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

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2answers
46 views

Area of regular n-gon without trig?

As the title suggests I'm trying to find a formula for the area of a regular n-gon that doesn't use trigonometry. I already know the trig formula and I realize that my question is simply asking for ...
0
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2answers
42 views

Max value of trignometric function

Question:The maximum value of $\sin \left(x+\dfrac{\pi}{6}\right)+\cos \left(x+\dfrac{\pi}{6}\right)$ is at what value of $x$. I solved the problem by setting the slope of the function to zero and ...
3
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1answer
34 views

Let $S$ be a set of $n$ points in the plane with min spacing of 1. Prove $S$ has a subset of $\ge n/7$ points with min spacing of $\sqrt{3}$.

I believe I have proven the case $n=8,|T|=2$, but welcome feedback. I need help proving the case for general $|T|>2$. From the 2003 Canada National Olympiad: Let $S$ be a set of $n$ points in ...
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0answers
32 views

Evaluating the indefinite integral $\int\sqrt{\cos2x}\sin^32x\,dx$

I have tried to integrate the following indefinite integral but I'm not sure if I get the right answer. Please tell me if I'm wrong and if so, please indicate what went wrong. $$ ...
2
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4answers
40 views

Find all values that solve the equation

For which values a, the equation $$ a\sin{x}+(a+1)\sin^2{\frac{x}{2}} + (a-1)\cos^2{\frac{x}{2}} =1 $$ has a solution? My idea: I think it's possible to factorize equation or reduce equation to the ...
1
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1answer
33 views

Prove that a trigonometric equation has six distinct roots

Show that,in general,the equation $A \sin^3x+B\cos^3x+c=0 $has six distinct roots,no two of which differ by $2\pi$,and that the tangent of their semi-sum is $-\frac{A}{B}$. My attempt: I tried to ...
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1answer
48 views

What textbooks should I use for Trigonometry and Calculus? My basics are terrible.

I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by ...
3
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2answers
35 views

Prove the relation for cos inverse

Prove the relation $\cos^{-1}x_0=\dfrac{\sqrt {1-x^2_0}}{x_1\cdot x_2\cdot x_3\cdots \text{ ad inf.}}$ where the successive quantities $x_r$ are connected by the relation ...
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1answer
32 views

Minimum value of trigonometric function

The minimum value of the expression $\left|\sin x+\cos x+\tan x+\cot x+\sec x+\mathrm{cosec} x\right|$ can be expressed as $(\sqrt a-\sqrt b)$ where a and b are natural number then find the value of ...
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2answers
28 views

Unique solution to sin(2a) and cos(2a)

I'm a bit confused as to how to solve for $2\alpha^*$ using the equation 6.36 in the excerpt below. I know how to solve for it individually (ie acos and asin) but how do I solve them together to get ...
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2answers
38 views

Finding a triangle ABC if $2\prod (\cos \angle A+1)=\sum \cos(\angle A-\angle B)+\sum \cos \angle A+2$

Find $\triangle ABC$ if $\angle B=2\angle C$ and $$2(\cos\angle A+1)(\cos\angle B+1)(\cos\angle C+1)=\cos(\angle A-\angle B)+\cos(\angle B-\angle C)+\cos(\angle C-\angle A)+\cos\angle A+\cos\angle ...
3
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0answers
57 views

Solving the trigonometric equation $\tan^2x+\cot^2x=2-\cos^{2014}(2x)$

I was solving the trigonometric equation $$\tan^2x+\cot^2x=2-\cos^{2014}(2x) $$ I solve it by inequality $|a|+\frac{1}{|a| }\geq 2$. $$ L.H.S=\tan^2x+\cot^2x =\tan^2x+\frac{1}{\tan^2x} ...
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1answer
11 views

Find point on circle's tangent based on point on circle, radius and angle

The circle is centered at (0,0)"P" with a radius of 5. I have a point on the circle at (4,-3)"A". How would I find the points "B1" and "B2" on the tangent through point "A" given an arbitrary angle ...
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3answers
56 views

how can i prove this trigonometry equation

I need help on proving the following: $$\frac{\cos {7x} - \cos {x} + \sin {3x}}{ \sin {7x} + \sin {x} - \cos {3x} }= -\tan {3x}$$ So far I've only gotten to this step: $$\frac{-2 \sin {4x} \sin {3x} ...
1
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6answers
167 views

