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

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48 views

symbolic solution to trig equation for a variable

Is it possible to solve the following singular transcendental equation in $a$ for the variable $a$? Any symbolic solution will do. $$\sqrt{s^2 - v^2} = 2a \, \sinh \left( \frac{h}{2a} \right)\,\,\,$$ ...
2
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2answers
632 views

Bilinear interpolation of angles

Is their a solution to do a bilinear interpolation in x,y of angles in [0°-360°[ ? The elementary formula of bilinear interpolation don't work on angles due to the discontinuity at 360°-0°. http://en....
2
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4answers
123 views

Limit involving sin

The question is $$\lim_{t\to0}{{1\over 2+ \sin (t)}-{1\over 2}\over \sin (t)}$$ since I cannot directly substitute, how would I go about factoring the $\sin t$. Any help is appreciated!
2
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3answers
49 views

Applying angle addition formulas for subtraction

The angle addition formula says that: $\sin(\phi + \theta) = \sin(\phi) \cdot \cos(\theta) + \cos(\phi) \cdot \sin(\theta)$ Why are the following steps valid?: $\sin(\phi − \theta) = \sin(\phi) \...
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1answer
43 views

Parabolic asymptote of $n\cot\frac\pi{2n}$

I have determined that $$\lim_{n\to\infty}\frac{n\cot\frac\pi{2n}}{n^2}=\lim_{n\to0^+}n\cot{\frac{\pi n}2}=\lim_{n\to0^+}\frac{n\cos{\frac{\pi n}2}}{\sin{\frac{\pi n}2}}=\frac2\pi$$ So that the ...
0
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1answer
130 views

A,B,P are three points on a circle having centre O. If angle OAP=25 and angle OBP=35 , then the measure of angle AOB is???

A,B,P are three points on a circle having centre O. If angle OAP=25 and angle OBP=35 , then the measure of angle AOB is???
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1answer
178 views

Limits involving theta, cos, sin

Question: $$\displaystyle\lim_{\theta \to 0^-} θ^3 \cos\left(\frac 2\theta \right)$$ also $$\displaystyle\lim_{\theta \to 0^+} θ^3 \cos\left(\frac 2\theta \right)$$ I have no idea where to begin ...
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2answers
183 views

Bound for $\left|\sin(x) +\cos(x)\right|$

I'm taking a numerical analysis class and i'm needing to bound $\left|\sin(x) + \cos(x)\right|$ quite often. So far i've been putting that this is always $\leq |1 + 1| = 2$. Is this the minimal bound? ...
2
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4answers
119 views

When is $\cos (x) \geq \frac{1}{2}$?

When is $\cos (x) \geq \frac{1}{2}$? I know the function repeats, so I know I should end up with an interval that allows for integer multiples. e.g. something like this (but obviously not this ...
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0answers
106 views

How to find the period of a exponential function? $5\cdot(-1)^k$

Currently in class we are learning Euler's formula and how it relates to imaginary numbers, exponentials, and trig functions. I know the left hand of the equation is $e^{jwt}$ and I could easily use ...
2
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3answers
57 views

How was this approximation of transcendent equation solution found?

I have an equation for $\xi$: $$\xi\gamma=\cos\xi,$$ where $\gamma\gg1$. I've tried solving it assuming that $\xi\approx0$ and approximating $\cos$ by Taylor's second order formula: $$\xi\gamma\...
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2answers
59 views

Limit with trigonometric function

I have this limit, I have resolved it until a part but I'm stucked now. $$\lim_{x \to \frac{\pi}{4}} \frac{\tan^2(x)-1}{\cos(x)-\sin(x)}$$ $$ \lim_{x \to \frac{\pi}{4}} \frac{\frac{\sin^2(x)}{\cos^2(...
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4answers
291 views

Sum of squares of sines.

Any ideas on how to compute this sum? I'm sure there's a simple trick to it, but I just can't wrap my mind around it at the moment. Some insight would be tremendously appreciated, thanks! $$\sum_{...
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3answers
110 views

How to integrate $\int\frac{t^2}{\sqrt{4t-t^2}} \, dt$ using trig substitution.

How do I integrate $\int\frac{t^2}{\sqrt{4t-t^2}} \,dt$? I solved this integral by a very long process(over 2 pages of work) and I got the answer of $6\sin^{-1}(\frac{t-2}{2})-\frac{t+6}{2}\sqrt{4t-t^...
2
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2answers
65 views

How to simplify $\sqrt{\tan^2\theta}\sin\theta\cos^2\theta$?

I am solving $\int_{7}^{14}\frac{\sqrt{x^2-49}}{x^4}$ and got the integral down to $\frac{1}{343}\int_{0}^{\frac{\pi}{3}}\sqrt{\tan^2\theta}\sin\theta\cos^2\theta$ and wolfram simplified $\sqrt{\tan^2\...
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3answers
4k views

Find all values for cos(i)

In my Differential Equations class recently we have learned about Euler's Formula and Fourier Series. I am given the problem ...
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2answers
80 views

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal?

