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
176 views

Applications of higher powers of trigonometric functions

I am after a reference (book, papers etc) about the practical applications of trigonometric functions raised to higher powers. An example is one that I have been using in my own studies: $\cos^4 ...
7
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1answer
247 views

Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+…+$

Find the sum of the following : (i) $$\frac{1}{\cos0^\circ \cos1^\circ}+\frac{1}{\cos1^\circ\cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+\frac{1}{\cos88^\circ \cos89^\circ}$$ I tried : ...
3
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2answers
235 views

Find the value of $\sin 2013^\circ$

How do I find the value of $\sin 2013^\circ$? A precise decimal is not required, but must be expressed with $\sin 30^\circ,$ $\sin 45^\circ,$ and $\sin 60^\circ$ (cosine is also fine). Hint: Use ...
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1answer
50 views

How to rearrange cosines to get angles

Given this equation: $$ Z = \frac{10}{\cos a} + \frac{10}{\cos(90-a)}$$ how would I go about rearranging it to get a in terms of Z?
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2answers
102 views

$2 \cos^{-1}x =\sin^{-1}(2x \sqrt {1-x^2})$ is valid for which values of $x $

Problem: $2 \cos^{-1}x =\sin^{-1}(2x \sqrt {1-x^2})$ is valid for which values of $x $ Solution: $2 \cos^{-1}x =\sin^{-1}(2x \sqrt {1-x^2})$ $2 \cos^{-1}x =2 \sin^{-1}x$ $ ...
12
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1answer
372 views

When $\sin x, \cos x$ are $\mathbb{Q}$-linear combinations of square roots

Suppose $x\in\Bbb R$ is such that $$\sin x=\sum_{i=1}^m x_i\sqrt{r_i},\quad \cos x=\sum_{j=1}^n y_j\sqrt{s_j}$$ for some $x_i, r_i, y_j, s_j \in\Bbb Q \ , \ |x_i|=|y_j|=1$. Show that ...
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2answers
375 views

Use graphs and standard triangles to evaluate $\sin(\frac{11}{6}\pi)$

Use graphs and standard triangles to evaluate $\sin(\frac{11}{6}\pi)$. I end up with $\sin(\pi + \frac{5}{6}\pi)$ which I can't use standard triangles on.
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1answer
68 views

If n=(sin^2(2x))/4cos^2(x))+1/(sec^2(x)) and x=2.01307, find 2013n^2013

If $n=\dfrac{sin^2(2x)}{4cos^2(x)+\dfrac{1}{sec^2(x)}}$ and $x=2.01307$, find 2013n^2013 Your edits are wrong! These are two separate fractions not together!anymore!
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1answer
50 views

Some basic trigonometry questions

These are all practise questions from my book, i figured out most of them but am stuck on a few. Hoping that someone else can help me If $\cos(A) + \sin(B) = m$ and $\sin(A) + \cos(B) = n$, prove ...
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2answers
87 views

Trigonometric function, with integration of definited integrals

I have worked up to this stage of the question : $$\int_0^{\pi/6}\frac{1-\cos2{(x/2)}}{2} dx$$ so that's where I worked up to. can someone please show me how to finish it off
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2answers
235 views

Evaluate $\cos (-2\pi)$ using radiants graph

I plotted $2\pi$ on the $\cos$ graph and the result was $1$, so I assumed $-2\pi$ would be $-1$, but it turned out to be $1$. Can someone explain how to work this out?
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4answers
1k views

Integral of $\sin^2 \pi x$

Evaluate $$\int_0^{1/4} \sin^2 \pi x \; dx$$ Can someone please explain what to do if theres a power and how to do it in general thanks
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2answers
863 views

Proof of sum of sinus terms with mathematical induction.

Hello dear community I have some issues in solving the following problem: Proposition: If $x$ is a real number not divisible by $\pi$ ($x\notin \pi\mathbb{Z}$), then for all integers $n \ge 1$ ...
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2answers
97 views

Reduce formula using Euler's?

