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

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19 views

How fast is this dot moving when the angle $θ$ between the beam and the line through the searchlight perpendicular to the wall is $π/6$?

A searchlight rotates at a rate of $4$ revolutions per minute. The beam hits a wall located $11$ miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per ...
1
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2answers
45 views

Identities on $\cos n\theta$ and $\sin n\theta$

How to prove that: $$\cos{n\theta}=\cos^n{\theta}- \binom {n} {2}\cos^{n-2} \theta \cdot \sin^2 \theta+ \binom {n} {4}\cos^{n-4} \theta \cdot \sin^{4} \theta -\cdots$$ $$\sin n\theta = \binom {n} ...
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3answers
31 views

How to convert this particular expression into some desired form?

The parametric equations of a curve are $$x=\cos(t) \cdot e^{-t} $$ $$y=\sin(t) \cdot e^{-t} $$ Show that $dy/dx =tan(t-\pi/4) $. how to solve this? I can get a $dy/dx$ but i cannot convert into the ...
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2answers
26 views

Solving Trigonometric Equations with a ranging from $-2\pi$ to $2\pi$

Solve the following for $-2\pi \le \theta \le 2\pi$: $$12\cos(2\theta) – 6 = \sin\theta$$ What I did was used trig identities to make cosine into sine by using $\cos^2\theta = 1 - ...
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1answer
49 views

proof trigonometric identitie

We just started to learn trigonomety so far we have learned $\sin^2 a + \cos^2 a = 1$, $\tan a = \frac{\sin a}{\cos a}$ how can I proof this identity with that $(\sin a - \sin b + 1)^2 = (\sin a + ...
1
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1answer
46 views

Value of $cos\theta$ when $\theta$ is very small

What happens (actually why) to the value of $cos\theta$ when $\theta$ is small enough that its higher powers that is $\theta^2$ (and more) can be neglected?
3
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3answers
51 views

Factorize Trigonometric Equation

I have a problem with the following trigonometric equation: $$3\sin(x)^2 - 2\sin(x)\cos(x) - \cos(x)^2 = 0$$ It's from the book Engineering Mathematics 7th edition by Stroud. The book is giving the ...
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1answer
50 views

Trigonometric identities with multiplication

Why aren't there Trigonometric identities with multiplication inside the function? For example for $\sin(xy)=?$.
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1answer
25 views

Trigonometric Equation, quadratic using two functions

I am struggling to know how to solve this equation as it involves more than one type of trigonometric function, I know how to do it with one repeated function. If a solution could be explained, that ...
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3answers
88 views

Does $\tan x\cdot \cos x$ equal $\sin x$?

Is it true that $\tan x\cdot \cos x = \sin x$? If I put $x=30$ in my calculator then I don't get the same answer as $\sin 30$, why is this? Don't the two cosines cancel out? I'm probably missing ...
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1answer
28 views

$K$ is a region in $\mathbb{R}^2$ where the area is $5$

Say that $K$ is a region in $\mathbb{R}^2$ where the area is $5$. Let B = \begin{pmatrix} 3 & 8 \\ 4 & 6 \end{pmatrix} Find the area of the region B$K$. Any starting hints? Is it possible ...
2
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1answer
24 views

Proving $\sin ((n-1/2)\phi) + \sin(\phi/2)=\sin({n+1 \over 2}\phi)$

I am trying to show that $\sin ((n-1/2)\phi) + \sin(\phi/2)=\sin({n+1 \over 2}\phi)$ I tried to apply $\sin (x-y) = \sin x \cos y - \cos x \sin y$ to it and I got $\sin ((n-1/2)\phi) + ...
3
votes
2answers
46 views

How to prove this trigonometry

I need to prove that $$\cos^2(\beta -\gamma)+ \cos^2( \gamma - \alpha) +\cos^2(\alpha -\beta) = 1+2 \cos(\beta- \gamma) \cos( \gamma - \alpha)\cos(\alpha -\beta) $$ To do this I have used the ...
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1answer
22 views

Find the largest segment

I have seven lines with different measures. The length of each line it's a positive integer and the shortest length is equal to 1 cm. It is known that's impossible to choose three of them that makes a ...
4
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2answers
34 views

Roots of unity, where $\omega^3 = 1, \omega \neq 1$.

Say that $\omega^3 = 1$ and $\omega \neq 1$. Find the value of $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$. I'm not very good at the roots of unity. May I have a couple of hints to get started? ...
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1answer
31 views

Proof of Compound Angle from Ptolemy's Theorem

I have a query regarding a proof I'm reading on the additive Sine compound angle formula, which uses Ptolemy's theorem. http://www.cut-the-knot.org/proofs/sine_cosine.shtml I'm looking at the ...
2
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3answers
50 views

How can I solve the simultaneous equations that arise in solving $\cos(z)=2$.

