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

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4answers
71 views

What is $g'(x)$ if $g(x) =x^2 \int_{x-2}^{\sin x} \cos^2t dt $?

What is $g'(x)$ if $$g(x)= x^2 \int_{x-2}^{\sin x} \cos^2t dt?$$ So i get $g'(x) = 2x(\int_{x-2}^{sinx} cos^2t dt ) + x^2(cos^2(sinx)-cos^2(x-2))$ as my final answer. Is this right?, thanks. Use ...
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1answer
54 views

Solution of two equivalent quadratic equation solutions

I have two solutions to quadratic equations, based on the quadratic formula. The solutions are equivalent. Additionally, one of the variables (Tx, ...
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1answer
186 views

De Moivre's theorem question.

State De Moivre's theorem and use it to find integers $ A,B,C$ such that $$\sin^5\theta=A\sin5 \theta + B\sin3\theta + C\sin\theta.$$ I know De Moivre's theorem, how to prove it, and converting to ...
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1answer
112 views

$\cos$ not a contraction on $\mathbb R$

I know that $\cos$ is a contraction mapping on $[0, a]$ with $a<\pi/2$. I also know that the proof of this uses the mean value theorem and it fails on $\Bbb R$. However, this is not a proof to the ...
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3answers
2k views

Prove that the Tangent of 75 degrees equals 2 plus the square-root of 3

My (very simple) question to a friend was how do I prove the following using basic trig principles: $\tan75^\circ = 2 + \sqrt{3}$ He gave this proof (via a text message!) $1. \tan75^\circ$ $2. = ...
10
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2answers
143 views

How to find the minimum of $f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$?

I need to find the minimum of $f(x)$ with $$f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$$ Could you help me with some clues?
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1answer
92 views

Maximum and minimum of $y = 4x-8*(\cos(x))$ between $-\pi$ and $\pi$

I have found that the maximum of this function is at $\pi$, where the function will equal $$4\pi+8,$$ which is approximately $20$. However, I tried to get the minimum value, and it was incorrect. The ...
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4answers
172 views

$\lim_{x\rightarrow\infty}\sin(x)$?

In physics I came across these kind of equations when I am trying to find the asymptotic behaviour of some function. Can anyone explain if there is any sense in talking about $\sin(x)$ or $\cos(x)$ ...
6
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4answers
1k views

Finding the limit of $\frac{1-\cos(2x)}{1-\cos(3x)}$ for $x \to 0$

As $x$ goes to $0$, what is the limit of $$\frac{1-\cos(2x)}{1-\cos(3x)}$$ Thanks.
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0answers
112 views

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at ...
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1answer
193 views

Power series for $\cos(n\theta)$ in terms of $\sin^{2i}(\theta/2)$?

Does anybody know an expression for the weights in $$ \cos(n\theta) = \sum_{i=0}^n c_i \sin^{2i}(\theta/2) $$ I checked the standard sources (Abramowitz & Stegun, Gradshteyn & Rhyzik) and ...
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1answer
154 views

Solve the integral $\int_{x=0}^{\infty}\frac{1}{x}\int_{y=0}^{x}\frac{\cos{(x-y)}-\cos{x}}{y}dydx$

Find the value of: $$I=\int_{x=0}^{\infty}\dfrac{1}{x}\int_{y=0}^{x}\dfrac{\cos{(x-y)}-\cos{x}}{y}dy \ dx$$ I think we could take: ...
3
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3answers
233 views

What is -cos(t) equivalent to in terms of cos(t)

I want to know if, $-\cos(t) = \cos(t+180)$ or $-\cos(t) = \cos(t-180)$ Please guide me. Thanks
7
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3answers
312 views

Find this $a,b,c$ such that $\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$

It is known that$$\sqrt{9-8\sin 50^{\circ}}=a+b\sin c^{\circ}$$ for exactly one set of positive integers $(a,b,c)$ where $0<c<90$ find the value $$\dfrac{b+c}{a}$$ my idea,$ \sin 50^\circ ...
0
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1answer
53 views

Inverse trigonometric question

Prove that $\tan^{-1}(\frac{x\sin\alpha}{1-x\cos\alpha})-tan^{-1}(\frac{x-\cos\alpha}{\sin\alpha})$ is independent of $x$ and is equal to $\frac{\pi}{2} - \alpha$ Please guide how to proceed in this ...
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1answer
28 views

What is the amplitude of this function

What is the amplitude if I have these two functions? $$y1: y=\cos(x)$$ $$y2: 2y=3\cos(3x/2)$$ I answered 3 but it was wrong
10
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2answers
1k views