I am performing a self-study, and I am lost as to a derivation that has taken place. I basically started with this equation: $$ \Upsilon(\phi) = e^{-j\frac{N-1}{2}\phi} \ \Big[ \frac{1 - e^{j N ...
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1answer
123 views

Trigonometric Functions. Definite Integrals

Find, correct to one decimal place, the value of $$\int_{0}^{60} 2\sin(x/2) \, dx.$$ Can someone please show me how this question is done. It would be very helpful thanks!
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1answer
155 views

Arctanh to exp: Prove two equations are equivalent

For some peace of mind in a project, I am trying to prove two equations are somewhat equivalent. I have these two equations. $$ i_{1} = ...
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4answers
109 views

How to find the value of $a$ for which $\;\tan^2x + (a+1)\tan x-(a-3)<0$ is true

I wanted to know, how can I find the value of $a$ for which the inequality $\tan^2x + (a+1)\tan x-(a-3)<0$ is true for at least one $x\in(0,\pi/2)$. I don't know how to proceed, any help is ...
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1answer
105 views

How to solve these trigonometric functions

The functions that I need to solve are 1) $A\sin x + B \sin x \cos x + C \cos x=0$ and 2) $A\sin x + B \sin x \cos x + C \cos x+D=0$
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8answers
3k views

Reasoning that $ \sin2x=2 \sin x \cos x$

In mathcounts teacher told us to use the formula $ \sin2x=2 \sin x \cos x$. What's the math behind this formula that made it true? Can someone explain?
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2answers
102 views

Simplifying second derivative using trigonometric identities

Given that $x=1+\sin(t)$ , $y=\sin(t) -\frac{1}{2} \cos(2t)$ show that $\frac{\text{d}^2y}{\text{d}x^2}=2$. I am having trouble proving this. Here is my working so far: ...
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1answer
264 views

Get legs length from right angle and hypotenuse

Here I have a 2 points. $A$ (lat,lng) and $B$ (lat,lng), this two point is hypotenuse of right triangle. How I can get legs length if I know angle, hypotenuse and these points?
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1answer
42 views

Get location from sun's position and time?

I once heard it was possible to obtain your position from the suns azimuth and elevation and the current time. also, the time from the your and the suns position. I am currently in upper high and ...
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1answer
1k views

How to find the area of an isoceles triangle when only the length of the two equal sides is given?

I'm trying to apply the method of ehxaustion to have an approximation of the area of the circle: I know that the task is about decomposing the circle into a great number of triangles and then ...
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0answers
264 views

Determining similarity between paths (sets of ordered coordinates).

With limited knowledge of mathematics, I am not sure what tags to use for this question. I have a path on a 2D surface called $(p1)$. A path consists of a set of ordered $(x,y)$ coordinates. By ...
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2answers
311 views

Convert $\sin\theta/2$ using trig identities

$$\sin\dfrac \theta2 = \sin^2θ+\cos^2θ-1$$ $$\sinθ = 2\sin^2θ+\cos^2θ-1$$ Am I on the right track?
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1answer
132 views

Help needed on graphing on the unit circle?

The problem is as follows: Graph $f(x) = \csc x$ on the interval $[0,2\pi]$. (Original image here: http://i.stack.imgur.com/kB97e.png) Any advice on how to graph would be appreciated.
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2answers
342 views

Solve for $\theta$ in$\sqrt 3 \csc \theta =2 $

I got $\dfrac{1}{\sin(\theta)}=\dfrac{2\sqrt {3}}{3}$ then $\sin (\theta) = \dfrac{2\sqrt {3}}{3}$ The question is that the answer is not in the Unit circle I believe I made a error in my problem.
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4answers
425 views

Solve for $\theta$ in $2\sec^2θ-4 =0$

Solve for $\theta$ in $2\sec^2θ-4 =0$ I have gotten toward $\sec^2θ=2$ Then $\dfrac{1}{\cos^2θ} = 2$ What is the next step to this problem?
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2answers
41 views

Trigonometric functions

I wonder how does a WolframAlpha get this relation where input is a LHS and output is RHS: $$\cos^2(x)\cos(2x) = \tfrac{1}{4}\cos(4x) + \tfrac{1}{2}\cos(2x) + \tfrac{1}{4}$$
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1answer
78 views

A problem of triangles.

