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

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832 views

To the nearest degree, the solution of the equation $3\space\sin^2(x ) - 2\sin(x) = 1$ in the interval $[270^\circ, \space 360^\circ]$ is $x = ?$

This is the problem I have in my trigonometry packet. Can anyone give it a try? I factored by doing: $$\sin x(3\sin x - 2) = 1$$ Then divided $\sin x$: $$ 3\sin x - 2 = \frac{1}{\sin x}$$ But ...
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1answer
6k views

Solving the ArcTan of an angle (Radians) by hand?

How do you solve $\arctan(n)$ to radians by hand? I. e. $\arctan(1)$ >> process >> $\pi/4$ ::EDIT:: I have this taylor expansion that allows me to calculate an approximate value for arctan, but am ...
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2answers
333 views

Verifying - Trigonometry Homework

I have a major problem with verifying a Trigonometric identity. (My teacher couldn't really get it, so I would like to find it just in case it appears on a test) The problem goes like this: ...
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1answer
240 views

Solving the equation $(3x + \sin x) \cos x = -3$

I got stuck on solving the following equation. I try all the identities, but hopeless. Do you have any suggestion? $$(3x + \sin x) \cos x = -3, \quad\pi/2 \leq x \leq \pi$$ Solve for $x$. ...
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1answer
240 views

How do I calculate if two moving objects will hit each other in three dimensions?

Position, speed and heading are known, and the size and shape of these objects are also known. Gravity is irrelevant, so these objects will be moving in straight lines. I tagged this with geometry ...
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0answers
106 views

Functional inverse of $(a + b\sin\theta)^2\tan\theta$

So, I revisited to the situation involved in my previous question, with the intent of generalizing it to any two masses and charges. When I started going through the model again, beginning with the ...
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3answers
1k views

What is the periodicity of the function $\sin(ax) \cos(bx)$ where $a$ and $b$ are rationals?

So, I have a general question first. What happens to the periodicity when we multiply two periodic trig functions with one another ? The next one is very specific, what is the period of the function ...
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1answer
358 views

Integration Problem Proof ($\sin x$)

Problem: Integration of $\displaystyle\int_{-1}^1 {\sin x\over 1+x^2} \; dx = 0 $ (according to WolframAlpha Definite Integral Calculator) But I don't understand how. I tried to prove using ...
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2answers
595 views

Show $f(x) = x^3 - \sin^2{x} \tan{x} < 0$ on $(0, \frac{\pi}{2})$

This is the last of a homework problem set from Principles of Mathematical Analysis (Ch. 8 #18(a)) that I've been working/stuck on for a few days: Define $f(x) = x^3 - \sin^2{x}\tan{x}.$ Find ...
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3answers
4k views

How to Derive a Double Angle Identity.

How does one derive the following two identities: $$\begin{align*} \cos 2\theta &= 1-2\sin^2\theta\\ \sin 2\theta &= 2\sin\theta\cos\theta \end{align*}$$
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2answers
62 views

Find the Cartestian form of $6 - 7i$ rotated anticlockwise through $\frac{3\pi}{4}$ about the origin

Find the cartestian form of $6 - 7i$ rotated anticlockwise through $\frac{3\pi}{4}$ about the origin I realize that I am going to be doing something like: $\sqrt{85}e^{i\alpha}.e^{i\frac{3\pi}{4}}$ ...
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1answer
166 views

Evaluate $\sin(\frac{\pi}{8})$ and $\cos(\frac{\pi}{8})$

Evaluate $\sin(\frac{\pi}{8})$ and $\cos(\frac{\pi}{8})$ I was just wondering what I am doing wrong, as I don't seem to be arriving at the correct answer for $\sin(\frac{\pi}{8})$ What I did: ...
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2answers
212 views

rational triangles and cosines

I've recently started to try working on exercises from this book on Diophantine equations before I need to return it to the library. This one has me slightly stumped. It asks to show that the cosine ...
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1answer
126 views

Given an exact velocity and a “velocity range”, what is the relative velocity range?

I'm trying to calculate the relative velocity ($V_R$) between an exact velocity ($V_0$) and a velocity range ($V_1$). The exact velocity ($V_0$) is represented simply by ($course$, $speed$). The ...
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1answer
503 views

Finding the hypotenuse of Right Angle Triangle

The perimeter of a square is 48 inches. What would be the length, in inches, of its diagonal?
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3answers
702 views

Law of Sines will give a unique solution iff a > b?

