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

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1answer
13 views

Rotating an object around any origin

I want to extend my program that generates PDF and I need like to rotate an object (for example -30deg clockwise): 1: original 2: rotated object (origin is bottom left) The first problem is, that ...
0
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1answer
11 views

Find catesian coordinate of T-point $P(-\frac{65\pi}{2}) $

Find the Cartesian coordinates of T-point $P(-\frac{65\pi}{2}) $. It is easy when there is no negative sign. I don' t know how to do with negative sign.
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2answers
42 views

Why must closest approach occur when relative velocity is perpendicular to motion?

The first part i) I can solve correctly, but I need some advice and intuition on how to solve the second part ii). Here is the mark-scheme for the question: But for part ii) I do not understand ...
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0answers
47 views

Would this thinking about the dot product hold?

Background today I completed the chapter on the dot product of vectors. But in trying to figure out exactly what the dot product is. I came to the conclusion that it can be interpreted as the length ...
1
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1answer
31 views

Trigonometric inequality question [on hold]

Let $0 < A < \frac {\pi}{2}$ and $0 < B < \frac {\pi}{2}$. (a) prove that $\sec^2 A + \csc^2 A \cdot \csc^2 B \cdot \sec^2 B \geq 9.$ (b) determine values of $\sec A$ and $\sec B$ when ...
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5answers
47 views

If angular velocity $\omega=\sqrt{\frac{3g\sin\theta}{2a}}$ can I find angular acceleration $\alpha$ by differentiating $\omega$?

It was my understanding that angular acceleration is the derivative of angular velocity. The reason I ask is Thanks.
2
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5answers
96 views

Showing the $n$-th derivative of $\cos x$ by induction

I was asked to show that the $n$-th derivative of $\cos x$ is $\cos(\frac{n\pi}{2} + x)$. My progress : By induction, I proved it was true for $n=1$. Then I assumed it was true for $n = k$ so now I ...
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1answer
23 views

How to find slope on line that known only point and angle

How to find slope on line that known only point and angle Image will describe more clearly I'm wont to find the orange line slope to find point on it ( b , c , d ) suppose that A and angle are ...
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4answers
55 views

Why $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$?

I am working out a Fourier Series problem and I saw that the suggested solution used $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$ to simply the expressions while finding the Fourier Coefficients ...
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2answers
38 views

How do I change $\cos(\frac{\theta}{2})$ into $\cos(\theta)$ in an equation?

Just give me an example. eg. $\cos(\frac{\theta}{2})=\frac{1}{2}.$ I want to make $\cos(\frac{\theta}{2})$ become $\cos(\theta)$. Thanks.
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2answers
30 views

simplification of the area of a hyperbolic circle (BONOLA, S 53)

I'm trying to understand the S-53 of "Non-Euclidean Geometry" (BONOLA, R.) in which the formula for the area of a circle of radius r: $$2\pi k^2(\cosh\frac rk -1)$$ is somehow reduced by only applying ...
7
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3answers
98 views

Evaluate trig functions without a calculator

My precalculus test asked me to determine which was greater: $\tan (53)$ or $\sec (38)$. I looked at it like this, but it seems so close that it's difficult to imagine that they would ask this: ...
9
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1answer
63 views

How were the sine, cosine and tangent tables originally calculated?

As I understand it... ahem... the (cosine, sine) vector was calculated for (30 degrees, PI/6), (45 degrees, PI/4) and (60 degrees, PI/3) angles etcetera, however, I would like know the original ...
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2answers
50 views

Induction Proof of trig inequality

This is for a course, I don't want the answer just a prod in the right direction! I've got a problem that states let n be an integer such that $$n\gt0$$ $$\text{Prove: }\sum_{k=0}^n |\cos k| \ge ...
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4answers
90 views

Find $\sec \theta + \tan \theta$.

If $\tan \theta=x-\frac{1}{x}$, find $\sec \theta + \tan \theta$. This was the question ask in my unit test. My Efforts: $\tan^2 \theta=(x-\frac{1}{x})^2$ $\tan^2 \theta= (\frac {x^2-1}{x})^2$ ...
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2answers
21 views

Finding Principale period of $\cos$ function

Find principle period of $3\cos (2x-3)$. Today I have learned about principle period of various trigonometric function. I know that principle period of cos is $2 \pi$. Please someone can help me ...
3
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2answers
53 views

can't seem to understand $\sin{\theta} = y$ on a unit circle

So I've been working very hard on my trigonometry on khan academy. However I'm constantly getting stumped by one type of question in particular. There is some fundamental flaw in my understanding. ...
3
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1answer
55 views

Nonsensical result in the midst of calculating an integral via substitution.

