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

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3k views

What does $\sin(\sin(x))$ mean?

What does an equation like $\sin(\sin(x))$ mean? I know it can be seen as a composite function $f(f(x))$, where $f(x)=\sin(x)$. Is there a way to simplify functions like this, and where will this be ...
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3answers
2k views

How do I find the maximum and minimum of a sinusoidal function?

I understand basic $\sin x$ and $\cos x$ min/max, but I am having a problem solving the minimum and maximum of the following: $f(x) = \sin^2 x - \sin x$ Oh, and the range is $0 \le x \le ...
4
votes
4answers
249 views

How do I prove this trigonmetric identity?

I need to prove that the following identity is true: $$ \frac{\cos^2x-\sin^2x}{1-\tan^2x}=\cos^2x $$ This isn't homework; just a practice exercise. But I keep getting stuck! Thanks much.
0
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1answer
319 views

Are Euclidean distances a monotone function of inner products?

Does the sum of all pairs of inner-products of k vectors (real) have to decrease if the sums of Euclidean distances between all pairs of $k$ vectors happens to decrease? Similarly-if decrease is ...
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1answer
163 views

Perimeter of a right-angled triangle

I am a student of Class VII and studying for my examination. Can you please help me in solving this question? Is the perimeter of a right-angled triangle of base $b$, hypotenuse $l$ and height $h$, ...
3
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2answers
142 views

Integration of $1/(1+\sin x)$

I solved it using $t=\tan(\frac{x}{2})$ substitution and got $-2/(1+\tan(x/2))+C$, but in my math book solution is $\tan(x/2-\pi/4)+C$. Are those the same expressions and if they are, how do I ...
3
votes
1answer
210 views

How to prove $\sin{b}\sin{c}\sin{(b-c)}(\sin^2{b}+\sin^2{c}+\sin^2{(b-c)})+\dots=0$

If $a,b,c\in (0,\pi)$ and $a+b+c=\pi$, show that: $$\sin b \sin c \sin(b-c) \left(\sin^2 b + \sin^2 c + \sin^2(b-c)\right) \\ + \sin c \sin a \sin(c-a) \left(\sin^2 c + \sin^2 a + ...
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1answer
363 views

The Pythagorean Theorem a special case of law of cosines

I heard that the Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines?
0
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1answer
56 views

implicit differentiation using trigonometry functions

xcos(4x+3y)=ysinx I have been stuck on this problem for the longest. I have the answer but I don't know how to get to it. I have used the product and chain rule ...
3
votes
3answers
148 views

Proof this curious trigonometric identity

Proof that $$\cos^2{10^\circ} + \cos^2{50^\circ} - \sin{40^\circ}\sin{80^\circ} = \frac{3}{4}$$ I notice that $10^\circ + 80^\circ = 90^\circ$, and $50^\circ +40^\circ = 90^\circ$. I tried doing some ...
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3answers
50 views

Trigonometry Identity homework help

Could you please help me prove this: $${2\cos(\theta/2)-1-\cos\theta\over2\cos(\theta/2)+1+\cos\theta}={1-\cos(\theta/2)\over1+\cos(\theta/2)}$$
2
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1answer
148 views

Partial fraction decomposition of $\frac{1}{x^{2n}+a^{2n}}$

I came across a formula for the partial fraction decomposition of $ \displaystyle \frac{1}{x^{2n}+a^{2n}}$. It seems correct (at least for $n=1,2$, and $3$). But how is it derived? ...
2
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2answers
166 views

Simplifying the expression $\sqrt{(1+\cos t)^2+(-\sin t)^2}$

I wish to find the length of the parametric curve $$x = f(t) = t + \sin t, y = g(t) = \cos t, t \in [0, \pi]$$ The length $L$ is given by $$\int_0^\pi\sqrt{(f'(t))^2+(g'(t))^2}dt = ...
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1answer
58 views

Positive sinus on given points

I have $N$ points: $x_1, \dots, x_N $ How to prove that there exists $\alpha$ such that $\sin (\alpha x_i) > 0 \quad \forall i \ \in \{1,\dots,N\}$
4
votes
2answers
163 views

Find the integral $\int \frac{1}{x^2 \cdot \tan(x)} \ dx$

This problem seems pretty tricky. I need to find the integral of $$\int \dfrac{1}{x^2 \cdot \tan(x)} \ dx$$ Any help would be greatly appreciated!
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1answer
33 views

Multiply same trig functions

I apologize if this is a duplicate of another question in advance. I did not find any in the search. Does $\sin(x)\sin(y)=\sin(xy)$? I am wondering because I need to know if ...
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0answers
41 views

limitations of non linear multivariant equation solvers

I have a system of non-linear multivariate equations. I am only interested in the roots of the system in an interval of each variable. For example, $$ \begin{align} \frac{1}{10} \sin \left( \frac{x ...
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8answers
1k views

Solve $\sin x - \cos x = -1$ for the interval $(0, 2\pi)$

We have an exam in $3$ hours and I need help how to solve such trigonometric equations for intervals. How to solve $$\sin x - \cos x = -1$$ for the interval $(0, 2\pi)$.
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3answers
1k views

How do I find the direction angle of a vector?

