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

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126 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
109 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
734 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
1k 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, ...
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1answer
166 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
93 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 ...
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2answers
120 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
196 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
225 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
168 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
71 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}) = ...
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3answers
877 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)$.
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1answer
88 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!
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2answers
103 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!
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1answer
325 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
138 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?
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4answers
224 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
104 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
945 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 $180^\circ<\theta<270^\circ$.
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2answers
134 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.
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3answers
303 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}, ...
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4answers
225 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
66 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 ...
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2answers
104 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 ...
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0answers
312 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
113 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 ...
3
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1answer
165 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
82 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 ...
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1answer
93 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 $x=\dfrac{1}{2} \pi$. ...
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4answers
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 ...
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1answer
338 views

Unit circle table of values

I want some way to find the unit circle table of values without having to learn it by heart. Is there a way to do it? I thought you could for example just calculate using your calculator: $sin(0.5) = ...
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3answers
711 views

Show that $\tan {\pi \over 8} = \sqrt 2 - 1$

Show that $\tan {\pi \over 8} = \sqrt 2 - 1$, using the identity $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan }^2}\theta }}$ Using $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan ...
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1answer
53 views

Solve $x_0 = a\cos(x) + b\cos(x+y-\pi) + c\cos(x+y+z-2\pi)$ and $y_0 = a\sin(x) + b\sin(x+y-\pi) + c\sin(x+y+z-2\pi)$

It has been a while since last time I have tried to solve a trigonometric problem $x_0 = a\cos(x) + b\cos(x+y-\pi) + c\cos(x+y+z-2\pi)$ $y_0 = a\sin(x) + b\sin(x+y-\pi) + c\sin(x+y+z-2\pi)$ Is it ...
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7answers
262 views

Solve $\cos^2x - \sin^2x = 0$ for $x\in [0,2\pi]$ [closed]

How can we find $x\in [0,2\pi]$ such that $\cos^2x - \sin^2x = 0$?
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2answers
314 views

Sin inverse of a complex number

Is it possible to calculate the value of $\delta$ from the relation $\delta=\sin^{-1}(5.4i)$ ? where $i=\sqrt{-1}$
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1answer
111 views

What is $\tan^3 x$?

I can't find how to calculate $\tan^3 x$. I don't even know how to use it on a calculator and have no idea what it means. If $\tan x$ is the ascending of the angle $x$, is $\tan^3 x$ the ascending^3. ...
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1answer
37 views

Finding the angle.

first question here. I ran into a problem where my math skills are just not enough, probably it's simple, but I don't know how to approach it. I'll show you this graphic so you can understand what I ...
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3answers
510 views

Trigonometrical limit $\lim\limits_{ x\to 0 } \frac{\sin x - x\cos x}{x^3}?$

Can you help me solve this without using de l'Hôpital's rule (just using Standard rules): $$ \lim_{ x\to 0 } \frac{\sin x - x\cos x}{x^3}? $$
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2answers
239 views

Hard proof concerning the periodicity of trigonometrical functions. Is that a challenge or just trivial

i want to know if exist or if you can develop or give me ideas of a proof to show that the least number for which sine is periodic is $2\pi$ $$\neg \{\exists n\in \mathbb{R} \wedge n < 2\pi: ...
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1answer
232 views

Angle between 2 faces of a pyramid

The problem: Given a pyramid with $P_0=(0,0,0)$, $P_1=(1,1,1)$, $P_2=(2,-1,2)$, $P_3=(3,0,1)$, find the angle between the $P_1P_2P_3$ face the $P_0P_1P_2$ face. My idea for the solution is to ...
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3answers
149 views

How do I prove that $2\cos (2\theta + {\pi \over 3}) \equiv - 2\sin(2\theta - {\pi \over 6})$

Using the identity $\cos (\theta + {\pi \over 2}) \equiv - \sin\theta $
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3answers
140 views

Proving $1-\frac{1}{2}\sin(2x)=\frac{\sin^3x+\cos^3x}{\sin x+\cos x}$ without factoring

Is there a way to prove this identity without factoring? $$1-\frac{1}{2}\sin(2x)=\frac{\sin^3x+\cos^3x}{\sin x+\cos x}$$
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6answers
2k views

How do I prove: $\cos (\theta + 90^\circ) \equiv - \sin \theta $

How do I go about proving this? I know one method is: $\eqalign{ \cos (90^\circ + \theta ) &\equiv \cos90^\circ \cos\theta - \sin90^\circ \sin\theta \cr & \equiv (0)(\cos\theta ) - ...
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6answers
2k views

Proofs of $\cos(x+y) = \cos x\cos y - \sin x \sin y$

Define $\sin x $ and $\cos x$ via their infinite series: $$ \sin x = \sum_n (-1)^{n}\frac{x^{2n+1}}{(2n+1)!}, \qquad \cos x = \sum_n (-1)^n \frac{x^{2n}}{(2n)!}. $$ Is there a short, clever proof that ...
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1answer
138 views

Is this a valid proof of the derivatives of the trigonometric functions?

For the sake of this proof, the trigonometric functions $\cos$ and $\sin$ are defined as the coordinates of a point on the unit circle, rather than any of the modern analytic definitions. Let $\vec ...
3
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2answers
332 views

Equation $(8\cos^3x+1)^3=162\cos x-27$

Solve equation $$(8\cos^3x+1)^3=162\cos x-27$$ I saw this equation before 5 month, and I couldn't solve it. This isn't homework, etc. (I don't do stuff like this anymore). I am just curious.
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1answer
61 views

Why is $\lim\limits_{h\to 0}h\cos\frac1h\stackrel{?}=0 $?

Can someone explain why is this true? $$\lim_{h\to 0}h\cos\frac1h\stackrel{?}=0$$ $\lim_{h\rightarrow0}\cos{\frac{1}{h}}$ is undefined (limit does not exist), right? So how can the above be true?
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4answers
645 views

Find the value of $\alpha $ given $2\sin\theta -\sqrt 5 \cos \theta \equiv - 3\cos (\theta + \alpha )$

Given: $$2\sin\theta -\sqrt 5 \cos \theta \equiv - 3\cos (\theta + \alpha ),$$ where $$0 <\alpha < 90^\circ, $$ find $α.$ The issue I have with this question is the $-3$ on the right hand ...