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

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2
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3answers
86 views

meaning of powers on trig functions

I always forget this, when a trig function has an exponent does that mean multiply itself or apply itself to the result recursivly? e.g. does $\sin(x)^2=\sin(x)\sin(x)$ or $=\sin(\sin(x))$? What ...
0
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2answers
272 views

What is the difference between the geometric and trigonometric definition of an angle?

I vaguely remember reading that there is a difference between the geometric definition of an angle and the trigonometric definition of an angle. I've tried to search everywhere I can think of but I ...
0
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4answers
166 views

How to solve $a \cos \alpha + b \sin \alpha = c$ for $\alpha$?

I'm solving a physics problem and I came down to solving an equation of the form $$a \cos \alpha + b \sin \alpha = c$$ Can someone help me to solve this? Thanks in advance!
1
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1answer
40 views

Product of projections of equispaced rotating vector

When equal and equi-spaced forces are summed on y-axis what is vector sum? How do we derive the formula $$ \sum_{k=1}^{n-1}\sin\frac{\pi k}{n} = \cot \frac{\pi}{2 n} $$ ( Formula given by ...
0
votes
2answers
297 views

Bearing of a line or a point

Rochelle is 25 miles due south of Rockford,and North Chicago is 65 miles due east of Rockford.Find the bearing of North Chicago From Rochelle. I used Pythagorean theorem in solving this because i ...
2
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2answers
94 views

Geometric proof of $\frac{\sin{60^\circ}}{\sin{40^\circ}}=4\sin{20^\circ}\sin{80^\circ}$

It is well-known that $$\sin{20^\circ}\sin{40^\circ}\sin{80^\circ}=\frac{\sqrt{3}}{8}$$ It follows that $$\frac{\sin{60^\circ}}{\sin{40^\circ}}=4\sin{20^\circ}\sin{80^\circ}$$ But how to prove this by ...
1
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3answers
60 views

Maximum of a trigonometric expression

Maximize $f(x)=4\sin x+48\sin x\cos x+3\cos x+14\sin^2x$ I broke it in 2 parts but then realized they don't have their maxima at the same points. So I am stuck. This is what I did. I wrote it as $$3\...
0
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2answers
111 views

If $\tan x=\sin x/\cos x$ then what is $\tan 3x$ equal to?

Would $\tan 3x$ be equal to $\sin 3x/\cos x$? Or perhaps $\sin 3x/\cos 3x$? Regards, Tom
3
votes
1answer
76 views

Evaluate a rational function of $x,y,z$ given two polynomial equations in $x,y,z$

Let $x, y, z$ be real numbers. Given that $$2x(y^2−1)+2y(x^2−1)=(1+x^2)(1+y^2)$$ and $$4z(1−y^2)+4y(1−z^2)=(1+z^2)(1+y^2)$$ Find the value of the following expression: $$\Bigg(\frac{2x}{1+x^2}−\frac{...
0
votes
1answer
56 views

In an equilateral spherical triangle, show that SecA=1+Seca

Q. In an equilateral spherical triangle, show that $SecA=1+Seca$ So A is the vertex or the angle of the triangle and a is the side of the equilateral spherical triangle. I started off the proof by ...
0
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2answers
60 views

How to solve $ dx = (\sin y + 3 \cos y + x) dy $

How could I solve this? I guess I need to use integration factor, but I do not understand it very well. $ dx = (\sin y + 3 \cos y + x) dy $
3
votes
2answers
56 views

Prove that $ \cos x - \cos y = -2 \sin ( \frac{x-y}{2} ) \sin ( \frac{x+y}{2} ) $

Prove that $ \cos x - \cos y = -2 \sin \left( \frac{x-y}{2} \right) \sin \left( \frac{x+y}{2} \right) $ without knowing cos identity We don't know that $ \cos0 = 1 $ We don't know that $ \cos^2 x + \...
5
votes
3answers
140 views

Integral of root of $\sec x$

How to evaluate the integral $$\int \sqrt{\sec x} \, dx$$ I read that its not defined. But why is it so ? Does it contradict some basic rules ? Please clarify it .
1
vote
1answer
65 views

Finding range of $f(x) = \sin^4 x\tan x + \cos^4 x\cot x$

I got to a certain step and couldn't continue. I can't fully understand the provided solution... $$ f(x) = {\sin^6x+\cos^6x \over \sin x \cos x} = {2-1.5\sin^2 2x \over \sin 2x}$$ Let$$ t=\sin2x, t \...
3
votes
7answers
803 views

Why is the tangent of 22.5 degrees not 1/2?

Sorry for the stupid question, but why is the tangent of 22.5 degrees not 1/2? (Okay... I get that that the tangent of 45 degrees is 1 ("opposite" =1, "adjacent" =1, 1/1 = 1. Cool. I am good with ...
4
votes
4answers
858 views

Prove that the set $ \{\sin(x),\cos(x),\sin(2x),\cos(2x)\}$ is linearly independent.