I was just calculating an integral via a trigonometric substitution and ended up with $\color{red}{ \text{something pretty nonsensical} }$ but $\color{blue}{ \text{reversing the substitution} }$ ...
5
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1answer
47 views

To find a trigonometric limit without Wallis' integrals

What is the limit $$ \lambda =\lim\limits_{n \to \infty}{n\int_0^{\frac{\pi}{2}}(\sin x)^{2n} dx}$$ I would like to find it without Wallis' integral formula: I mean without evaluating the closed ...
1
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2answers
38 views

Naive proof that $\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$ [duplicate]

As part of a larger proof, I must show that: $$\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$$ I have thought about this but can't figure out how to get my hands on the value since I don't know any ...
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2answers
75 views

Proof of trigonometric identity $\sin(A+B)=\sin A\cos B + \cos A\sin B$

All the proofs I've seen are geometrical, assuming that $A+B$ is less than $90$ degrees. How can you prove this identity for $A+B$ greater than $90$ degrees, or more generally, any arbitrary value?
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5answers
49 views

Solve for $\theta$: $a = b\tan\theta - \frac{c}{\cos\theta}$

This question was initially posted on SO (Link). I'm not sure the answer given there was correct. I cannot get the results from those expressions to match my CAD model. The title pretty much sums ...
2
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3answers
87 views

Trigonometric equation $2\sin x+\cos x+1=0$

I have to calculate $\dfrac{d}{dx}\dfrac{1+\cos x}{2+\sin x}=0$. I have already simplified to: $2\sin x+\cos x+1=0$, but I have no idea how to go further.. Could someone give a hint?
2
votes
1answer
34 views

At the instant of release of an object from rest. Is the only force that can act its weight? [on hold]

This is the third question from a mechanics exam past paper: I can do parts i) and ii) but for iii) in finding the angular acceleration, i used $C=I\alpha$, where $C$ is the applied couple or ...
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2answers
56 views

Trigonometry and triangle proof

Question: Prove that in an acute angle triangle ABC: $$\tan A\tan B +\tan A \tan C + \tan B \tan C \geq 9$$ I have no idea where to even begin this question. Please help me!
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2answers
41 views

Expressing $ 12\sin( \omega t - 10) $ in cosine form

$$ 12\sin( \omega t - 10) $$ I understand how it's solved when using the graphical method, however I'm having trouble understanding something about the trigonometric identities method. The solution ...
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6answers
264 views

Range of a trigonometric function

Question: Prove that: $$0 \leq \frac{1 + \cos\theta}{2 + \sin\theta}\leq \frac{4}{3}$$ I have absolutely no idea how to proceed in this question. Please help me!
3
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2answers
34 views

Solving for an angle

I was never good in trigonometry. I have a rectangle with dimensions $L_1$ and $W_1$. I want to rotate it so that it fits inside another rectangle with dimensions $L_2$ and $W_2$. I need to find the ...
3
votes
6answers
135 views

Evaluate$ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$

Find the value of the integral $ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) $ I tried putting $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but am unable to proceed further. I think the following integral can be ...
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1answer
26 views

Use $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial?

I am reading Barbeau's Polynomials and I found the following problem: Use the identity $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial. But I really have no idea on how ...
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2answers
40 views

Why does resolving forces in one direction give a completely different answer to resolving the opposite way?

I can solve parts i), ii) and am able to show that $R=0$ for part iii). In this question $g$ is the acceleration of free fall taken to be $9.8$ Using Newtons 2nd law [$F=ma$] for the last part I ...
0
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1answer
24 views

In a triangle ABC,a:b:c is 4:5:6.The ratio of the radius of the circumcircle to that of incircle is

In a triangle ABC,a:b:c is 4:5:6.The ratio of the radius of the circumcircle to that of incircle is
-5
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1answer
30 views

In a triangle ABC,a:b:c is4:5:6.The ratio of the radius of the circumcircle to that of incircle is [on hold]

In a triangle ABC,a:b:c is 4:5:6.The ratio of the radius of the circumcircle to that of incircle is
1
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2answers
35 views

When can I and when can I not use complex replacement?

