{2} + \cos \frac{\theta}{2} = \sqrt{2}$$ I know it's supposed to be simple, but I can't figure out ...
0
votes
1answer
24 views

A trigonometric expression with the angles being in Arithmetic progression

Prove that $$\cot\theta\cot2\theta +\cot2\theta\cot3\theta +2 = \cot\theta(\cot\theta -\cot3\theta)$$ Well I have already proved it by expresing in terms of sin and cos and taking it from there I ...