Let $v= -3 i-10j$. I found the dot product to be $109$ and the magnitude to be $\sqrt{11881}$. I divided it out and it came out to be one. I don't know what I'm doing wrong. Please help.
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1answer
38 views

factoring trig functions

I'm having an issue with factoring trig functions. For example the following: $$ x^2 \ cos(2x) + 2x \ sin(2x) \\ $$ I thought it was $$ x(x\ cos(2x) + 2 \ sin(2x)) \\ $$ But this is what my books ...
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0answers
62 views

Triangle rotating freely around origo, need to calculate corners.

Lets say I have a triangle with corners $(-1, -2)$, $(0, 2)$ and $(1, -2)$. I specify a line that is exactly one side of the triangle, for example $(-1, -2)$, $(0, 2)$. Now, I rotate the triangle ...
30
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4answers
1k views

Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3 $ is close to $ \pi $?
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3answers
45 views

Determining all right-angled triangles with sides $a$, $a+k$ and $a+2k$

Determine all triangles that are right-angled which has the sides: $$a, \quad b = a+k, \quad c = a+2k.$$ (a, k are natural numbers!) So does this mean I have to take a look at: $a^2+b^2=c^2 ...
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2answers
92 views

Meaning of “edge of the building”?

I'm hoping someone can tell me what they mean "edge of the building" in the following word problem: ...
0
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1answer
87 views

solving trigonometric equation $ \sqrt 2 \sin^2 x+\cos x=0$

$$\sqrt 2 \sin^2 x+\cos x=0$$ Does anyone have an idea to solve this? I tried to do like : $\sqrt 2 \sin^2 x+\cos x=0/2$. But can't get anything.
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4answers
136 views

Approximation to $\sqrt{\cos(\theta)}$?

I have this formula, (it is just the law of cosines angle formula): $$ d = \sqrt{a^2 + b^2 - 2ab \ cos(\theta)} $$ Here is my issue. I am wondering if there is a way to 'extract' the $cos$ term. My ...
0
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1answer
27 views

solving a trigonometric equation Cos(x+30°)=cos(x+60°)

$cos(x+30°)=cos(x+60°)$ In the start i used the addition formula and i get : $√3/2cosx-1/2sinx-1/2sinx+√3/2cosx=0$ and at the end i get like $cos(x+30°)=0$ but i don't know is it right. Can ...
1
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1answer
40 views

Trigonometric functions (complex)

I have to find $sen^3{5a}$ and $cos^2{5a}$ considering that $sen{a}=\displaystyle\frac{1}{2}$ and $a$ belong to the first positive quadrant. I tried to apply De Moivre formula to find ...
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4answers
102 views

Help solve a trigonometric equation,

$$3\sin^2(x)+5\sin( x)\cos( x) = 2$$ Can't seem to solve it. Anyone can help me?
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2answers
170 views

Help with the proof of the Witch of Agnesi curve

$a=1$ (The radius is 1). How do I prove that if we talking about $P=(x,y)$, then: $$y=\frac{8}{x^2+4}$$ I'd like to get any help! Thank you!
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2answers
175 views

solving this equation $\sin x-\cos x=1/\sqrt2$

I need some help on solving this equation: $$\sin x-\cos x=\frac{1}{\sqrt2}$$ If I do $\sin x=\sqrt{1-\cos^2x}$ and then $\cos x=t$ but don't get anything. Or $\sin x-\cos x=1/√2)/√2$ Or $\sin ...
3
votes
4answers
678 views

solve a trigonometric equation $\sqrt{3} \sin(x)-\cos(x)=\sqrt{2}$

$$\sqrt{3}\sin{x} - \cos{x} = \sqrt{2} $$ I think to do : $$\frac{(\sqrt{3}\sin{x} - \cos{x} = \sqrt{2})}{\sqrt{2}}$$ but i dont get anything. Or to divied by $\sqrt{3}$ : $$\frac{(\sqrt{3}\sin{x} - ...
0
votes
1answer
109 views

Probability Distribution for angle between two random planes

Given two randomly placed planes what is the probability distribution dependence for the angle $\theta$ between them? I believe it to be simply $\sin\theta$ but does anyone have a simple proof? ...
0
votes
1answer
30 views

Trigonometric Integral of variable function.