If I have $\cos(z)=2$ I can say $\cos(a+ib)=2$ using double angle ideas $\cos(a)\cos(ib)+\sin(a)\sin(ib)=2$ using Euler's formula $\cos(a)\cosh(b)+i\sin(a)\sinh(b)=2$ equating real and imaginary ...
3
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3answers
37 views

Find the least degree Polynomial whose one of the roots is $ \cos(12^{\circ})$

Find the least degree Polynomial with Integer Coefficients whose one of the roots is $ \cos(12^{\circ})$ My Try: we know that $$\cos(5x)=\cos^5x-10\cos^3x\sin^2x+5\cos x\sin^4x$$ Putting ...
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2answers
34 views

Finding the measure of a base angle of an isosceles triangle given its side lengths [on hold]

I know this is looked down upon, but it is a simple question I got jammed on and can't figure out why I kept getting the wrong answer. I just want to see solution, simple question for most of you ...
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5answers
47 views

Find an algebraic expression for sin(arccos(x))

I have a question that asks me to find an algebraic expression for sin(arccos(x)). From the lone example in the book I seen they're doing some multistep thing with the identities, but I'm just not ...
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5answers
49 views

How to prove $\displaystyle\lim_{x \to 0} \dfrac{\sin^{-1} x}{x} = 1$?

How to prove this? Is there any geometrical proof? I have proved , btw, $\displaystyle\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ by Sandwich Theorem and little geometry.
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1answer
37 views

trigonometry - identities and formula, proving

Prove: $$\sin(x) - \sin(x) \cos^2(x) = \sin^3(x).$$ I swear it is easy, but I don't know what I'm forgetting to look at?
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1answer
36 views

Solving triangle

If side $a$ is known and the angles are given as functions of two variables (let's call them $x$ and $y$), what is the easiest way to find $y$ as a function of $x$. To make things easier, let one of ...
3
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1answer
68 views

How prove that $\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$ [duplicate]

How prove that sum $$\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$$
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2answers
60 views

show that $(1 +\sin{A}) + \cos^2{A} = 2(1 + \sin{A})$? [on hold]

It is an identity. The question is asking me to show that the left hand side is equal to the right hand side. $$(1 +\sin{A}) + \cos^2{A} = 2(1 + \sin{A})$$
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3answers
50 views

Show that: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 +1 } )$

could someone Please give me some hint of how to do this question thanks
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1answer
191 views

Express $C_n = \cosh(0) + \cosh(1) + \cosh (2) + \dots + \cosh(n)$

Could someone give me some hint of how to do this question please. I've been stuck for more than $3$ hours on this question.
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1answer
27 views

Trig and Geometry problem

I have this problem to solve. There is a triangle ABC containing a line segment bisecting Angle C with length s. The side opposite angle A is length a, across angle B is length b and the measure of ...
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3answers
59 views

Complex Number to a power

I asked this question yesterday, but the answers did not actually answer what I wanted to know since I asked the question in the wrong way. I have $e^{i\frac{2014\pi}{12}}$. I know Euler's formula, ...
0
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1answer
25 views

Find an equation for a sinusoid with minimum and maximum

Here's my problem: Find an equation for a sinusoid that has a minimum at (30°,-1) and an adjacent maximum at (75°,7). Please help! I've tried everything I can think of, but I'm really drawing a ...
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2answers
52 views

finding angle value inside this triangle

I need a method to calculate the angle X in the image below, I know its value (30 degree) but how ?!! thank you.
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0answers
21 views

Can trigonometric functions for double precision be implemented in terms of those for single precision?

In some program environments like GLSL there is support for single and double precision numbers for arithmetic and square roots computation, but only single precision trigonometric functions are ...
4
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0answers
47 views

How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin z}=\root{12}\of{\sin 2y} $?

Let $\sin(x+y) = 2\sin\left(\dfrac{x-y}{2}\right)$ and $\sin(y+z) = 2\sin\left(\dfrac{y-z}{2}\right)$. How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin ...
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0answers
18 views

Coordinate Geometry Help (circles + trigonometry)

Question : Find all points $(x, y)$ {if there are too many then number of points is enough} which lie on or inside the circle $x^2 + y ^2 = 9$ and satisfying the equation $\tan^4 (x) + \cot^4 (x) + 2 ...
0
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0answers
24 views

Dirichlet integral using real Analysis

The teacher made this approach to solve the Dirichlet integral , $$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin ...
2
votes
4answers
46 views

Simplifying the expression $2\cos^{2}6x-1$

I am trying to simplify the expression $2\cos^{2}6x-1$. The book got the answer of $\cos 12 x$ by doing $2\cos^{2}6x-1 = \cos2\left(6x\right) = \cos12x$ It said the double angle is $12x$. I don't ...
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1answer
30 views

Textbook error or my own error, Law of sines question.

