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

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-4
votes
1answer
42 views

Prove this trigonometric inequality [closed]

We know this : $0<\theta <(\pi/2)$ How to prove this inequality : $0<\sin^6 \theta + \cos^6 \theta<1$ I used basic inequalities but I can't get answer. Please help!
0
votes
1answer
20 views

Trigonometric substitution [illustration / right triangle derivation]

Real quick: If I have the function $$\int { \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } dx$$ I can easily substitute by setting $x$ equal to $a\sin \theta$. But why actually is that? If I draw a right ...
3
votes
2answers
741 views

What is $\int x\tan(x)dx$?

I have a problem when trying to solve this question Question. What is the answer of the indefinite integral $$\int x\tan x \; dx?$$ Maple gives a complicated answer based on the series. Is there any ...
1
vote
2answers
827 views

Can you help me reverse the Minimum Curvature Method?

The minimum curvature method is used in oil drilling to calculate positional data from directional data. A survey is a reading at a certain depth down the borehole that contains measured depth, ...
2
votes
1answer
1k 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
4answers
36 views

Solve for $\theta$: $2 \sin \theta = 2 - \cos \theta$

Actually , I'm new to trigonometry .. So i want help in this question $$2 \sin \theta = 2 - \cos \theta $$ My attempt -> $$\begin{align} 2 \sin \theta &= 2 - \cos \theta \\ 2 \sin \theta &...
7
votes
1answer
105 views

What's the explanation for these (infinitely many?) Ramanujan-type identities?

Define the function, $$F(\beta) = \sqrt[3]{\beta+x_1}+\sqrt[3]{\beta+x_2}+\sqrt[3]{\beta+x_3}\tag1$$ where, $$x_1 =2\cos\Big(\frac{2\pi }{7}\Big),\;x_2 =2\cos\Big(\frac{4\pi }{7}\Big),\; x_3 = 2\...
2
votes
2answers
31 views

Prove: $\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$, for circumradius R, inradius $r$, and exradii $r_x$

In $\triangle ABC$, prove: $$\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$$ for circumradius $R$, inradius $r$, and exradii $r_a$, $r_b$, $r_c$ in the standard ...
1
vote
4answers
44 views

$\cos2\theta +\cos\theta +k = 0 $ - set of all values of $k$ for which there is a solution

The set of all values of $k$ (real), such that the equation $\cos2\theta +\cos\theta +k = 0 $ admits a solution for $\theta$ is? MY ATTEMPT: I substituted $\cos2\theta$ with $2\cos^2\theta - 1 $. On ...
1
vote
2answers
39 views

Evaluate using complex numbers: $\prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right)$, where $m=2n+1$

Evaluate using complex numbers: $$\prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right)$$ where $m=2n+1$. $\bf{My\; Try::}$ Let $\displaystyle P = \prod^{n}_{k=1}\cos\left(\frac{k\pi}{m}\right).$ Now let $\...
0
votes
2answers
8 views

Given point before and after rotation at given axis calculate the angle of rotation

I have a point at 2d space in only positive x and y axis, point P(x1, y1) is rotated along axis point C(x3, y3) to reach at point P2(x2, y2). Now I just need to calculate the angle of rotation. If ...
0
votes
2answers
23 views

Difference between $\cos(x)$= positive and $\cos(x)$= negative

I'm confusing myself here so need some clarification. If I was to work out the solutions to $\cos(x) = 1/\sqrt2$ , I know the solutions would be $\pi/4, 2\pi - \pi/4, 2\pi + \pi/4...$ etc. What is ...
-2
votes
0answers
27 views

Area of sector.

For a circle with radius $7$ and a chord $AB=10cm$ the area of the sector $AOB$ I keep obtaining is $38.955cm^2$ while the answer in my book is $14.5cm^2$ I don't know where I keep going wrong.
-2
votes
2answers
34 views

$3A$ and $4A$ are supplementary angles…

If $3A$ and $4A$ are supplementary angles,then find the exact value of : $4\cos A - \sec 2A$ I tried breaking the angles into $ A + 2A + 4A$ and then using the formulas for angles of a triangle. ...
-1
votes
2answers
39 views

If $\frac{\cos(\alpha+\gamma)}{\cos(\alpha-\gamma)} = \cos(2\beta)$, then prove that $\tan(\alpha)$, $\tan(\beta)$ and $\tan(\gamma)$ are in G.P.

If $\frac{\cos(\alpha+\gamma)}{\cos(\alpha-\gamma)} = \cos(2\beta)$, then prove that $\tan(\alpha)$, $\tan(\beta)$ and $\tan(\gamma)$ are in G.P.
4
votes
1answer
114 views

If $\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\gamma-\alpha)+1=0$,show that $\alpha-\beta$ or $\beta-\gamma$ or $\gamma-\alpha$ is multiple of $\pi$.

