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
42 views

How is $\tan^{-1}(a/b) = \tan^{-1}(a) - \tan^{-1}(b)$?

I'm having a problem proving: $\tan^{-1}(a/b) = \tan^{-1}(a) - \tan^{-1}(b)$ Thanks!
1
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1answer
35 views

Geometric deduction that $\pi$ is lower bound of the circumscribed polygons and upper bound of the inscribed ones

I know some proofs that $2r\pi$ is the upper bound of the perimeters of the polygons inscribed in a circle and the lower bound of the circumscribed ones, but they are all very laborious. Can you show ...
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0answers
11 views

Calculating originally arc approximated by cubic bezier curve

I have an cubic bezier curve, which is representing an arc by an approximation. The approximation was calculated with the kappa constant: $$ \\k = \frac43*(\sqrt{2}-1) $$ This means, that the ...
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3answers
13 views

Cyclic quadrilaterals - finding the size of an angle

I know this might seem like a really simple question, but I really don't understand where I am going wrong. I am familiar with cyclic quadrilaterals as well as their properties, but this question ...
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1answer
19 views

Finding Y coordinate of third triangle point when X coordinate and two other points are already known

Suppose you know the coordinates for points A and B of a triangle. We can refer to those coordinates as (Ay,Ax) and (By,Bx). Also, suppose you know the X coordinate for point C (Cx) but do not know ...
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1answer
9 views

Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi)

This is for trigonometric equations and inequalities: Find all of the exact solutions of the equation and then list those solutions which are in the interval [0, 2pi) Cos(9x)=9
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1answer
16 views

Finding Trig Functions

I am given $$\cos(\beta) = \frac{\sqrt{15}}{8}$$ and I am asked to find the indicated trig function of $$\sin(90^{\circ}- \beta)$$ I know $$\sin(\beta)= \frac{7}{8}$$ but I don't know how to go about ...
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0answers
15 views

The cosine rule, Bearings [on hold]

two forest rangers leave base and travel to different parts of the forest. Simone averages 70km/hr along a straight track in a direction 25 degrees, While bruce averages 80km/hr in a direction 135 ...
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2answers
26 views

Limit as x approaches 0 from the left: $\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$

Help me find the limit as x approaches 0 from the left: $$\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$$ Thanks,
3
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2answers
51 views

Prove $\lim_{x\rightarrow 0}\cos (x)=1$ with the epsilon-delta definition of limits

Prove $$ \lim_{x\rightarrow 0}\cos (x)=1 $$ with the epsilon-delta definition of limits
2
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1answer
40 views

Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
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0answers
12 views

How to compute uniformly distributed points on an ellipse

The ellipse can be parametrized in polar coordinates by $$r(\theta)=\frac{1}{a+\cos\theta}$$ up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the ...
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1answer
19 views

Trigonometric problem with $\cos{\alpha}$ and $\sin{\alpha}$

The problem is $$3 \cos{\alpha} = 3 - 2\sin{\alpha}.$$ Find the value of angle alpha I have some trouble solving this. I don't know where to start.
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0answers
1 views

Taking components of a system containing multiple vectors.

Q. In the arrangement shown in fig. the ends P and Q of an inextensible string move downwards with uniform speed u. Pulleys A and B are fixed. The mass M moves upwards with a speed. My text ...
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1answer
30 views

If $y = 2\sin(x)-\sin^2(x)$ and $x = 2\cos(x)-\sin(x)\cos(x)$ what is $\frac {dy}{dx}$? [on hold]

If $y = 2\sin(x)-\sin^2(x)$$\ \ \ x = 2\cos(x)-\sin(x)\cos(x)$ What would $\frac {dy}{dx}$ equal to? so $\frac {dy}{dx}=2\cos(x)-\frac {2\cos(x)\sin(x)}{-2sin(x)}$ ... ? what would $y'$ of ...
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0answers
20 views

Determine sine wave frequency from two arbitrary points

If I have only two arbitrary points on a sine wave, what would be the simplest method for determining the frequency of the sine wave? The frequency is unknown. The bandwidth is restricted, the time ...
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2answers
114 views

How does one evaluate $\int \frac{\sin(x)}{\sin(5x)} \ dx$

The below problem is taken from Joseph Edwards book Integral Calculus for beginners. How does one show: $$5 \int \frac{\sin(x)}{\sin(5x)} \ dx= \sin\left(\frac{2\pi}{5}\right) \cdot ...
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1answer
27 views

Is $\cos(\arctan(-x) - \arcsin(y))$ the same as $\cos(\arctan(-x)) - \cos(\arcsin(y))$?