A triangle with side 8 inches and corresponding angle 13 degrees, side x inches and corresponding angle 120 degrees. To answer this I set up a proportion: $sin(13)/8=sin(120)/x$ ...
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3answers
42 views

Write the product of two trig equations equal to one?

Solve: Write the product of two trig equations that is equal to one. This one confuses me because I can think of trig equations that equal one, but I can't think of trig equations that I could ...
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1answer
61 views

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? [duplicate]

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? So if it converges then $\lim_{b \to\infty}\int_{0}^{b}\sin(x^2)\;\mathrm dx$ exists and our integral converges to this ...
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1answer
58 views

How can you find the integral of $\frac{cos(2t)}{2t^2}$ between 1 and infinity?

How can you find the integral of $\frac{\cos(2t)}{2t^2}$ between 1 and infinity? $$ I = \int\limits_1^\infty \frac{\cos(2t)}{2t^2} dt $$ My problem is that I just simply do not know how to handle ...
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1answer
63 views

Don't understand proof of why $\cos x$ is a contraction mapping on $[0, 1]$

I've read a couple proofs of why $\cos x$ is a contraction mapping on $[0,1]$ but none of them are clear enough for me to understand. What if we have something like $\lvert \cos x - \cos y \rvert = w ...
2
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6answers
73 views

Solve for $x:1 + \tan^2(x) = 8\sin^2(x)$

I have a tricky problem , I tried several methods and I can't seem to get a definite answer. $1 + \tan^2(x) = 8\sin^2(x), x \in [\frac{\pi}{6} , \frac{\pi}{2}]$ I got to ...
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2answers
31 views

Solutions for the following equation.

Let us have an $n$ positive integer. How many solutions do we have for the following equation in the interval $(0,\frac{\pi}{2})$? $$\underbrace{\cos(\cos(\ldots(\cos x)\ldots))}_{n\text{ times ...
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3answers
41 views

Trig and derivatives: If condition holds for derivative, does it hold for the original equation?

Let's say I have some trigonometric identity such as $\sin(x) + 1 = -\cos(y)$. As we can see, the derivative of this identity gives $\cos(x) = \sin(y)$, which implies that $x + y = \pi/2$. Does that ...
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2answers
1k views

What is cosine to the power of zero?

I was doing a question relates to substitution rule under integration. The question is as follow: Evaluate $\int{1\over{(1+x^2)^n}}dx, n\in \mathbb{Z}^+$ We have seen that ...
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5answers
47 views

Expressing $\cos^5(x)$ using trigonometric addition formulas

If $\cos(3x) = 4\cos^3(x)-3\cos(x)$, and $\cos^3(x) = \frac{1}{4}(\cos(3x) + 3\cos(x))$, how can we express $\cos^5(x)$ in the same way?
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1answer
26 views

Integration: Find length of curve using NINT

Here are the questions - For question 4, part (b) gives a unit circle. But I'm unable to proceed with parts (a) and (c), since the curve is double valued for -0.5 Also, for question 6, integration ...
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2answers
57 views

Method to integrate $\cos^4(x)$

Here my attempts for integrating $\cos^4(x)$ in few methods. 1st method. $(\cos^2x)^2=(\frac{1}{2})^2(1+\cos2x)^2$ $=\frac{1}{4}(1+2\cos2x+\cos^22x)=\frac{1}{4}(1+2\cos2x)+\frac{1}{4}(\cos^22x)$ ...
2
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1answer
32 views

If $ \cos(\theta) = - \frac{2}{3} $ and $ 450^{\circ} < \theta < 540^{\circ} $, find…

If $ \cos(\theta) = - \frac{2}{3} $ and $ 450^{\circ} < \theta < 540^{\circ} $, find: The exact value of $ \cos \! \left( \frac{1}{2} \theta \right) $. The exact value of $ \tan(2 \theta) $. ...
2
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0answers
29 views

Trigonometrical functions and complex numbers

(This question will at first appear too broad. However, the overall philosophy will be explained below in a way that asks specific questions, which I hope will be conducive to this being a reasonable ...