Given a triangle ABC, with known sides a=BC and b=AC, and known angle A, we wish to find angle B. This is a typical application of the Sine Rule (Law of Sines). In some circumstances, the sine rule ...
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5answers
6k views

How to expand $\cos nx$ with $\cos x$?

Multiple Angle Identities: How to expand $\cos nx$ with $\cos x$, such as $$\cos10x=512(\cos x)^{10}-1280(\cos x)^8+1120(\cos x)^6-400(\cos x)^4+50(\cos x)^2-1$$ See a list of trigonometric ...
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1answer
1k views

$\sin(2\arccos(x))$, please help me understand how to do these kind of problems.

We need to be able to transform this equation to get rid of the trig functions. To better explain this, this is how the problem before this one was done. (I checked the answer, i got this one right.) ...
5
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2answers
775 views

Power-reduction formula

According to the Power-reduction formula, one can interchange between $\cos(x)^n$ and $\cos(nx)$ like the following: $$ \cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} ...
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2answers
1k views

Linear Algebra - Linear Transformations

Let $V$ be the space spanned by the two functions $\cos(t)$ and $\sin(t)$. Find the matrix $A$ of the linear transformation $T(f(t)) = f''(t) + 7f'(t) + 4f(t)$ from $V$ into itself with ...
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6answers
2k views

If we define $\sin x$ as series, how can we obtain the geometric meaning of $\sin x$?

In Terry Tao's textbook Analysis, he defines $\sin x$ as below: Define rational numbers Define Cauchy sequences of rational numbers, and equivalence of Cauchy sequences Define reals as the ...
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5answers
4k views

how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$

$$\sin x<x\,(0<x<\frac{\pi}{2})$$ In most textbook,to prove this inequality is based on geometry illustration(draw a circle, compare arc length and chord ),but I think that strict prove ...
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1answer
123 views

$C^2=A^2+B^2-2AB \cdot\cos(c)$ getting a different answer than creating a third triangle with the distance formula?

I have the following triangle: The side going up has a length of 96, the side going down has a length of 112. The angle closest to the center is 91 degrees broken up into 62 and 29 degrees from ...
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2answers
2k views

How to simplfiy $ \cos^{2}(\theta) \sin(\theta) $ to terms of $\sin$ or $\cos$ only?

Is it possible to simplify $$ \cos^{2}(\theta) \sin(\theta) $$ to terms of only $\sin$ or $\cos$? I need to simplify this to take the integral.
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1answer
167 views

Converting pixels to millimeters

I am trying to write a program that can show the worth of a coin in an image. I have already took care of finding radius of coins in pixel. But I am stuck here: if the image has been taken from a ...
6
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4answers
289 views

Let $a,b$ be positive real numbers. Prove $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}} \geq \frac{2}{\sqrt{1+ab}}$

Let $a,b$ be positive real numbers. Prove $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}} \geq \frac{2}{\sqrt{1+ab}}$ if either $(1) 0 \leq a,b \leq 1$ OR $(2) ab \geq 3$ Since this question ...
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3answers
213 views

What was this theorem called

Back at the university we have proven (lot of work) that if $$S(X)C(Y)+C(X)S(Y) = S(X+Y)$$ and $$C(X)C(Y)-S(X)S(Y) = C(X+Y)$$ then $S(X)$ is $\sin(x)$ and $C(X)$ is $\cos(x)$ (or constant $0$, meh). ...
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3answers
837 views

If $\cos\theta = \sin\theta$, then what is $\cos2\theta$?

If $\cos\theta = \sin\theta$, then what is $\cos2\theta$? I am stuck on this problem, please help.
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2answers
502 views

Is the Maclaurin series expansion of $\sin x$ related to the inclusion-exclusion principle?

When I see the alternating signs in the infinite series expansion of $\sin x$, I'm reminded of the inclusion-exclusion principle. Could there be any way to visualize it in such a way? Also, is there ...
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2answers
6k views

Determining angle to cut two boards

I am building a small roof and I need to determine how to find the distance/angle in that I need to cut two 2x4's in order to make them fit together perfectly. I've drawn a crude image below to ...
3
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2answers
883 views

Showing the identity: $\tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha = \cot \alpha − 8 \cot 8\alpha$

My knowledge of trigonometry are still insufficient to resolve this problem. Any help would be greatly appreciated. $$\tan \alpha + 2 \tan 2\alpha + 4 \tan 4\alpha = \cot \alpha − 8 \cot 8\alpha$$
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2answers
973 views

How to calculate angles and X,Y coordinates for drawing a hand of playing cards on a canvas

Scenario: I'm programming a module to draw a deck of cards on a canvas as though they were being held by a person. Edit: I've cleaned up the question as best I can to be clearer. What I'm looking ...
2
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1answer
199 views

Finding launch angle help.