Prove that the set $ \{\sin(x),\cos(x),\sin(2x),\cos(2x)\}$ is linearly independent. I have tried plugging in values for $x$ but where does this lead? I know what the Wronksian is and it would give ...
1
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1answer
58 views

How to best simplify a chain/product rule with lots of trig functions?

I've found the derivative of the following: $$g(x) = \sec(8x)\tan(5x^9)$$ to be $$g'(x) = 8\sec(8x)\tan(8x)\tan(5x^9) + 45x^8 \sec(8x)(\sec(5x^9))^2$$ I'm aware that the trig identities are ...
3
votes
4answers
97 views

Is it true that $a\cos \alpha \theta = b \cos \beta \theta \implies a=b$ and $\alpha = \beta$

Is it true that $\forall \theta:a\cos \alpha \theta = b \cos \beta \theta \implies a=b$ and $\alpha = \beta$? If so, how do I prove it? I know it isn't true for the sine case since we could have $a=-...
2
votes
2answers
856 views

Formula to best fit a rectangle inside another by scaling

I am very week in Math. I am a web programmer, and usually my work does not involve too much math - its more of putting records into database, pulling out reports, making those fancy web pages etc etc....
0
votes
1answer
47 views

Solve integral analytically

Could this integral be solved analytically? $$ \int_0^a \frac{\cos(\omega t+\phi)}{b - \sin(\omega t + \phi)} \, dt $$ So far I solve it numerical. If anyone could hint substitutions or ...
0
votes
1answer
102 views

conversion of Cartesian to spherical

$$(x,y,z) = (-0.000088,-0.180976,0.930832)$$ For finding $\theta$ I'm using the conversion formula $\theta = \tan^{-1} \dfrac{y}{x}$ I'm getting a value of $1.57031007451$ radians ($89$ degrees). ...
9
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3answers
132 views

How to prove $x=120^\circ$

Let $ABC$ and $CDE$ be equilateral triangles. How to prove that $x=120^\circ$? Thank you.
3
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4answers
310 views

How to solve this trigonometric integral $\int \sin^2t\cos^2t\,dt$?

$$\int \sin^2t\cos^2t\,dt$$ Since both exponent are pair and $\ge 2$, according to my understanding I should use one of these equality to solve : $\sin^2t = \dfrac{1-\cos2t}{2}$ $\cos^2t = \dfrac{1+...
0
votes
2answers
938 views

Finding number of solutions for a trigonometric equation.

How exactly would one go about solving the following math question. I know the answer is 3 but I don't get how to arrive at this answer. A step by step explanation is greatly appreciated. Please keep ...
0
votes
1answer
52 views

Verifying Euler's Formula from trigonometry

I know the proof for the Euler's formula by writing $e^{iz}$ as a Taylor series and arrange the brackets so that I get: $e^{iz}=cos(z) + isin(z)$. But I wonder if there is another way from going from $...
0
votes
2answers
200 views

Trigonometry - how to find angles of triangle within another triangle?

What are the angles of angle 1 and 2? I don't see how any of them could be corresponding angles... The adjacent side of angle 2 is parallel to the hypotenuse of the bigger triangle, just to make it ...
3
votes
2answers
95 views

Evaluating the sum $ \sum_{n = 1}^{44} {\sin^{2}}(n^{\circ}) ~ {\cos^{2}}(n^{\circ}) $.

I want to find the sum $$ {\sin^{2}}(1^{\circ}) ~ {\cos^{2}}(1^{\circ}) + {\sin^{2}}(2^{\circ}) ~ {\cos^{2}}(2^{\circ}) + {\sin^{2}}(3^{\circ}) ~ {\cos^{2}}(3^{\circ}) + \cdots + {\sin^{2}}(44^{\...
-1
votes
1answer
153 views

calculating x y coordinates of a moving object

I'm am trying to figure out how to calculate an expected x y position given starting x y, velocity and angle. If velocity is a maximum of 4 pixels per turn how can i estimate where finishing position ...
2
votes
1answer
122 views

Area of a segment of a circle

If I know the radius of a given circle, the length of the chord of the segment, and the height of the segment can I find the area of the segment?
0
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2answers
55 views

Trigonometric Homogenous Differential Equation

I have the following nonlinear differential equation (I am using $y$ as shorthand $f(x)$): $$\sin(y - y') = y''$$ I have tried the following $$\cos(y - y')(y'-y'') = y'''$$ $$-\sin(y - y')(y'-y'')^...
0
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1answer
95 views

Graphing a Triangle Given the Side Lengths/Angle Measures

Are there any other ways than converting from polar to rectangular coordinates to graph the three points of a triangle on a Cartesian coordinate system if you are given the values of all three side ...
-1
votes
1answer
317 views

Find Rotation Angle between two points on Circle

I am trying to find the rotation angle between two points on a circle in clockwise sense. I have the formula for rotating a point around a circle: $$x'=x\cos\theta+y\sin\theta\\y'=-x\sin\theta+y\cos\...
0
votes
3answers
748 views

Why are the units Radians? Related rates with an angle and $\frac{d\theta}{dt}$

Consider on of those rising balloon related rates Calc problems. Based on the actual problem, you'd label a triangle with a few sides and one of the angles as $\theta$. You set up some trig ...
0
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0answers
428 views

Find Z component of a 3-dimensional vector's magnitude

So, I'm quite confused. I'm currently working with 3-dimensional vectors in an attempt to model an object's local axes in Unity. I already understand how to find the X and Y components of said vectors'...
1
vote
1answer
66 views

Solve 2nd order ordinary differential equation by Laplace transforms and convolution of their inverse functions. (5.6-40)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
0
votes
2answers
142 views

Critical Points of $f(x) = \sin(3x)$ — Help!