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal? I know both of these are the answer to $\int \csc x \space dx$, and I am able to work them out with calculus ...
3
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6answers
100 views

Proving a trigonometric inequality $(1-\sin a)x^2 -2x\cos a + 1+ \sin a \ge 0$

$(1-\sin (a))x^2 -2x\cos(a) + 1+ \sin( a) \ge 0$, where $a,x$ are any two real constants. Any suggestions on how to prove this ? I tried playing with it, but nothing useful came out.
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1answer
31 views

Differentiating this trigonometric function

Differentiating $$ L = \frac{2v_o^2\cos^2\theta}{g\cos\alpha}\cdot(\tan\theta-\tan\alpha) $$ with regard to theta. I know I have to use trig. idendities, but I'm just completely stuck.
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2answers
93 views

The supremum of a sequence of definite integrals

I am interested to find the supremum of the following sequence of definite integrals: $$ I_n=\int_0^\pi\sqrt{4\cos^2((2n+1)x)+4\cos((2n+1)x)\cos(x)+1}\,\textrm{d}x,\ n\ge 1. $$ One of my ideas was to ...
0
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1answer
142 views

Find the side of an equilateral triangle given only the distance of an arbitrary point to its vertice [duplicate]

Triangle ABC is an equilateral triangle and P is an arbitrary point inside it. The distance from P to A is 4 and the distance from P to B is 6 and the distance from P to C is 5. How to find the side ...
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3answers
67 views

Evaluating $\int \:\sqrt{1+e^x}dx$ , why I got different answers?

I've got 2 steps to evaluating $\int \:\sqrt{1+e^x}dx$ which lead to different values first step : $\int \:\sqrt{1+e^x}dx$ let $u\:=\:\sqrt{1+e^x}$ , $du\:=\:\frac{e^x}{2\sqrt{1+e^x}}dx$ , but $...
2
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3answers
77 views

How to Integrate $\int\frac{(x^2)}{\sqrt{7-x^2}}dx$.

I am trying to Integrate $\int\frac{(x^2)}{\sqrt{7-x^2}}dx$ and I have worked this problem a couple times and keep getting the same answer. So I will show my process and please point my errors out. $$...
2
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1answer
151 views

Integrating $\int\frac{\sqrt{16x^2-9}}{x}dx$?

I am trying to differentiate from my previous question, but I am having trouble in the finishing steps. I have the integral $\int\frac{\sqrt{16x^2-9}}{x}dx$. $$v=4x \hspace{15pt}dv=4dx$$ $$\int\frac{\...
2
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1answer
493 views

Trigonometric limit: $(1-\sqrt{\cos x})/x^2$ as $x\to 0$, without using L'Hopital

I have to evaluate this limit without using L'Hopital. Could you help me $$\lim_{x \to 0} {1-\sqrt{\cos(x)}\over x^2}$$ I already rationalized it: $$\lim_{x \to 0} \left({1-\sqrt{\cos(x)}\over x^2}\...
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1answer
40 views

how should I think about the complex-exponential form of sinusoid waves?

Say there's a sinusoid wave with amplitude $A$, frequency $\omega$, and phase shift $\psi$, then one way to write it is $A cos(\omega t - \psi)$. But it can also be written as $Re(Ae^{i(\omega t - \...
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2answers
361 views

How to integrate $\int\frac{\sqrt{16x^2-9}}{x}dx$?

I have the integer; $\int\frac{\sqrt{16x^2-9}}{x}dx$, and I am having trouble doing the trigonometric substitution. So for integrals in the from of $\sqrt{x^2-a^2}$ where $a$ is a constant is by ...
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3answers
74 views

Solve $\lim_{x\to 0} \frac{\sin 2x}{4x}$

$$\lim_{x\to 0} \frac{\sin(2x)}{4x}$$ In this form, it would be undefined, so how would you change it so that the denominator would not be $0$?
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4answers
106 views

Is $\sin 2x$ the same thing as $2\sin x$?

Is $\sin2x$ the same thing as $2\sin x$? I am unsure whether it is valid to bring out the two outside the sine.
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1answer
49 views

Differentiation Proving

Can someone please help me solve this question. Provide a hint? If $$\cos\frac x 2\cos\frac x 4\cos\frac x 8\cdots=\frac{\sin x}x$$ then prove that $$\frac{\sec^2(x/2)}4 + \frac{\sec^2(x/4)}{16} +\...
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0answers
16 views

Nonnegativity on a special domain entails nonnegativity on the whole plane

Let $Q$ be a real bivariate polynomial such that $Q(x,\tan(x))\geq 0$ for any $x\not\in\{\pm\frac{\pi}{2}\}+(2\pi){\mathbb Z}$. Does it necessarily follow that $Q$ is nonnegative on the whole of ${\...
2
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3answers
108 views