Is $\cos(\arctan(-x) - \arcsin(y))$ the same as $\cos(\arctan(-x)) - \cos(\arcsin(y))$? Because if I put this on my calculator I got different solutions.
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1answer
33 views

How to prove this inequality relating to trigonometric function?

In a triangle, A, B, C are three corners of the triangle, try to prove that : $$\root 3 \of {1 - \sin A\sin B} + \root 3 \of {1 - \sin B\sin C} + \root 3 \of {1 - \sin C\sin A} \geqslant {3 \over ...
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4answers
45 views

$ \sin x \geq \frac{2x}{\pi}, x \in [0, \frac{\pi}{2}]$

I have tried the following: $$ f(x) = \sin x-\frac{2x}{\pi} \\ f'(x)= \cos x-\frac{2}{\pi} \\ f''(x) = -\sin x \leq 0 $$ But this doesn't seem to be heading in the right direction as it would appear ...
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0answers
19 views

Is a sine wave plus the sum of its odd harmonics symmetrical around the x axis at half the period of the fundamental?

I have a function such that $$x(t)=A_1 \sin(2 \pi f t+\phi_1)+A_2 \sin(2 \pi (3f) t+\phi_2)+...+ A_n \sin(2 \pi ((2n+1)f) t+\phi_n)$$ Is such a function symmetric around the point that is half ...
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3answers
44 views

Using $x=\tan \theta$ to solve $\int x\sqrt{1+x^2}\,\mathrm dx$

I'm having a lot, I repeat, a lot of trouble with Calculus II, particularly trigonometric substitution. At the moment, I'm extremely confused as to how to integrate $\int x\sqrt{1+x^2}\,\mathrm dx$ ...
4
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1answer
28 views

Find the missing angle of similar triangle

Find the missing angle $\theta$ in the triangle below given that $R>r$, $l\geq R$, $0< \theta < \frac{\pi}{2}$. Attempted Solution I attempted to use similar triangles to find the angle ...
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5answers
53 views

The limit as $x$ goes to infinity of $x-x\cos(4/x).$

I would like to determine $$\lim_{x\to \infty} \left(x-x\cos \frac4x\right)$$ How do I even start this? I can't plug in infinity to $4/x,$ can I? That will make it zero? Is the answer infinity then?
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2answers
40 views

How to find the exact value of $\tan(\sec^{-1} 4)$?

I would like to know if there is a general method to solve equation looking like this: $$\tan(\sec^{-1} 4)$$ without using a calculator (you have to find the exact value)? How to proceed?
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1answer
22 views

Prove this trig identity?

I'm having trouble proving the following identity: $$A\cos(\omega t) + B\sin(\omega t) = \sqrt{A^2 + B^2} \cos(\omega t - \arctan(\tfrac{B}{A}))$$ Does anyone know how this can be done? Thanks!
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0answers
31 views

Robotic Kinematics - Differential drive

I'm new to Kinematics and my robotic book assumes I have a base knowledge which I do not. The book doesn't do a very good job explaining for someone with minimal math background. The question is: ...
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0answers
41 views

How to evaluate the following integrals

$$\int\limits_0^{\frac{\pi }{2}} {{x^2}{{\ln }^2}\left( {\sin x} \right)\ln \left( {\cos x} \right)dx} ,\int\limits_0^{\frac{\pi }{2}} {x\ln \left( {\sin x} \right){{\ln }^2}\left( {\cos x} \right)dx} ...
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2answers
126 views

Need some help solving high-school level trignometry question.

here it is. I've tried solving it multiple ways but it gets too complicated. Is there any way to solve this?
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1answer
33 views

Trigonometry rewrite

How do you go from $$ \frac{ \sin z }{e^z -1} $$ to $$ \frac{\cos z}{e^z}$$ My first approach was to use eulers formula. But that made the calculations quite hard. I'm guessing this can be done ...
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2answers
46 views

Some help with sin and cos

I'm having trouble to understand the following equalities in these two equations, i.e. how to apply the addition formulas. Firstly: $$ \frac {1- \frac {sin^2(\frac x2)} {cos^2(\frac x2)}} {1+ \frac ...
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1answer
30 views

Trigonometric ratios

I'm stuck with a problem. Given is a triangle $\Delta ABC$ with $\angle A = 35°, BC=3$ and $AC=5$. I need to find the two possible values for $\angle C$. I only managed to found one angle. I did the ...
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1answer
24 views

$\tan \{\frac{1}{2} \sin^{–1} (2x/ 1 + x^2) + \frac{1}{2} \cos^{–1} (1 – y^2/1 + y^2) \}$ is equal to.

