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

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15
votes
3answers
1k views

Find the value of a function whose derivative is zero

The initial function is $$h(x)=\arcsin x + \arccos x$$ The derivative of this function is $0$ since $$h'(x)=\frac{1}{\sqrt{1-x^2}}-\frac{1}{\sqrt{1-x^2}}\equiv0$$ This means that $h(x)$ is a ...
0
votes
2answers
28 views

How to derive this trig identitfy?

$$\large y = A\cos(\omega t - kx +\phi_1)+A\cos(\omega t + kx + \phi_2)\\ \large= 2\cos\Big(\omega t+\frac{1}{2}(\phi_1+\phi_2)\Big)\times \cos\Big(kx+\frac{1}{2}(\phi_2-\phi_1)\Big)$$ Could someone ...
0
votes
0answers
25 views

Can anyone help to find the value of h from the given equation?

$V = \frac{0.5r^{2}\cdot \cos^{-1}(\frac{r-h}{r})\cdot 2-\sin(\cos^{-1}(\frac{r-h}{r})\cdot 2)}{10^{6}}$ This is the equation to find the volume of liquid in a tank in the shape of a capsule. Where h ...
5
votes
4answers
151 views

Infinite limit of trigonometric series

The value of $\displaystyle\lim_{n\to\infty}(\sin^4x+\frac{1}{4}\sin^4(2x)+\cdots+\frac{1}{4^n}\sin^4(2^nx))$ is (A) $\sin^4x$ (B) $\sin^2x$ (C) $\cos^2x$ (D) does not exist My attempt: ...
1
vote
2answers
12 views

Eliminate the parameter of a

Eliminate the parameter to find a description of the following circles or circular arcs in terms of $x$ and $y$. Give the center and radius, and indicate the positive orientation. ...
-1
votes
4answers
322 views

Trig to determine distance: boat on course parallel to shore.

A boat going parallel to shore spots a lighthouse ahead on shore. The angle of the line from lighthouse to boat is 30 degrees. The boat sails 3mi, and now angle is 90. How far offshore is boat?
1
vote
4answers
44 views

Solution for the trignometric equation

I am looking for a solution for an equation of the form : $ax - \sin(bx) + c = 0$. Without the constant term $c$, I can easily take a derivative to get the solution. But how do I take into account the ...
1
vote
3answers
3k views

Does the Law of Sines and the Law of Cosines apply to all triangles?

Do the Law of Sines and the Law of Cosines apply to all triangles? Particularly, could you use these laws on right triangles? That is, could you use these laws instead of the ...
0
votes
1answer
18 views

Question based on incenter and excenter

In a $\bigtriangleup ABC $,$sin\frac{A}{2}+sin\frac{B}{2}+sin\frac{C}{2}=\frac{6}{5}$ and $II_1+II_2+II_3=9$ where I is incenter and $I_1,I_2,I_3$ are the excenters of $\bigtriangleup ABC $.Then find ...
18
votes
11answers
3k views

What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
0
votes
1answer
54 views

What is meaning of this question and how to solve it?

I am stuck with understanding the meaning of the question, which states: Show that $\cos(n\theta)=f_n(\cos\theta)$ for polynomials $f_n(x)$ satisfying $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
2
votes
2answers
96 views

Finding $4$ variables using $3$.

if I have: $ x=\dfrac{a-.5b-.5c+.25d}{a+b+c+d}$ $ y=\dfrac{\dfrac{b\sqrt{3}}{2}+\dfrac{c\sqrt{3}}{2}+\dfrac{d\sqrt{3}}{4}}{a+b+c+d}$ $ z=a+b+c+2d $ Then how do I get back to: $ a= $ , $ b= $ , $ ...
0
votes
0answers
70 views

Is there a space in which the $\vec a$ in $\sin(a_1)+\sin(a_2)$ is linear?

