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
157 views

Integral of $\log(\sin(x)) \tan(x)$

I would like to see a direct proof of the integral $$\int_0^{\pi/2} \log(\sin(x)) \tan(x) \, \mathrm{d}x = -\frac{\pi^2}{24}.$$ I arrived at this integral while trying different ways to evaluate $\...
1
vote
2answers
40 views

Show that $e^\mathbf{iA} + e^\mathbf{iB} = 2e^\frac{i(A+B)}{2}\cos(\frac{A-B}{2})$

Where $i=\sqrt{-1}$ and $A,B\in \mathbb{R}$ are constants. I've tried already with Euler's formula, but cannot prove the equation above. Best Regards, Thanks.
5
votes
0answers
170 views

resizing rectangle within triangle

Imagine I have a parking lot that changes in width and length and in number of levels, and all of the levels need to be visible to a cctv camera at a fixed position, and I would want the camera to see ...
0
votes
1answer
102 views

express tan(x) as a power series using maclauran's theorem. [duplicate]

the theorem states that if f(x) can be expanded as a power series for a given range of values of x then: $$f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+\frac{x^3}{3!}f'''(0)+\cdots$$ ($'$ means derivative) ...
2
votes
5answers
421 views

How to solve integration of a product of an exponential and a trigonometric function?

Preparing for the exam I bumped into this integral and I just can't get hold on it. It's an integration of a product of an exponential and a trigonometric function. It's going in an endless loop for ...
3
votes
2answers
124 views

Product of repeated cosec.

$$P = \prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$$ I realize that there must be some sort of trick in this. $$P = \csc^2(1)\csc^2(3).....\csc^2(89) = \frac{1}{\sin^2(1)\sin^2(3)....\sin^2(89)}$$ I ...
0
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1answer
48 views

Trigonometric identities: need a simple product form

Is there is a simple product form for $\cos(at) + \cos(bt+\alpha)$??
0
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3answers
56 views

Can anyone explain how this is concluded?

$$z=\sqrt{x^2-y^2}\tan{z\over \sqrt{x^2+y^2}}.$$ $$\tan{z\over \sqrt{x^2+y^2}}={z \over \sqrt{x^2-y^2}}$$ $$\cos^{-2}{z \over \sqrt{x^2-y^2}}={z^2 \over {x^2-y^2}}+1$$ I\m aware that the derivative of ...
0
votes
0answers
70 views

Osborne's rule for hyperbolic functions?

I am confused as to why you only change the sign for powers of sine that are 4n+2. As I understand, $sin(i\theta)=isinh(\theta)$ $sin^2(i\theta)=-sinh^2(\theta)$ $sin^3(i\theta)=-isinh^3(\theta)$ ...
1
vote
0answers
166 views

How to find XYZ Coordinates of the Major and Minor Axis end-points in an orbit?

To give some context to this problem, I'm attempting to convert an orbit into a Cubic Bezier Spline, by first plotting four points around the Orbits Ellipse and then computing the Control points of ...
1
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2answers
96 views

Evaluating: $\int^\pi_0 \cos(a(\sin(x))e^{a\cos(x)} dx$

How can I evaluate the following integral? I don't know how to start... $$\int^\pi_0 \cos(a(\sin(x))e^{a\cos(x)} dx$$ Where $a \ne0, a\in R $
0
votes
1answer
45 views

Trigonomnetric equality involving tg

I need help proving the following identity. $\tan^210^\circ+\tan^250^\circ+\tan^270^\circ=9$. I am not sure if it is even true.
0
votes
2answers
51 views

Is it true that $\int_{0}^{1}(1+x^{2})^{-1/2} = \log (1 + \sqrt{2})$?

Since $$D^{-1} (1 + x^{2})^{-1/2} = \sinh ^{-1} (x) + C,$$ is it true that $$\sinh ^{-1} x + C \big|_{0}^{1} = \log (1 + \sqrt{2})?$$ What relates $\sinh^{-1}(\cdot )$ to $\log(\cdot )$? Here $D^{-1}$...
0
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0answers
28 views

How to calculate the product of $\sin\frac{\pi}{n}$ to $\sin\frac{(n-1)\pi}{n}$ [duplicate]

How to calculate $\sin\frac{\pi}{n}\sin\frac{2\pi}{n}\cdots\sin\frac{(n-1)\pi}{n}$?
1
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1answer
82 views

Definition and characterization of trigonometric functions

I was wondering, throughout one's mathematical education, one is introduced to a various equivalent definitions of sine and cosine, beginning from that of right triangle, then unit circle, then series,...
1
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1answer
348 views

Finding the the radius of an arc between two lines.