$$ \tan \left\{ \frac{1}{2} \arcsin \frac{2x}{1 + x^2} + \frac{1}{2} \arccos \frac{1 – y^2}{1 + y^2} \right\} $$ is equal to. Note: i think $\sin a=2x/1+x^2$, $\cos b=(1 – y^2/1 + y^2)$
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1answer
17 views

How many milliliters of liquid to fill [duplicate]

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more millilitres of liquid need to be added in order to fill the ...
0
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1answer
93 views

Translation request: geometry problem stated in Korean [on hold]

Please im a foreing studying in south korea.. so i dont understand nothing in class... can any one tell me what is this called in english? thanks
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0answers
21 views

In need of formula: Gravity at Specific Coordinates [on hold]

Doing Research on the gravitational pull at a specific set of coordinates. Does anyone know how to solve this mathematically? Please Help. Thanks
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0answers
25 views

Find the length of the longer diagonal on a trapezium with only 2 sides stated.

Im at a loss here, i know i have to divide the trapezium, but im still not sure which calculation is relevant to it then. Thanks in advance.
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2answers
34 views

Trigonometry textbook or tutorial

Is there an actual textbook or online resource that has a tutorial to solve $a\sin x+b\cos x=c$ for $a, b, c$ being either positive or negative? I tried to find these types of equations/functions in ...
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0answers
25 views

trigonometric equation solution

I want to solve -4sinx-11cosx=8 I convert -4sinx-11cosx=Rsin(x+p) R=sqrt of (-4)^2 + (-11)^2 = 11.7 P=invtan(-11/-4)=70.01 So 11.7sin(x+70.01)=8 So I get x= 66.85 or 333.13 for x between 0 and ...
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4answers
54 views

Simplify: $\sin \frac{2\pi}{n} +\sin \frac{4\pi}{n} +\ldots +\sin \frac{2\pi(n-1)}{n}$. [duplicate]

Can you help me solve this problem? Simplify: $\sin \dfrac{2\pi}{n} +\sin \dfrac{4\pi}{n} +\ldots +\sin \dfrac{2\pi(n-1)}{n}$.
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1answer
27 views

Find angle and hypotenuse of right angled triangle

Find the missing side and the hypotenuse of a right triangle that has a side length of 5 cm and a perimeter of 30 cm. I'm confused. Can somebody please explain to me how to do this step by step? Not ...
6
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2answers
55 views

Infinite Sum of Sines With Increasing Period

A while ago, I was thinking about the Weierstrass function, which is a sum of sines with increasing frequencies in such a way that the curve is a fractal. However, I wondered what would happen if one ...
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1answer
23 views

How to calculate point on circumference of circle given radius

I am trying to come up with a formula to calculate the y co-ordinate of the point on the circle in the attached picture (i.e. delta y) based on the circumference of the circle and the distance x. ...
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0answers
10 views

Angle of elevation problem

the angle of elveation on the top of a tower at a point A on the ground is $30$ degrees. On walking $20$ m toward the tower,the angle of elevetion becomes $60$ degrees.Find the height of the tower and ...
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3answers
41 views

Sum of $\sin$ and $\cos$

We are given a trigonometric equation to solve: $$a\sin x+b\cos x=c$$ with $a,b,c$ nonzero real numbers. We are also given that $$a\sin x+b\cos x=R\sin(x+\varphi)$$ with $R^2=a^2+b^2$ and ...
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2answers
10 views

Verify the trig formula with complex representation

Verify the trigonometric formula for sin(a+b)=sinacosb+sinbcosa and cos(a+b)=cosacosb-sinasinb by using complex representation. I tried to use Euler's formula to start but I am unsure how to use ...
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3answers
30 views

Verify Euler's formula

Verify Euler's formula for $e^{ix}$ by considering $\frac{dz}{dx}$ where $z=r(\cos x+i\sin x)$ I tried taking the derivative of z but could not get to Euler's from there.
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1answer
31 views

Express $\sinh x$ in terms of the exponential function

I found that $\cosh x=\frac{e^x + e^{-x}}{2}$ but I am unsure how to find $\sinh x$ in terms of the exponential function by using Euler's formula.
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3answers
17 views

Finding Lipschitz for trigonometric functions

how would i find the Lipshitz constant for $$\sin(x)\times \cos(x)$$ or other trigonometric functions? How would I get my $\operatorname{abs}{x_1 - x_2}$
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2answers
37 views

trouble solving the integral of $\cos(x^2)$

No, I really mean the integral of $\cos(x^2)$, not $[\cos(x)]^2$. Can the chain rule be applied here?