I have equations of the form $\sin(a_1)+\sin(a_2)=y$ (actually more complicated, but that's the general essence). I want to solve for $\vec a$ using linear regression instead of non-linear regression ...
1
vote
2answers
54 views

Solve $2\sin^3x + \sin3x +3\sin^2x \cos x + \cos^3x=0$

$2\sin^3x + \sin3x +3\sin^2x\cos x + \cos^3x=0$ My try: $$2\sin^3x +3\sin x - 4\sin^3x +\cos x(3\sin^2x+\cos^2x)=0 $$ $$ \cos x(2\sin^2x+1) - 2\sin^3x+3\sin x=0.$$ And then i have no idea.
1
vote
3answers
73 views

what is the definition of cosine , sine [duplicate]

I know that sine is the ratio of the perpendicular to the hypotenuse of an acute angle. Similarly cosine is the ratio of the base and hypotenuse . But now I found that there is sine and cosine of an ...
2
votes
2answers
54 views

Question based on triangle inscribed in unit circle

$ \bigtriangleup ABC $is inscribed in a unit circle.If angle bisectors of internal angles at A,B and C meet the circle at D,E and F respectively then value of $\frac{AD \cos\frac{A}{2}+BE ...
0
votes
3answers
34 views

Trigonometical identity proof

I was given a proving sum: $\sec(x) + \tan (x) = p$, prove $\frac{p^2-1}{p^2+1} = \sin (x)$ I went head on and tried to directly do it by solving the LHS: $\sec(x) + \tan(x)$ = $\frac{1}{\cos(x)} ...
0
votes
3answers
60 views

$a \cos(v) + b\sin(v) = A\sin(u+v)$ proof

I'm trying to find an $A$ and $u$ that satisfy: $a\cos(v) + b\sin(v) = A\sin(u+v)$. However, my result gets me $\sqrt{(a^2 + b^2)}\sin\big(v+\tan^{-1}(a/b)\big)$ which is incorrect according to the ...
0
votes
0answers
42 views

exam for my subject trigonometry [on hold]

from A, a pilot flew a course 60° for 500 km to B. from B, he flew a course 150° for 800 km to C , what is the direction and distance of A Flight from C to A
0
votes
1answer
22 views

Complex number identity by trigonometry

Show that $\lvert e^{i\theta} - 1\rvert = 2\lvert\sin(\theta/2)\rvert$ by using the geometry of the triangle with vertices 0, 1, and the midpoint of the line joining 0 and $e^{i\theta}$. I have been ...
3
votes
1answer
83 views

Prove that $\tan\alpha =\tan^{2}\frac{A}{2}.\tan\frac{B-C}{2}$

Given a triangle ABC with the sides $AB < AC$ and $AM, AD$ respectively median and bisector of angle $A$. Let $\angle MAD = \alpha$. Prove that $$\tan\alpha =\tan^{2}\frac{A}{2}\cdot ...
4
votes
7answers
1k views

How many ways are there to define sine and cosine?

Sometimes there are many ways to define a mathematical concept, for example the natural base logarithm. How about sine and cosine? Thanks.
0
votes
1answer
35 views

Show that $y=\frac{4\sin\theta}{2+\cos\theta}-\theta$ is increasing function when $\theta \in [0,\frac\pi2]$

Show that $$y=\dfrac{4\sin\theta}{2+\cos\theta}-\theta$$ is increasing function when $\theta \in [0,\frac\pi2]$ What I have done If $\theta_1,\theta_2\in[0,\frac\pi2]$ then $$\sin\theta_1 < ...
0
votes
0answers
20 views

Calculating point following with rotation

as my question my sound about programming it's really just a math. I just want to know how to calculate it not write it in programming language. So, I want to create effect, which looks like this: ...
1
vote
5answers
104 views

Prove that $\alpha + \beta=\frac {\pi}{2}$

It is given that- (1) $0<\alpha,\beta<90$. (2) $\sin^2\alpha+\sin^ 2\beta=\sin(\alpha+\beta).$ Prove that $\alpha + \beta=\frac {\pi}{2}$
1
vote
0answers
30 views