Essentially, I'm trying to recreate the 'Mini' logo in a program. I know that one line extends from the centre 50. There is a line 25 below it, which extends 25 from the centre. I'm trying to find ...
0
votes
1answer
71 views

Whats the difference between 5π/3 vs -π/3?

The question says: Evaluate arctan( -√3) and I got 5π/3, but the back of the book says the answers is -π/3. Isn't that basically the same thing?
2
votes
3answers
94 views

Solve Trig Equations with Different Arguments

How are systems such as solved analytically: $$0.2187=\cos x-\cos y$$ $$-0.469=\sin x-\sin y$$ Wolfram alpha gives the analytical solution so there has to be away, but I cant figure it out. From ...
1
vote
2answers
64 views

How to turn sin(arcsinh(x)) into algebraic form?

How can I turn $\sin({\sinh^{-1}{x}})$ into explicit algebraic form ? I've tried to plug in $\sinh^{-1}{x}$ into sine's exponential form $\frac{e^{ix} - e^{-ix}}{2i}$, but then I cannot think of any ...
0
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1answer
632 views

Writing 1+1= 2 in a complicated way

I am learning Unit Circle at the moment and I am using this source as an education tool Trigonometry: Unit Circle (Starts at 20:00). The author solves these simple equations like below: ...
2
votes
2answers
71 views

Find a formula for $\sin(3a)$ and use to calculate $\sin(π/3)$ and $\cos(π/3)$?

Problem: Find a formula for $\sin(3a)$ in terms of $\sin(a)$ and $\cos(a)$. Use this to calculate $\sin(π/3)$ and $\cos(π/3)$. My attempt: $$\sin(3a) = \sin(2a + a) = \sin(2a)\cos(a) + \cos(2a)\...
0
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1answer
98 views

Area Enclosed By Tangents and Major Arc

A chord AB of a circle of radius $5a$ is of length $3a$. The tangents to the circle at A and B meet at T. Find the area enclosed by TA, TB and the major arc AB. I keep getting an answer of $81.20a^2$ ...
1
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0answers
48 views

Prove $\frac{d}{dx}[\csc^{[-1]}{x} = \frac{-1}{|x|\sqrt{x^2-1}}$

Show that the $\frac{d}{dx}[\csc^{[-1]}{x} = \frac{-1}{|x|\sqrt{x^2-1}}$. $$ \begin{align*} y &= \csc^{[-1]}{x} \\ \csc{y} &= x \\ \frac{1}{\sin{y}} &= x \\ \frac{1}{x} &= \sin{y}...
1
vote
2answers
104 views

Integral solutions to inverse trigonometric equation.

The original question is this : I have to find number of ordered pairs of integral solutions to this : $$\tan^{-1}{x} + \cos^{-1}{\frac{y}{\sqrt{1+y^2}}} = \sin^{-1}{\frac{3}{\sqrt{10}}} $$ I ...
0
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0answers
66 views

Trigonometry - word problem - functions cosθ = adjacent/hypotenuse

I'm not sure what trigonometric equation I should use for this problem: At a certain instant, a ship was 5km south of a lighthouse. The ship was travelling westward and after 30 minutes it's bearing ...
1
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2answers
41 views

Trigonometric equation; finding all solutions

I'm having a hard time with trigonometric equations. I need to find all solutions to the following equation: $$4\sin^2 θ=3$$ Any help will be appreciated.
0
votes
3answers
39 views

Prove Trigonometric Identitiy

So we just have started to learn Trigonometric and so far we have learn these basic dentitiy $$ \sin^2{a} + \cos^2{a} = 1 $$ $$\tan{a}=\frac{\sin{a}}{\cos{a}} $$ ...
4
votes
2answers
98 views

Find the values of m and n(Trigononetry in series)

$$\sin ^6(1)+\sin ^6(2)+... ...+\sin ^6(89)=\frac{m}{n}$$ Find $\frac{m}{n}$ in its simplest form , and hence find both values. (All angles are in degree) I've no idea how to start to solve this ...
2
votes
1answer
48 views

estimation of a unit circle - how to show a relationship

It has been eons since I've done any trigonometry, but I just can't prove how this following relationship holds for $n = 4, 8, 16, 32, \dots$ The relation is: $$ 2 \biggl( \! \frac{A_{2n}}{n} \! \...
3
votes
1answer
313 views

Calculating angle on ellipse

This is a really basic question, yet I can't remember my old geometry classes nor could I find an answer via google. Given a circle "tilted" at angle a to the horizontal plane, and given angle b ...
0
votes
3answers
53 views

Cosine and Sine Angle Addition Intuition [duplicate]

I am lacking in understanding in the cosine and sine angle addition formulas. I have seen several questions similar to this but I have not seen an answer that explains how this conclusion can be ...
0
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5answers
20k views

how do you type sec^2(0) on a calculator?

