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

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1answer
43 views

Trigonometric equation

I was solving this trigonometric equation $$\tan^2x+\cot^2x=2-\cos^{2014}(2x) $$ I solve it by inequality $|a|+\frac{1}{|a| }\geq 2$. I provided an answer below. Can someone give an alternative ...
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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
46 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 + \sin x}{ 2 \sin ...
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6answers
166 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
125 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
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7answers
290 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. ...
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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
16 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. ...
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4answers
69 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
41 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
38 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 ...
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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 ...
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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 ...
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3answers
86 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 $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
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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.
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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 ...
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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 ...
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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
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 ...
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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
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0answers
34 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 < ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\sec$?
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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 ...
3
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4answers
72 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 ...
1
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3answers
30 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 $\cot(x)$.
2
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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.
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3answers
131 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 ...
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2answers
63 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
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3answers
197 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
68 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 ...
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 ...
19
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 ...
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 ...
0
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0answers
41 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} ...