Integral of an expression involving sine and cosine powers

For integers $a,n\in \mathbb N$, consider the following integral $$ I_n(a) = \frac{(-i)^x}{\pi}\int_0^\pi e^{i\theta(n-2a)} \sin^x \theta \cos^{n-x} \theta\; \mathrm d\theta\;. $$ How would one go ...
0
votes
4answers
59 views

How to prove that $\sin(180^\circ-\theta)=\sin\theta$

Mi question is: How to prove $$\sin(180^\circ-\theta)=\sin\theta$$ ? Here, sine is defined for any angle such as 'alpha' This is the question mi college teacher asked me to derive it but i could ...
-1
votes
1answer
26 views

For acute $\theta$, write $\cot\theta$ in terms of $\sin\theta$

For acute $\theta$, write $\cot\theta$ in terms of $\sin\theta$. I know that's $\cot\theta = \frac{\cos\theta}{\sin\theta}$ but why is the answer $\cot\theta= \frac{\sqrt ...
2
votes
2answers
84 views

Prove that given a triangle satisfying $8\prod \sin\frac{A}{2}=\prod \cos(A-B)$ then that triangle is equilateral.

Prove that given a triangle $ABC$ satisfying $$8 \sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2} = \cos(A-B)\cos(B-C)\cos(C-A)$$ then that triangle is equilateral.
2
votes
0answers
24 views

Existence of formulae for sines/cosines of products of angles in terms of sines/cosines of original angles? [duplicate]

There was something that I was getting a little curious about. We know that there are the so-called compound-angle formulae for calculating sines and cosines of sums of angles in terms of those of the ...
3
votes
6answers
66 views

Solve $\sin A +\sin 2A +\sin 3A + \sin 4A = 0$, for $0 \leq A \leq 180$

I've tried using factor formula but still did not manage to get the answer, not sure if factor formula is the right method. I rearrange to $\sin 4A + \sin 2A + \sin 3A + \sin A = 0$, and after ...
3
votes
4answers
71 views

Given $\tan A + \tan B = 3x$ and $\tan A \tan B = 2x^2$, find $\tan A - \tan B$ [on hold]

Given $$\tan A + \tan B = 3x$$ and $$\tan A \tan B = 2x^{2}$$ How to find $\tan A - \tan B$ ? I've tried substitution but still couldn't find. EDIT: Can you solve this problem using the formulas for ...
0
votes
2answers
26 views

Question on trigometric graph sketching

Say I have to sketch the graph of √2 sin A + √7 cos A By R-formula, √2 sin A + √7 cos A = 3 sin (A + 61.9 degrees) So basically I have to sketch, 3 sin (A + 61.9 degrees) Question 1: Do I ...
0
votes
0answers
25 views

Trigonometric Substitution Method to solve Cubic Equation.

Here are the questions. IN the wiki page, it says p has to be smaller than 0. But they didnt really explain why... Therefore, I assume it is impossible to have a complex number inside arcosine, is ...
0
votes
1answer
54 views

Could someone help me solve this trigonometry problem?

If $\theta \in \mathbb{R}$ such that $$\frac { 2\sec { \theta } +3\tan { \theta } +5\sin { \theta } -7\cos { \theta } +5 }{ 2\tan { \theta } +3\sec { \theta } +5\cos { \theta } +7\sin { \theta ...
2
votes
3answers
130 views

Trigonometric equation with sine and cosine

So the equation is $3\cos ^2t + 5\sin t = 1$ Now I have simplified this to $$3(1-\sin ^2t) + 5\sin t -1 = 0$$ which leads to $$-3\sin ^2t + 5\sin t + 2 = 0$$ Then I get $$-3t^2 + 5 t +2 = 0$$ Is ...
10
votes
6answers
553 views

Wanted : for more formulas to find the area of a triangle?