$\int \dfrac{\cos x}{\left(\cos (2x)\right)^{3/2}} dx$

Wolfram gives this nice result: $$\int\frac{\cos x dx}{\cos^{3/2}2x}=\frac{\sin x}{\sqrt{\cos 2x}}+\text{constant}$$ I have tried writing $\cos 2x = \cos^2x - \sin^2x $ and doing Weierstrass ...
10
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5answers
160 views

Prove $3(\sin x-\cos x)^4 + 6(\sin x+ \cos x)^2 + 4(\sin^6 x + \cos^6 x) -13 = 0$

Q) Prove that $3(\sin \theta-\cos \theta)^4 + 6(\sin \theta+ \cos \theta)^2 + 4(\sin^6 \theta + \cos^6 \theta) -13 = 0$ Source: Trigonometric Functions, Page 5.9, Mathematics XI - R.D. Sharma My ...
0
votes
1answer
31 views

When total time gets minimized?

We want to get from $\displaystyle{C}$ to $\displaystyle{A}$. The path $\displaystyle{C \to B \to D \to A}$ can be done with constant velocty $\displaystyle{w}$. So that the time to get there gets ...
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0answers
67 views

Calculate area of this figure

I have an homework assignment where I have to calculate area of the figure underneath. I used the following formula to calculate the result $\frac {130 \cdot 55 \cdot sin35}{2} = 4188 m$ and then $...
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2answers
61 views

Why are the sines and cosines of something resulting in the wrong anser?

For any developers, I use the following code: ...
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3answers
354 views

Find the sum $1+\cos (x)+\cos (2x)+\cos (3x)+…+\cos (n-1)x$ [duplicate]

By considering the geometric series $1+z+z^{2}+...+z^{n-1}$ where $z=\cos(\theta)+i\sin(\theta)$, show that $1+\cos(\theta)+\cos(2\theta)+\cos(3\theta)+...+\cos(n-1)\theta$ = ${1-\cos(\theta)+\cos(n-1)...
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3answers
910 views

What is the order of operations in trig functions?

Is $\sin(x)^2$ the same as $\sin^2(x)$ or $\sin(x^2)$? I thought it would mean the former interpretation, $\sin^2(x)$, rather than the latter, but my teacher and I had a long argument on this and in ...
0
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1answer
70 views

3 snooker balls in a box

A regulation snooker ball is 52.5 mm in diameter. What are the minimum internal dimensions of a cube that can exactly contain 3 of them? I'm sure there must be an easy answer, but I'm not a ...
1
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1answer
66 views

Evaluation of $\int_{-\pi}^{\pi} \cos(ax) \sin^n(bx) dx$

As it is a kinda famous integral I thought I would find something on MSE but I didn't so here I am. If there is, link it in the comments and I will delete the question. How do I evaluate $$\int_{-\pi}...
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2answers
76 views

Simplify the trigonometric expression

Simplify the expression $$\left(1-\frac{\cos61^{\circ}}{\cos1^{\circ}}\right) \left(1-\frac{\cos62^{\circ}}{\cos2^{\circ}}\right)\cdot ...\cdot \left(1-\frac{\cos119^{\circ}}{\cos59^{\circ}}\right)$$
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1answer
52 views

How many points to span a goniometric wave and how to construct the goniometric function

I have two questions concerning the spanning of a simple trigonometric function: What is the minimum number of points to define/span a "simple" trigonometric wave in two dimensions? Is it possible ...
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1answer
37 views

Solve the equation $\left | \tan x \right | = 2 \cos^2x$

Solve the equation $\left | \tan x \right | = 2 \cos^2x$
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1answer
81 views

Find the speed of an object given two vectors

Here's the question: An airplane flies horizontally from east to west at 304 miles per hour relative to the air. If it flies in a steady 50 mile per hour wind that blows horizontally toward the ...
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1answer
52 views

Figuring out the radius from chord or arc

Are you able to figure out the radius of a circle by any chord and or arc?
2
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0answers
57 views

Polygons inscribed in circles, with integer sides and integer radius

Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius? This is what I got so far: All ...
2
votes
1answer
41 views

Find the angle of a triangle

I'v tried to solve this problem but did not get the right result. Triangel PQR is PQ = 5,0 cm, QR = 6,3 cm and RP = 7,4 cm. Calculate angle P. I tried to solve it by using by using the following ...
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4answers
2k views

Find the midpoint between two points on the circle

I want to place a new point in the middle of the two points which are on the circle outline (Arc). I have the coordinates $(x,y)$ of the center of the circle, the two red points and the radius of the ...
2
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1answer
41 views

How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - \cot A)$?

Consider all triangles $ABC$ where $A < B < C \leq \frac{\pi}{2}$. How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - ...
6
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1answer
149 views

Prove |cos(x−1)|+|cos(x)|+|cos(x+1)|≥3/2

I'm working on an induction proof, but I keep coming up against a brick wall. While working through the induction proof process I keep ending up with $$|\cos(m)|\ge\frac12$$ ,but clearly this isn't ...