The problem is asking me to find the Critical Points of : $$f(x) = \sin(3x)$$ on the closed interval: $$[\frac{-\pi}{4},\, \frac {\pi}{3}]$$ I know that $f'= 3\cos(3x)$. the problem I seem to be ...
0
votes
1answer
70 views

How to solve arcsin(x1/x2 * sin(theta)

$$f = \arcsin\left(\frac {x_1}{x_2} * \sin(\theta)\right)$$ I think this is a simple question, but is there an easy way of solving this equation? Is there a way to solve without having to sin both ...
1
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2answers
51 views

Closed-form solution for 3D rotation angles given pre- and post-image

I'm working on some math involving a pinhole camera model, and I've run in to the following problem: given only $x$, $y$, $z$, $A$, $B$, and $C$, I need to solve for the angles $\theta$ and $\phi$ in ...
1
vote
1answer
191 views

Proving the Sine Rule with one line.

Working on a general proof of the Law of Sines for ALL Euclidean triangles. Right triangles are easy. Acute triangles are just two proofs of the right triangle. But this is not sufficient for me. I ...
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2answers
50 views

How do I solve $\sqrt {1/2} = \cos (a)- \sin (a)?$

I have no idea how to do it... I need to find $\alpha$ in this equation $$\sqrt{1/2} = \cos(\alpha) - \sin(\alpha)$$ Thanks for your help
0
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3answers
94 views

Simplify $(\cos \theta \div\sec \theta) + (\sin \theta \div\csc \theta)$

Please show how to simplify these types of expressions without using a calculator. I'm new to trigonometry and I don't know how to simplify these expressions. $$\frac{\cos \theta}{\sec \theta} + \...
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2answers
110 views

Trig Identity / Pythagorean Theorem confusion?

I run into a problem when I'm trying to prove how $\tan^2x+1 = \sec^2x$, and $1+\cot^2x=\csc^2x$ I understand that $\sin^2x+\cos^2x = 1$. (To my understanding 1 is the Hypotenuse, please correct me ...
0
votes
0answers
41 views

Convex Constraint on Sine Wave Simularity

So lets say you have a vector X = [x1 x2 x3 ..... xn] You want to optimize a cost function over X. However you want to constrain the vector X to look like a sine wave. Say you can parameterize a ...
2
votes
1answer
62 views

What's wrong in this integral

Where's the mistake in this solution? $$\int \tan^3x\sec^2xdx = \int \frac{\sin^3x}{\cos^5x}dx=\int\frac{\sin x(1-\cos²x)}{\cos^5x}dx=\int\frac{u^2-1}{u^5}du$$$$=\frac{1}{4u^4}-\frac{1}{2u^2}+C=\frac{...
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3answers
90 views

Basic trigonometry with popsicle sticks

I am trying to make a model out of popsicle sticks of half a cycle of a sine wave. It is easy to do it in a crude way. Just stack up the sticks, draw half a cycle of a sine wave on the sticks, then ...
0
votes
2answers
97 views

solve integral of $\frac{\sin (ax)}{ \sin(x)}$

I would like to find the area under the curve of $\frac{\sin(ax/2)}{\sin(x/2)}$, namely between the first zero crossing on the left and right: $$ \int_{-\frac{2\pi}{a}}^{\frac{2\pi}{a}} \frac{\sin(\...
3
votes
1answer
117 views

geometric proof of $2\cos{A}\cos{B}=\cos{(A+B)}+\cos{(A-B)}$

I have seen geometric proof of identities $$\cos{(A+B)}=\cos{A}\cos{B}-\sin{A}\sin{B}$$ and $$\cos{(A-B)}=\cos{A}\cos{B}+\sin{A}\sin{B}$$ By adding two equation, $$2\cos{A}\cos{B}=\cos{(A+B)}+\cos{(A-...
1
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1answer
406 views

How do you find the radius of an arc given arc length and height?

. Please excuse the poor drawing, but how would you go about solving this problem? Known: Arc length Height (I'm not sure what the proper term for this parameter is.) Unknown: Sagitta Chord ...
0
votes
1answer
34 views

$\tan(\arg(1+ix))=x$

Let $\log$ be the inverse function to $e^x$ with branch cut on the negative real axis, and define $\arg z=\Im\log(z)$. The basic relation between the $\arg$ function and the arctangent is $\arg(x+iy)=\...
0
votes
2answers
37 views

Stuck on Circles question on tangents

Here the length of AO is equal to diameter of circle. AB and AC are tangents from A. The triangle ABC has to be proved equilateral. I put it in geogebra and it was indeed equilateral. I can't find ...