I press cos^-1 then ^2 then brack (o) but then it comes up with syntax error
0
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0answers
35 views

Pre Calculus Simplifying Trigononmetric Expressions

I am trying to simplify this trigonometric expression in terms of sine and cosine. The equation that I have to simplify is 7 cos t tan t.
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1answer
38 views

How does one derive the trigonometric parametric equations for a circle w/o trigonometry?

This question seems simple to me, but I can't figure it out. I know the parameterizations for a circle probably better than I know the back of my hand. I know why, geometrically, it works. I can ...
0
votes
1answer
46 views

How to find vector that is in the same plane and perperndicular to a side of a triangle?

Suppose we have a triangle ABC. How does one find a vector E that is in the same plane as the triangle and is perpendicular to segment BC? I know that a dot product is 0 when two vectors are ...
0
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2answers
31 views

Explain how solution got $c_1$ and $c_2$

Can someone explain how the solution manual got $c_1$ and $c_2$ in this:
-1
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3answers
43 views

Arc- trigonometric functions

This is, perhaps, too simple of a question for here, but I'd love it if someone helped me out. I'm just learning about arc- trigonometric functions (because I failed both calculus exams) and my ...
0
votes
1answer
114 views

Maximum and minimum of this complex periodic function

I came up with this function by using fourier transform. My only problem is how to get the amplitude of this function. Im planning to get the difference between their maxima and minima. I get its ...
1
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1answer
144 views

How to evaluate $\int \sqrt{\sin^{-1}(\sqrt{\phi})} d\phi $?

How do I go about solving the below integral? $$I_1=\int \sqrt{\sin^{-1}(\sqrt{\phi})} d\phi $$ Background: I came across the simpler version of this, which required me to evaluate: $$\int\sin^{-1}...
0
votes
1answer
34 views

Integral of fraction of product of trigonometric functions

I have encountered a problem with integrating $$ \int \frac{1}{\sin(4x) \sin(2x)}dx $$ I thought about substituting for $2x$, but I don't know, what to do after the substitution.
3
votes
1answer
30 views

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x?

If $\sin^{-1}\frac{2a}{1+a^2}-\cos^{-1}\frac{1-b^2}{1+b^2}=\tan^{-1}\frac{2x}{1-x^2}$ then what is value of x? Solution $\tan^{-1}x=\tan^{-1}a-\tan^{-1}b=\tan^{-1}\frac{a-b}{1+ab}$ x=$\frac{a-b}{1+ab}...
2
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0answers
27 views

How do you integrate $2\sqrt{-x^2+1}dx$ step by step? [duplicate]

How do you integrate $$2\sqrt{-x^2+1}dx$$ to achieve the answer of $$\sqrt{-x^2+1}*x+\arcsin(x)$$ like Wolfram Alpha does? Wolfram showed no steps.
6
votes
1answer
69 views

Tan inverse summation

$$S=\sum\limits_{i=1}^{4}\tan^{-1} x_i$$ How to simplify this ? I think I will have to use this : but it looks too long a method . Is there a method or symmetrical way which yields ...
5
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0answers
115 views

Graphs of interesting integrals of the form: $\int \sin^a(x^a)\cos^a(x^a)$

Here are a few graphs of the form:- $$\int \sin^a(x^a)\cos^a(x^a)dx$$ Where $a$ is an even, positive integer. $a = 2$ $a = 4$ $a = 6$ Now, a few graphs of the form:- $$\int \sin^a(...
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0answers
27 views

Rearranging equations using hyperbolic transcendental functions [duplicate]

I have tried and tried but cannot for the life of me see how one equation follows onto the other... can anybody help?? $$\Omega(\theta)=-b.\coth(\operatorname{arsinh}(\exp a\theta . \sinh(c_0)))$$ $$...
0
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0answers
54 views

Constructing a pentagon from a circle

To my understanding you can create any regular N sided shape by using a circle. I decided to give this an attempt from the equations/ formulas given from the internet. Just a side note - is it ...
1
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3answers
3k views

How do you find the sine of the angle between two vectors

This is a homework question and do not know what the sine of the angle between two vectors is, I think it may be the vector created by connecting the tips of the two vectors but I am not sure. So How ...
1
vote
5answers
61 views

Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $

Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $ Can someone please explain to me how to go about doing this?
0
votes
1answer
29 views

Where can I find the proof of trigonometrical functions for complementary angles.

For example sin a=cos(90-a), sin a=cos(90-a)... I found it in one book, but I can't find it anywhere. Can you tell me a website where I can find the proof?
1
vote
1answer
36 views

how can I get the value of x

I've trying to get the x value of the : sin(x+90) / sin(x+210) = 0.2222 is there a way to solve this equation, not numerically !