This question is from SL Loney. If $\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\gamma-\alpha)+1=0$, then show that $\alpha-\beta$ or $\beta-\gamma$ or $\gamma-\alpha$ is a multiple of $\pi$. My try: ...
2
votes
1answer
702 views

Real life math to explore/solve [on hold]

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
1
vote
0answers
15 views

Graphs of the hours of daylight in certain latitudes

Problem Attempt We are given that Graph A is North. Since South is "dark" when North "light" I picked the graph that is mirrored horizontally: Graph D. For East and West I can only guess. ...
1
vote
4answers
216 views

why $\lim\limits_{x\to-\infty}(\sin x+2)\ln(-x)=\infty$?

Why does $\lim\limits_{x\to-\infty}(\sin x+2)\ln(-x)$ equal $\infty$? Breaking up the limit: $\lim\limits_{x\to-\infty}(\sin x+2)$ DNE because it oscillates between 1 and 3 $\lim\limits_{x\to-\...
3
votes
0answers
87 views

Sine identity involving (3/p) for prime p greater than 3.

I am working through Ireland and Rosen's "Classical Introduction to Modern Number Theory" and am very stuck on this problem (#34 in Chp 5, 2nd edition): Note that $(a/b)$ is the Legendre symbol (or ...
10
votes
4answers
4k views

Solving trigonometric equations of the form $a\sin x + b\cos x = c$

Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos ...
2
votes
1answer
32 views

Is there a proof that the sum of the trihedral angles of a tetrahedron is minimal when the latter is regular?

Since the sum of the 6 dihedral angles is always 1 sphere more than the sum of the 4 trihedral angles, both sums are maximized or minimized at the same time. I showed that for all four extremal ...
-3
votes
0answers
34 views

Clock angles question. [on hold]

At 4:00 oclock, what will be the angle between the two pointers ? Adding a picture :
0
votes
2answers
45 views

How can I solve this trigonometry question? [on hold]

$ABC$ is a triangle $m( A \widehat BC) = m( A \widehat CB) + 90^\circ$ $3 \lvert AC \rvert = 7 \lvert AB \rvert$ area of the $ABC$ triangle is $4{,}2$ cm$^2$ $\lvert BC \rvert =$ ?
4
votes
2answers
85 views

'Strange' trigonometric roots of $x^5-4x^4+2x^3+5x^2-2x-1$ - could someone explain?

This quintic equation has $5$ real roots: $$x^5-4x^4+2x^3+5x^2-2x-1=0 \tag{1}$$ The roots are, from left to right: $$x_1=\frac{\cos \frac{19}{22} \pi}{\cos \frac{1}{22} \pi}$$ $$x_2=\frac{\cos \...
2
votes
1answer
516 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
17
votes
5answers
1k views

Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt[3]{\cos(2\pi/7)}+\sqrt[3]{\cos(4\pi/7)}+\sqrt[3]{\cos(8\pi/7)}$

Ramanujan found the following trigonometric identity \begin{equation} \sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}= \sqrt[3]...
0
votes
4answers
48 views

How do I remember trigonometric angles?

I am really stuck here, I need help remembering the trigonometric angles, Can someone point me to a good place to learn them? I often confuse when trying to implement them in my problems
5
votes
3answers
134 views

Calculate $\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$

How can we evaluate the following integral: $$\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$$ I know basically how to calculate by using the substitution $x=\tan{\theta}...
0
votes
1answer
12 views

Sin wave that has half wave length of $ a - b$

How can I make a sin wave that has double the wavelength of $a - b$, such that two consecutive zero points on the line are through a and b. And that the peak of the wave in between $a$ and $b$ is at $$...
-2
votes
0answers
46 views

how to solve $\sin ( 2 \theta)=\tan (2\theta + 2 \beta)$

if we have the equation : $$ \sin ( 2 \theta)=\tan (2\theta + 2 \beta )$$ then, what's the direct relation between $ \theta$ and $ \beta $? What do I do to solve for $ \theta$ as a function of $ \...
-1
votes
1answer
43 views

Calculating height of a viewpoint given 3 angles of depression [on hold]

Can not solve this problem I can help English Translation: From the top of a hill, a person finds the angles of depression of three consecutive kilometer markers on a straight road to be $\alpha,...
2
votes
1answer
424 views

how to solve $a\sin x+b\cos x$

Let's solve: $\sqrt{3}\sin x - \cos x=2$ The left hand side may be expressed as $R\sin(x+ \phi)$ We know that $R=\sqrt{3+1}=2$ We also know that $\tan \phi= \frac{-1}{\sqrt{3}}$ The solution to $\...
1
vote
3answers
7k views

“Show” that the direction cosines of a vector satisfies…

"Show" that the direction cosines of a vector satisfies $$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$$ I am stumped on these things: "SHOW" that the direction cosines corresponds to a given ...
2
votes
2answers
57 views