Let for any $n \in \mathbb Z$, define a function $f_n \text { on } [0,1]$ as follows: $$f_n(x) = \begin{cases}0 &\text{if} &x=0 \\ \sin ...
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1answer
653 views

Proof: The coordinates of the witch of Agnesi curve

I need to prove that the coordinates ofthe witch of Agnesi curve is: $$x=2a\cot \theta$$ and $$y=2a\sin ^2 \theta$$ Any idea how to prove it? And I don't understand how we got $a$... (because the ...
3
votes
1answer
98 views

Find $\int \cos^4(x)dx$

We have: $\int \cos^n x\ dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n}\int \cos^{n-2} x\ dx.$ Find $\int \cos^4x\ dx$ by using the formula twice What I have so far is: $\int \cos^4 x\ dx = ...
2
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2answers
35 views

Precalc - Trig Identities

I only need a hint as to where to go from here. My problem is this: $$ \dfrac{1+\tan(x)}{\sin(x)}-\sec(x) $$ Here's my work trying to solve the problem, up until I got stuck. Did I make a mistake ...
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4answers
68 views

Solve Precalc Trig Identity

I don't need the answer, I just need to be pointed in the right direction. My problem is the following: $$ \dfrac{\sec(x)-\cos(x)}{\tan(x)} $$ Here's my work trying to solve the problem. Did I make ...
2
votes
1answer
337 views

Speed of a bicycle wheel

OKAY I corrected a blunder and still have problems I'm following this diagram from my book: http://i.imgur.com/04I9YvA.jpg It asks that if the chainring is 150 mm in diameter, the sprocket is 80 ...
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3answers
144 views

Precalc Trig Identities

I started off with this problem: $$ \dfrac{1}{\sec(x)+1} + \dfrac{1}{\sec(x)-1}$$ I attempted to solve it by multiplying the denominators, which gave me this: $$ \dfrac{2}{\sec^2(x)-1} $$ I then ...
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1answer
78 views

Why does $-\sec(x) = 1/\cos(x)$?

Why does $-\sec(x)$ become positive $1/\cos(x)$? I thought that the negative would carry and it would be $-1/\cos(x)$.
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1answer
54 views

How can you spot what trigonometric substitution to use in an integral?

So I seem to be coming across a lot of integrals requiring trigonometric substitutions. However, it's becoming tiring, because I have no idea how to spot what substitution should be used - i.e. ...
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0answers
45 views

Which figure provides the greatest change in angle per change in distance? (trigonometry)

I have been having a lively discussion with others about the following: We (myself and others) are using triangulation to measure distance to an object with a linear image sensor (CCD) and a ...
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4answers
605 views

Trigonometry inside a trapezium

I have the following image, and it's asked to find the values of $X$ and $Y$. I've managed to find it using the this idea: Divide the image in two right triangles and let's call the height of the ...
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1answer
71 views

A trigonometric equation with many cases

I have to solve the equation $4\cos^m(x)+3\sin^n(x)=5 $ where $m$ and $n$ are non-negative integers. So here comes my question: The case $m=n=0$ is trivial. The case $m=n=1$ is easy to solve. We ...
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2answers
152 views

solving 3-d coordinates from x, y and z distance

I working with WiFi positioning and i want to know the exact coordinate where I am. Here is the problem..
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4answers
693 views

Precalc - Prove Trig Identity

I'm trying to prove the following problem: $$\dfrac{\sec^2(x)}{\cot(x)} - \tan^3(x) = \tan(x)$$ I thought I had the answer, but the internet says otherwise. Unfortunately I have been unable to find ...
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1answer
85 views

Trigonometric Substitution

I am having trouble with this problem even though everything I did seemed right to me since we went over a similar one in my class. I used the method of setting up a triangle, my hypotenuse is ...
0
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2answers
57 views

Differentiate the function

$$v=\left(\sqrt{x} + {1\over x^{1\over 3}} \right)^2$$ We are working on differentiating functions. This one I have tried everything on and my teacher keeps saying I'm wrong. I'm just not seeing what ...
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1answer
47 views

Finding the angle

The center of the bottom square rotates so that a line cast from the highlighted corner perpendicular to the right face will intersect with the highlighted corner of the other square. I'm trying to ...