Integrate $2\int x^2\, \sec^2x \,\tan x\, dx$

$$ 2\int x^2\, \sec^2x \,\tan x\, \mathrm{d}x $$ How to solve this using integration by parts? WolframAlpha can solve it, but is unable to give a step-by-step solution, and has a different answer to ...
1
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1answer
52 views

inverse trigonometry derivation

Prove that : $sin^{-1}x+sin^{-1}y = sin^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^2}]$ If -1 $\leq x \leq 1; -1 \leq y \leq 1 $ and $x^2+y^2\leq 1$ or if $xy <0 $ and $x^2+y^2 > 1$ solution : Let ...
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1answer
129 views

looking for reference for 2 trig facts

Math people: I am looking for a reference for two trigonometry facts, one of which I proved myself, and another which a random person had posted on the Web. I have evidence to believe the second ...
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2answers
122 views

What is required to establish the law of cosines?

In my quantum computation course, we have been given nothing more than the basic axioms of a linear vector space, and and the properties of an inner product; but we have started referring to "the ...
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3answers
1k views

Solving the equation $\sin 40^{\circ}=\cos x$

How to solve this equation (finding $x$ ): $$\sin 40^\circ=\cos x$$
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2answers
2k views

Finding the 9th derivative of $\frac{\cos(5 x^2)-1}{x^3}$

How do you find the 9th derivative of $(\cos(5 x^2)-1)/x^3$ and evaluate at $x=0$ without differentiating it straightforwardly with the quotient rule? The teacher's hint is to use Maclaurin Series, ...
3
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1answer
177 views

Largest element of the set $\{ \sin{1}, \sin{2}, \sin{3}\}$

i have to find the largest element of the following set $\{ \sin{1}, \sin{2}, \sin{3}\}$. I converted every element to the first quadrant so i can use the monotony of cosine, the set becomes: ...
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2answers
107 views

How do I find the surface area of an angled conic base?

Thank you for viewing my question. I need help creating a formula for finding the surface area of a conic base. (eg. I install a flood light on my roof, I want to know how much surface area it will ...
1
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2answers
129 views

Finding an infinite trigonometric sum

Find the following infinite sum : $$q\sin a+q^2\sin 2a+\ldots+q^n\sin na+\ldots$$ where $|q|<1$ .It would be good if you could find it without the help of any auxiliary sequences using only ...
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1answer
237 views

Find area bounded by two unequal chords and an arc in a disc

Math people: This question is a generalization of the one I posed at Find area bounded by two chords and an arc in a disc . Below is an image of a unit circle with center $O$. $\theta_1, \theta_2 ...
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1answer
260 views

Upper and lower bound of $f(x)=(\tan x)^{\sin 2x}$ for $x \in (0, \frac{\pi}{2})$

Let define $f(x)=(\tan x)^{\sin 2x}$ for $x \in (0, \frac{\pi}{2})$ Please help me prove, that $f$ reaches its lower bound in only one point $x_1$ and reaches its upper bound $x_2$ also in only one ...
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1answer
185 views

A finite sum of trigonometric functions

By taking real and imaginary parts in a suitable exponential equation, prove that $$\begin{align*} \frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases} 1&\text{if } k ...
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4answers
73 views

Inverse trigonometric function

Prove that $$\tan^{-1}x + \tan^{-1}\frac{2x}{1-x^2}=\tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right)\;,\;\; |x| < \frac{1}{\sqrt{3}}$$ By taking R.H.S $\tan^{-1}(\frac{3x-x^3}{1-3x^2}) = ...
3
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3answers
1k views

How can I find the points of intersection between the curves $r=1+\sin\theta$ and $r=1-\sin\theta$?

Find the points of intersection for the curve $r=a(1+\sin\theta)$ and $r=a(1-\sin\theta)$ My book says the answer is $(0,0),(a,0),(a,\pi)$. However I calculated $ (a,0),(a,\pi),(a,2\pi)$.
6
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1answer
89 views

Sum of the roots equation

Need help! how to prove that equation have two roots on $(0,\frac{\pi}{2})$ and calculate $x_1+x_2$ $$\tan(x)^{\cos^2x}=\frac{\tan(x)^{\sin^2x}}{e}$$ That's what I tried : $ \tan(x)=t $ ...
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1answer
90 views

How do you simplify $\cos a\cdot(\cos(4a) + 2\sin^2 (2a))$?

Any help with this question will be greatly appreciated. Thanks!
1
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2answers
105 views

How do you simplify $\tan 10A$ in terms of $5A$?