I know some formulas to find a triangle's area, like the ones below. Is there any reference containing most triangle area formulas? If you know more, please add them as an answer ...
1
vote
4answers
34 views

question about trigonometric function transformation

what is the difference between $\cos^2(\theta - 180) $ and $\cos^2(180 - \theta) $ Does $\\cosec^2(450 + \theta)$ transform into $\sec$?
6
votes
7answers
666 views

Does $\operatorname{arcsec}(x) = 1 /\arccos(x)$?

Does $\operatorname{arcsec}(x) = 1 /\arccos(x)$? I have looked in a few books and Google'd it but I am not finding my answer.
2
votes
2answers
39 views

Trigonometric equation cos sin and power

The problem is $2\cos t - 3\sin^2t +2 = 0$. I get to $2\cos t -3\sin^2t =-2$ I think that I need to use a trigonometric identity like $\cos(x+y)$ and to divide $2\cos t -3\sin^2t$ with the ...
1
vote
3answers
29 views

Graphing of $y= \csc(x)+ \cot(x)$

What's the graph or table of values of $y=\csc(x) + \cot(x)$? I have already solved and graphed the values of $\csc(x)$ and $\cot(x)$.
1
vote
2answers
61 views

In a triangle, find the minimum and maximum of $\cos(A-B)\cos(B-C)\cos(C-A)$

In a triangle, with $A, B, C$ are three angles, find the minimum and maximum of $$\cos(A-B)\cos(B-C)\cos(C-A)$$
4
votes
3answers
196 views

Finding the area of a square that has a circle inside itself

I tried to solve the following problem: I think the image is self-descriptive. I tried to draw a vertical line from the top-end of $\theta$ angle to the horizontal line, then tried to use the ...
3
votes
3answers
67 views

Using the definition of derivative to find $\tan^2x$

The instructions: Use the definition of derivative to find $f'(x)$ if $f(x)=\tan^2(x)$. I've been working on this problem, trying every way I can think of. At first I tried this method: $$\lim_{h\to ...
0
votes
0answers
40 views

Limit approach to infinity [on hold]

During my studying to limits I find this limit but I want to know How we can know that this limit is exist??? $$\lim_{x\to \infty} \sqrt{1-\cos\frac{1}{x} \sqrt{1-\cos\frac{1}{x} ...
3
votes
3answers
103 views

Trigonometry question

If $$\frac{3-\tan^2\frac{\pi}{7}}{1-\tan^2\frac{\pi}{7}}=\alpha \cos\frac{\pi}{7}.$$ If $\alpha$ is a natural number.Find $\alpha$. My attempt is: ...
2
votes
2answers
63 views

Solve $\cos3x - 18\cos x +10 =0$

I want to solve $$\cos 3x - 18\cos x +10 =0 $$ I tried: 1) Replacing $\cos 3x$ to $\cos^3x - 3\cos x$ 2) Replacing $\cos x$ to $t$ we get: $$t^3 - 21t +10 = 0$$ So we get cubic equation. But I ...
2
votes
2answers
34 views

Range of an inverse trigonometric function

Find the range of $f(x)=\arccos\sqrt {x^2+3x+1}+\arccos\sqrt {x^2+3x}$ My attempt is:I first found domain, $x^2+3x\geq0$ $x\leq-3$ or $x\geq0$...........(1) $x^2+3x+1\geq0$ ...
1
vote
5answers
63 views

Simple trigonometry problem

It is given that, $ A+B +C= \pi $,and $\cos A = \cos B \times \cos C$ I have to prove: $\tan B \times \tan C= 2$ to prove that, this is what I did: $$\frac{\sin B}{\cos B} \times \frac{\sin ...
0
votes
1answer
43 views

Expressing $\cos(\varphi x)$ as a function of $x\sin\varphi,x\cos\varphi$

Let $\varphi,x\in\mathbb{R}$. I wonder if one can explicitly express $\cos(\varphi x)$ as a function of the variables $x\sin\varphi$ and $x\cos\varphi$. Suppose we denote ...