If I want to calculate: $$(2 cos(t))^3$$ Can I not replace cos(t) with $Re(e^{it})$ and calculate $(2e^{it})^3$ to be $8e^{3it}$ and thus the real part of this becomes 8cos(3t)? But that answer is ...
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2answers
26 views

Rearranging equation $t = \frac{T}{2\pi} (\psi - \epsilon \sin \psi)$ in terms of $\psi$

I was playing around with the maths for orbits and trying to make a parametric equation that, well.. worked. I found a worksheet with parametrics with another variable ($\psi$), but I wanted to be ...
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1answer
52 views

Is $f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$ strictly positive?

I would like to have an advice for this exercise. Let $x\in[0,\pi]$ For which values of $x$ this function $$f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$$ is strictly positive ? I tried to ...
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3answers
44 views

Implicit differentiation of trig functions

I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation: $$\cos\cos(x^3y^2) - x \cot y = -2y$$ I figured that the calculation requires the chain ...
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0answers
16 views

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle , then P^-1+P^-2+P^-3 is equal to [duplicate]

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle , then P^-1+P^-2+P^-3 is equal to note s=a+b+c/2 area of triangle rs/2
3
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3answers
308 views

In a triangle ABC, (b + c) cos A + (c + a) cos B + (a + b) cos C is equal to

In a triangle $ABC$ $$(b + c)\cos A + (c + a)\cos B + (a + b)\cos C=?$$
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2answers
67 views

In any triangle ABC, the expression (a + b + c) (a + b - c) (b + c - a) (c + a - b)$ is equal to

In any triangle ABC, give an equivalence to the expression $$(a + b + c) (a + b - c) (b + c - a) (c + a - b)$$ Can somebody help me? Note that ...
1
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0answers
34 views

Using two chords and an angle to find center and radius of a circle

Hello, I am trying to solve the problem below. Is it possible to solve for the Center and Radius of the circle given the information provided, or is there something missing? I know how it's simple ...
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5answers
55 views

If $\sin( 2 \theta) = \cos( 3)$ and $\theta \leq 90°$, find $\theta$

Find $\theta\leq90°$ if $$\sin( 2 \theta) = \cos( 3)$$ I know that $\sin 2\theta = 2\sin\theta\cos \theta$, or alternatively, $\theta = \dfrac{\sin^{-1}(\cos 3)}{2}$. Can somebody help me?
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1answer
37 views

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle, then P^-1+P^-2+P^-3 is equal to

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle , then P^-1+P^-2+P^-3 is equal to
0
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2answers
19 views

Vectors, How to measure total force and direction.

I am currently looking for some math help that I am quite struggling with. The problem is: (Vectors) A fisherman use his pole and line to pull a fish out of the water. The line exerts a force on ...
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0answers
32 views

Finding the widest angle to shoot a soccer ball from the sideline using optimization!! [duplicate]

I'm trying to do an independent project for my Math class, but I was stuck and couldn't figure out how to use optimization to find position along the sideline that gives the widest angle to shoot. ...
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2answers
29 views

Trying to solve a trig identity

Given $\sin 2x+\sin x=1$, find the value of $\cos 2x+\cos 4x$. I know $\cos 2x+\cos 4x \implies 1-2\sin^2x+1-2\sin^22x$, but didn't get the answer.
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2answers
42 views

How to solve these problems without using L'Hopital's Rule? [on hold]

$\lim _{x\rightarrow 0^{+}}\dfrac {\ln \left( \sin x\right) }{\ln \left( \tan x\right) }$ $\lim _{x\rightarrow 0}\left( \dfrac {\sin x}{x}\right) ^{\dfrac {1}{x}}$ $\lim _{x\rightarrow \dfrac {\pi ...
3
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0answers
55 views

Squeeze Theorem: Finding the limit of a trig function

I'm stuck on finding the limit of a complex fraction/trig function. Could someone please assist, or point out where I'm going wrong? Determine $$\lim\limits_{x \to 0} ...
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2answers
48 views

Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over ...
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6answers
120 views

Why $\cos^2 x-\sin^2 x = \cos 2x\;?$

I was hoping someone could explain how $\cos^2 x-\sin^2 x = \cos 2x$ After using the product rule to differentiate $\sin x \cdot \cos x$ I get the answer $\cos^2 x - \sin ^2 x$ I've come across this ...