A smooth spherical object (the first object) is projected horizontally from a vertical height of $26.83$ metres above horizontal ground with a launch speed of $23.44\textrm{ ms}^{-1}$. A second ...
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1answer
90 views

A problem with a trigonometric equation

I'm trying to solve this problem but I can't figure how. Can you help me? $$A=\frac{\sin \alpha+\cos(3\pi/2-\alpha)+\tan(5\pi+\alpha)}{\csc(2\pi-\alpha)+\sin(5\pi/2+\alpha)}$$ If $\tan \alpha=-2/3$ ...
4
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2answers
328 views

Prove that $\displaystyle{\frac{\cos A+\cos B - \cos C}{\sin A+\sin B - \sin C}} \geq -\frac{\sqrt{3}}{3}$

All the angles in a triangle $A,B,$ and $C$ are less than $120^{o}$ Prove that $\displaystyle{\frac{\cos A+\cos B - \cos C}{\sin A+\sin B - \sin C}} \geq -\frac{\sqrt{3}}{3}$
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1answer
512 views

Generalized Laws of Cosines and Sines

I wonder the "laws of sines and cosines" in the two cases below and how to derive them. (or any related sources) (i) For geodesic triangles on a sphere of radius $R>0$. (so constant curvature ...
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0answers
1k views

Direction ratio of a vector in 3d?

Suppose I have a vector whose starting end is the origin & the other end can be in any of the 8 quadrants (in 3d). I can easily get direction ratios for the vector & also know in which ...
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1answer
347 views

How did they get this solution?

I'm looking at the solution manual and I have no idea how they convert this. $$ k \cos {3\theta} = k [4\cos^{3} {\theta} - 3 \cos{\theta}] = k[\alpha P_{3}(\cos\theta) + \beta P_{1}\cos{\theta}] $$ ...
5
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1answer
488 views

Evaluate $\int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx$

How to evaluate $$ \int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx $$ I don't know how to deal with it.
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1answer
116 views

A computation with vectors

I'm going through these lecture notes, and I don't understand how one of the example problems was solved. Can anyone show me a step by step solution? Question: Suppose a Canada goose is flying ...
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2answers
109 views

What is $\cos^2(x)$ in relation to $\sin(x)$?

How would you find $\cos^2(x)$ in terms of $\sin(x)$? Please explain clearly, I am a high school student.
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3answers
145 views

Integral of a trigonometric function [duplicate]

Possible Duplicate: Evaluating $\int P(\sin x, \cos x) \text{d}x$ How do I integrate the following function? $$\frac{\sin 2x}{(1 + \cos^2x)^2}?$$ Thanks.
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4answers
7k views

What is $\sin^2(x)$ equal to?

Let's take the sine of $30^\circ$ which is one-half. If you take $\sin^2(30^\circ)$, would that just be the sine of $900$? Or would it be equal to one-quarter, or would it be equal to something ...
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1answer
69 views

Computing angle

See the drawing for the situation. Given lenghts a, b and c and also L, but k and angle alpha are unknown. How to compute this angle alpha? I know it is possible to compute if we first compute k in ...
3
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1answer
567 views

Distance between two gears surrounded by a known-length belt

This question is very similar (but not identical) to this one: Finding the distance between two gears (actually, we are trying to solve it on Bicycle Exchange: ...
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2answers
242 views

Find the length (in cm.) of the hypotenuse?

A right angled triangle has sides of length X, Y and Z (all lengths in cm.). It is known that Z is the length of the longest side. The lengths of the other two sides satisfy the inequality ...
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1answer
3k views

Reciprocal of sin(x)

What is the reciprocal of sin(x), or what is 1/sin(x) equal to in terms of trigonometric ratios? Please answer simply, as I am a high school student, not a mathematician.
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1answer
1k views

Finding the position of a point after rotation: Why is my result incorrect

I am attempting to calculate the position of a point after it has been rotated I have been using an algorithm but I am getting incorrect values which makes me think I am using the incorrect algorithm ...
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1answer
474 views

Where is zero degrees on a graph

I am using the following formula to calculate the position of a point after rotation in my web application. x' = xcos(0) - ysin(0) y' = xsin(0) + ycos(0) But ...
2
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2answers
646 views

Calculated rotated point coordinate: is my solution correct

Is my calculation correct for this rotation around a point? A point a(-19.94,392.11) is rotated -49.45 degrees, what is the new coordinates of point a? My solution: ...