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

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0
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2answers
25 views

Why is the sine and cosine always between $-1$ and $1$?

Why is the sine and cosine always between $-1$ and $1$? If I would have circle with a radius other than $1$, then it wouldn't be between $-1$ and $1$ anymore, would it? This also ties in with another ...
2
votes
5answers
41 views

Proof of $\arcsin x \le 2\arctan x$?

I am looking for a proof for the following 'fact': $$ \arcsin x \le 2\arctan x \quad \forall x\in[0,1). $$ I put fact between single quotes, as the only proof I found is a plot by wolframalpha. I know ...
-1
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0answers
22 views

Given that θ = tan^-1(4/3) find

Find the exact values of sin θ, cos θ, cot θ, sec θ and csc θ. Could anyone explain how I would find the values?
0
votes
1answer
14 views

Using sine law determine missing angel x

Okay so I need to use SINE to determine the missing angle. (X). I know one angle is 85 degrees. And the sides are 25mm, 43.8 mm, and 36 mm, can someone please tell me what I have to put into my ...
1
vote
3answers
55 views

The value of $\int_0^{2\pi}\cos^{2n}(x)$ and its limit as $n\to\infty$

Calculate $I_{n}=\int\limits_{0}^{2\pi} \cos^{2n}(x)\,{\rm d}x$ and show that $\lim_{n\rightarrow \infty} I_{n}=0$ Should I separate $\cos^{2n}$ or I should try express it in Fourier series?
1
vote
2answers
50 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!
0
votes
0answers
16 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 ...
32
votes
4answers
1k views

Integral $\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$$
11
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2answers
118 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 ...
0
votes
3answers
14 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 ...
7
votes
2answers
161 views
+50

Calculation of $\int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$

Calculate $$ \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2\cos^2 ...
0
votes
1answer
363 views

Determine sin cos and tan from slope? NON CALCULator

I am used to finding this by drawing the triangle and knowing the angle measurement. With the angle measurement i can find sin cos and tan. But i dont have angle all I have is line. I have put a ...
0
votes
1answer
22 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 ...
1
vote
1answer
18 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 ...
1
vote
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
0
votes
0answers
16 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 ...
3
votes
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
0
votes
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,
2
votes
1answer
44 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 ...
2
votes
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 ...
5
votes
3answers
679 views

Find $\int e^{2\theta} \cdot \sin{3\theta} \ d\theta$

I am working on an integration by parts problem that, compared to the student solutions manual, my answer is pretty close. Could someone please point out where I went wrong? Find $\int e^{2\theta} ...
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votes
1answer
20 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.
0
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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 ...
3
votes
2answers
205 views

Find this limit without using L'Hospital's rule

I have to find this limit without using l'Hôspital's rule: $$\lim_{x\to 0} \frac{\alpha \sin \beta x - \beta \sin \alpha x}{x^2 \sin \alpha x}$$ Using L'Hôspital's rule gives: ...
3
votes
3answers
45 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 $\int x\sqrt{1+x^2}\,\mathrm dx$ ...
-3
votes
1answer
31 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 ...
1
vote
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?
1
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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 ...
0
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0answers
21 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 ...
-1
votes
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.
9
votes
5answers
6k views

Best self study math books?

I graduated highschool a while ago, hardly remember anything and have no idea where to begin relearning. I am looking for math text/book recommendations from basic algebra to precalculus. cheers
3
votes
4answers
47 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 ...
4
votes
1answer
37 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 ...
6
votes
5answers
55 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?
4
votes
1answer
31 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 ...
0
votes
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!
0
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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: ...
2
votes
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?
3
votes
0answers
42 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} ...
0
votes
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 ...
9
votes
1answer
116 views

Proof of an equality involving cosine $\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{2}}}}\ =\ 2\cos (\pi/2^{n+1})$

so I stumbled upon this equation/formula, and I have no idea how to prove it. I don't know how should I approach it: $$ \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{\vphantom{\large A}2\,}\,}\,}\,}\ ...
0
votes
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 ...
1
vote
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 $\sin a=2x/1+x^2$, $\cos b=(1 – y^2/1 + y^2)$
5
votes
2answers
94 views

Evaluation of $\int \frac{x\sin(\sin x)}{x+5} \ dx$

How do we find $$\int \frac{x\sin(\sin x)}{x+5} \ dx\ ,$$ is there any way to take that $\sin x$ out from parent $\sin(\cdot)$ ?
0
votes
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
votes
1answer
918 views

How to remember trigonometric ratios for allied angles?

I just started studying trigonometry in unit circle and I want to know if there is some intuitive way to remember the value of $\sin(n\cdot\frac{\pi}{2}\pm\theta) \text{ and ...
0
votes
1answer
46 views
+100

Find the time interval between oscillations of SHM.

Parts i) and ii) I can solve. But for part iii) I can't do, as I don't know which equation describes the SHM motion? Is it $y=0.5sin(1.2t)$ or $y=0.5cos(1.2t)$ or $x=0.5sin(1.2t)+2.5$? I thought ...
2
votes
1answer
60 views

Prove $f$ isn't continuous at $\frac{1}{\pi}$

Let $f(x)=\left\lfloor {\sin {1 \over x}} \right\rfloor$ (meaning floor of $\sin x$). I need to prove that $f(x)$ isn't continuous at $x=\frac{1}{\pi}$. Proof: For a nehiborhood of $\frac{1}{\pi}$: ...
1
vote
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: $\sin \dfrac{2\pi}{n} +\sin \dfrac{4\pi}{n} +\ldots +\sin \dfrac{2\pi(n-1)}{n}$.