Sum of cosines of complementary/suplementary angles

Why are $(\cos(2^{\circ})+\cos(178^{\circ})), (\cos(4^{\circ})+\cos(176^{\circ})),.., (\cos(44^{\circ})+\cos(46^{\circ}))$ all equal zero? Could you prove it by some identity?
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7answers
132 views

Evaluating the indefinite integral $\int\sqrt{16-9x^2}\,dx$

I need to solve the integral below, but I just can't figure how. $$\int \sqrt{16-9x^2}\,dx$$ I have tried to replace $9x^2$ with $16\sin^2\theta$. I get to a point where I have the function ...
4
votes
7answers
293 views

Sine/cosine series

$$\frac{\sin²(1°) + \sin²(2°) + \sin²(3°) + .. + \sin²(90°)}{\cos²(1°) + \cos²(2°) + \cos²(3°) + .. + \cos²(90°)} = ?$$ I tried to use multiple identities but I couldn't simplify the expression. ...
0
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2answers
20 views

ind $\tan \alpha$ in the square

let say the square has sides of 2 units, $DM = DN = AN = AP = 1$, $NP = \sqrt 2$, $NQ = QP = \frac{\sqrt 2}{2}$, and $AR \ne AP$ (?) we have know that $\tan \alpha = \frac 2{RP}$, but what's the ...
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3answers
17 views

Determine if 2 points are horizontal without trigonometry

Let's say that I have 2 points: (c1X, c1Y) and (c2X, c2Y). I would like to consider these 2 points horizontal as long as their angle is below 45 degrees. I could accomplish this with trigonometry. ...
2
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4answers
70 views

Find $x$ in the triangle

the triangle without point F is drawn on scale, while I made the point F is explained below So, I have used $\sin, \cos, \tan$ to calculate it Let $\angle ACB = \theta$, $\angle DFC = \angle ...
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1answer
17 views

Polar conversions of coordinates and parametric equations

Express the polar coordinates $P\left(6, -\dfrac{\pi}{4} \right)$ in Cartesian coordinates. $\displaystyle x=r\cos{(\theta)} ,\ y=r\sin{(\theta)} \implies x^2+y^2=r^2 \wedge \theta = ...
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1answer
43 views

A triangle ABC with the internal bisector of $\angle A$, the median drawn from B and the altitude drawn from C meet at the same point.

A triangle $ABC$ with the internal bisector of $\angle A$, the median drawn from $B$ and the altitude drawn from $C$ meet at the same point. Prove that $$\tan A = \dfrac{\sin C}{\cos B}$$ I try to ...
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1answer
40 views

Geometric problem based on angle bisectors

I am not asking a question,i just want to conform,is my method of solving problem correct? Given a triangle ABC.It is known that AB=4,AC=2,and BC=3.The bisector of angle A intersects the side BC at ...
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2answers
41 views

Trig identity involving sum of cosines

$$\begin{align} y &= A\cos\left(\omega t - kx +\phi_1\right)+A\cos\left(\omega t + kx + \phi_2\right)\\[6pt] &= 2A\cos\left(\omega t+\frac{\phi_1+\phi_2}{2}\right)\cdot ...
0
votes
0answers
49 views

Find the value of $h$ from a Kepler-type 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 ...
2
votes
3answers
21 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. ...
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1answer
21 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 ...
1
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3answers
87 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 $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
1
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2answers
55 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.
0
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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)} ...
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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
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4answers
54 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 ...
0
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0answers
77 views

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

I have equations of the form $\sin(a_1\cdot x)+\sin(a_2\cdot x)=y$ (actually more complicated, but that's the general essence). I want to solve for $\vec a$ using linear regression instead of ...
2
votes
2answers
57 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 ...
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1answer
23 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 ...
0
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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: ...
2
votes
0answers
35 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 ...
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1answer
37 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 < ...
3
votes
1answer
84 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 ...
1
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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 ...
0
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4answers
62 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 ...
5
votes
4answers
152 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: ...
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1answer
27 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
0answers
26 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 ...
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
27 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 ...
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 ...
1
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4answers
35 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 $\sec$?
3
votes
6answers
67 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 ...