How do you simplify $\tan 10A$ in terms of $5A$? I just need a few steps to get me going. All help is appreciated. Thanks!
1
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1answer
355 views

Some tips on 3D Trig?

I understand this isn't a MATHS question specifically, however, sometimes I have trouble identifying when to use 3D trig, and trouble with it in general. Can some of the experienced/advanced people ...
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3answers
189 views

Could anyone explain how to simplify $2\sin(45^\circ-x)\cos(45^\circ-x)$?

$$2\sin(45^\circ-x)\cos(45^\circ-x)$$ I know you have to use the double angle formula for sine, but what next?
2
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4answers
258 views

Triangle proof using law of sines

In triangle $ABC$, suppose that angle $C$ is twice angle $A$. Use the law of sines to show that $ab= c^2 - a^2$.
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1answer
109 views

Size of the car in the picture

So I have this picture: If I print the picture, when I print this the mountain behind is around 1.3cm and the car in the lower left is around 0.4cm. I dont know how far away the car is from the ...
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3answers
1k views

Find $\sin(\theta/2)$, given that $\sin \theta = -4/5$ and $\theta$ terminates in $180^\circ<\theta<270^\circ$

Sorry, I'm having trouble with this trigonometry question Find $\sin(\theta/2)$, given that $\sin \theta = -4/5$ and $\theta$ terminates in $180^\circ<\theta<270^\circ$.
5
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2answers
139 views

Evaluating $\cos(A+B)$, given $\cos A$ and $\sin(B)$

Find the exact value: Find $\cos(A+B)$ given that $\cos A=1/3$, with $A$ in the first quadrant, and $\sin B = -1/4$, with $B$ in the fourth quadrant.
2
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3answers
334 views

Solving the equation: $\cos(x)= \cos(2x)$

I'll be glad if someone could explain the justification of this solution: $$\cos(x)=\cos(2x),\; [0^{\circ},360^{\circ})$$ $$\Rightarrow x=\pm2x+360^{\circ}k,\; k\in\mathbb{Z}\Rightarrow x=0^{\circ}, ...
3
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4answers
245 views

(Solved) Trigonometric equations confusion

Okay, there's this simple equation I've been looking into for a while and I don't know why one way of solving it is not correct. See: $$\sin(2x) + 3\cos(2x) = 0$$ Well, the most obvious would be to ...
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6answers
1k views

How to determine the exact value of $\sin(585^\circ)$?

I'm clueless on this question. Could someone explain how to do it?
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2answers
68 views

Help with $\arcsin(x)$ derivative and differentials.

I'm watching this video lecture http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-11-chain-rule/ and I'm stuck at around 3:40, I can't seem to ...
7
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2answers
107 views

Find eigenspaces using ruler and compasses

I think this is an interesting question: In the 2-dimensional real vector space, we are given a linear transformation $f$. Suppose we already know the images of the standard bases, say ...
3
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0answers
315 views

Trigonometric inequality proof

Can anyone help me in proving that $$\cos\theta > \frac{\left(x^a\cos\theta-(x-1\right)^a\cos\frac{\ln x\theta}{\ln(x-1)})\cos(\theta+\gamma)}{\cos\gamma},$$ where $a<1$, $x\in \mathbb{N}$, and ...
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2answers
115 views

Question on inverse trig functions and quadrants? Please Help!

Alright, I was doing a question in a book, and it said: $\displaystyle \cos(2x - \frac{\pi}{6}) = \frac{\sqrt{3}}{2}$ I proceeded and got: $\displaystyle 2x - \frac{\pi}{6} = \frac{\sqrt{3}}{2}.$ I ...
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1answer
252 views

Sine rule: Why doesn't it work in this scenario?

Why does: $$\frac{4.5}{\sin40^\circ} \not= \frac{3+3}{\sin(180^\circ - 58^\circ)}$$ Am I using the rule wrong? Any incorrect assumptions?
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3answers
88 views

Proofs on equilateral triangles

Let $\Delta$ be the set of all triangles with two equal edges and be inscribed in a circle of radius $R$. So, how do I show that: Equilateral triangle in $\Delta$ is maximizing the area? and this ...
2
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1answer
104 views

How to show that $f(x) = \cos^2(x)\sin(x)$ is symmetric about the line $x=\frac{1}{2} \pi$?

I really have trouble with making any exercises regarding point symmetry and line symmetry. For example: Show that $f(x) = \cos^2(x)\sin(x)$ is line symmetrical in the line $x=\dfrac{1}{2} \pi$. ...
12
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5answers
1k views

Show that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.

The following problem was on a math competition that I participated in at my school about a month ago: Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions. I will outline ...