Need help solving a trigonometric equation

I am preparing for finals and there is one exercise in my book that i don`t know how to solve. $$\frac{\sin a}{\sin \frac{a}{b}}=b$$ I just need to solve this for b. I tried wolfram alpha but it does ...
4
votes
4answers
42 views

Converting $\cos\phi$ into $\frac{1−t^2}{1+t^2}$, given that $t = \tan\frac{\phi}{2}$

I have to figure out the working to convert $\cos\phi$ into $\dfrac{1−t^2}{1+t^2}$, given that $t = \tan\dfrac{\phi}{2}$. It would be amazing if someone could help I've been trying to do it for ...
1
vote
3answers
1k views

Intuition around why domain of x of arcsine and arccosine is [-1;1] for “real result” & domain for arctangent is all real numbers

Context I'm working my way through basic trig (this question has a focus on inverse trig functions, specifically arcsine, arccosine and arctangent ), using Khan Academy, wikipedia and some of "trig ...
1
vote
1answer
35 views

Height of lighthouse based on angle difference

I have a question in my maths book: A lookout in a lighthouse tower can see two ships approaching the coast. Their angles of depression are 25° and 30°. If the ships are 100 m apart, show that the ...
-3
votes
0answers
22 views

Vectors applied on the arm [on hold]

I just received a maths question saying the body and its joints are subject to significant forces under load. Your challenge is to redesign a part of the body, and using vectors, explain how it could ...
2
votes
0answers
32 views

Number of solutions of some trigonometric equations

Let $N > 1$ and let $S$ be a subset of the integers in the (real) interval $[1, N]$. Can we prove that there are only finitely many solutions $x \in [1,N] \setminus \mathbb{Z}$ to the equation $$ \...
3
votes
1answer
49 views

Does this inequality involving inverse tangent (arctan) hold?

I am wondering if the following statement is true for $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $x,y\in\mathbb{R}$: $$\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)\leq\...
2
votes
3answers
85 views

Find the values of $x$ such that $2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ is independent of $x$.

Find the values of $x$ such that $$2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$$ is independent of $x$. Checking for $x\in [-1,1]$ In the taken domain $\sin^{-1}\left(\frac{2x}{1+x^2}\...
2
votes
2answers
61 views

Simplify $\arccos\left(2\cos x\right)$.

Let $x\in[\pi/3,2\pi/3]$. We know that $\arccos (\cos x)=x$ but what we can say about $\arccos\left(2\cos x\right)$? Are there, for example, any "half-angle formula" also for inverse trigonometric ...
0
votes
2answers
30 views

Find all values of parameter a, when sum of solutions of following equation is 100

Find all values of parameter $a$, when sum of solutions of following equation is $100$. $$ \sin(\sqrt{ax-x^2})=0 $$ I tried to get rid of that $sin$ and there was quadratic equation with two ...
0
votes
3answers
50 views

If $x^2 - 2x\cos\alpha + 1 = 0$ and $y^2 - 2y\cos\beta + 1 = 0$, then $2\cos(\alpha + \beta)$ is equal to?

If $x^2 - 2x\cos\alpha + 1 = 0$ and $y^2 - 2y\cos\beta + 1 = 0$, then $2\cos(\alpha + \beta)$ is equal to? MY ATTEMPT: Using the fact that $\cos\alpha$ and $\cos\beta$ must be real, I know that $x$...
0
votes
1answer
24 views

If $\cos\alpha = \frac{2\cos\beta - 1}{2-\cos\beta}$ , $(0<\alpha , \beta< \pi)$, then $\tan\frac{\alpha}{2}\cot\frac{\beta}{2}$ is equal to?

If $\cos\alpha = \frac{2\cos\beta - 1}{2-\cos\beta}$ , $(0<\alpha , \beta< \pi)$, then $\tan\frac{\alpha}{2}\cot\frac{\beta}{2}$ is equal to? MY ATTEMPT: I tried simplifying the equation to ...
13
votes
6answers
303 views

Is $ \sin: \mathbb{N} \to \mathbb{R}$ injective?

I was trying to show that $\sin(x)$ is non-zero for integers $x$ other than zero and I thought that this result might emerge as a corollary if I managed to show that the result in question is true. ...
2
votes
2answers
39 views

$\cos28 + \sin28= k^3\cos17=$?

If $\cos28^\circ +\sin28^\circ = k^3$ then $\cos17^\circ = $?. Find in terms of $k$. MY ATTEMPT: I tried finding $\cos28^\circ - \sin28^\circ$ in terms of $k$. Then I found out $\cos28^\circ$ with ...
29
votes
5answers
5k views

How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$

How can we prove the following trigonometric identity? $$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
1
vote
0answers
40 views

Evaluation of trigonometric function without complex numbers

We are asked to prove $$\tan{\frac{3\pi}{11}} +4\sin{\frac{2\pi}{11}}=\sqrt{11}$$ So far the solution that I have come across all use complex numbers to get to the result I am searching for a ...