Question: "In triangle $ABC$, angle $B$ measures $68°$ degrees and angle $C$ is $40°$ degrees. $AD$ and $BE$ are heights, $M$ is the midpoint of $BC$ and $N$ is the midpoint of $AC$.Calcule angles ...
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2answers
89 views

How to find $P_1$ in $(x,y)$ form

From following diagram, $A_1$ is center of circle of radius $r$. All distances are in coordinate system $(x,y)$. Distance from $A_1P_2$ is known. Distance $A_1,A_2,A_3$ is also known from origin. I ...
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1answer
131 views

An inequality involving arctan of complex argument

I have the following conjecture: \begin{equation} \text{Re}\left[(1+\text{i}y)\arctan\left(\frac{t}{1+\text{i}y}\right)\right] \ge \arctan(t), \qquad \forall y,t\ge0. \end{equation} Which seems to be ...
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4answers
1k views

Use the addition or subtraction formula for cosine to compute $\cos(-5\pi/12)$

Use the addition or subtraction formula for cosine to compute $\cos(-5\pi/12)$ (Leave your answer in exact form.) I have $$\begin{align}\cos(-5\pi/12)&=\cos((\pi/4)-(5\pi/6))\\ ...
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3answers
678 views

Use the half-angle formula for cosine to compute $\cos(\theta/2)$ given $\cos(\theta)=63/68$ where $0\lt\theta\lt\pi/2$

Use the half-angle formula for cosine to compute $\cos(\theta/2)$ given $\cos(\theta)=63/68$ where $0\lt\theta\lt\pi/2$. I know that $\cos(\theta/2)= \pm\sqrt{\frac{\cos(\theta)+1}{2}}$. Therefore ...
2
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3answers
101 views

A trigonometry problem

Solve θ in $\sin^2θ$ - $\cos^2θ$ = 1 $$-1 = \cos^2θ - \sin^2θ \\ -1 = \cos(2θ)$$ What would be the next step to solve this enigma?
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2answers
101 views

Geometry Problem with isobarycenter and isoceles triangle

$XYZ$ a triangle with $XY=YZ.$ $D$ middle of $XY,$ $G$ isobarycenter of $XYZ.$ a circle $O,$ with center $D, XY$ are on $O.$ a circle $O',$ with center $G,$ $X$ and $Z$ are on $O'.$ 1/the ...
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2answers
85 views

Matrix; Linear transformations

Let $ ( x , y ) $ be the co-ordinates of a point P referred to a set of rectangular axes $OX$, $OY$. Then its co-ordinates ($x^{'}$,$y^{'}$) referred to $OX^{'}$, $OY^{'}$, obtained by rotating the ...
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1answer
136 views

How to find distance between two points in shown diagram.

In figure below P1 is center of circle whose radius is known, say 'r'. Distance of point M from P1,P2 and P3 is known. How to find out length of D (shown in red line). @EDIT : All distances are in ...
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1answer
246 views

Integrating $\int_0^{\pi/2} \cos^a(x) \cos(bx) \ dx$

Please help me in this integral : $$\int_0^{\pi/2} \cos^a(x) \cos(bx) \ dx \quad \text{if}\; b>a>-1$$ Please help me I used everything and can't evaluate it.
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4answers
1k views

Find all solutions of $4\cos^2(x)-4\sin(x)-5=0$

Find all solutions of $4\cos^2(x)-4\sin(x)-5=0$ in the interval $(6\pi, 8\pi)$. I tried to work it out and got: $4y^2-4y -9 = 0$, but I can't figure out what $\cos x = $from there to finish the ...
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1answer
326 views

Higher Derivatives of trigonometric functions

The position of a particle is given by $s = 5 \cos (2t+ (\pi/4))$ at time $t$ . What are the maximum values of the displacement,the velocity,and the acceleration? The answers are displacement: $5$ ...
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2answers
232 views

Find all solutions of $\cos (x) + 1/2 \sec (x) = -3/2$ in the interval $(2\pi, 4\pi)$

Find all solutions of $\cos (x) + 1/2 \sec (x) = -3/2$ in the interval $(2\pi, 4\pi)$ (Leave your answers in exact form and enter them as a comma-separated list.)
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5answers
355 views

What's the importance of the trig angle formulas?

What's the importance of the trig angle formulas, like the sum and difference formulas, the double angle formula, and the half angle formula? I understand that they help us calculate some trig ratios ...
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2answers
125 views

Sequence and Series - If $a_n =\int^{\frac{\pi}{2}}_0 \frac{\sin^2nx}{\sin^2x}dx,$…

If $\displaystyle a_n =\int^{\frac{\pi}{2}}_0 \frac{\sin^2nx}{\sin^2x}dx, $ then find the value of $$\begin{vmatrix} a_1 & a_{51} & a_{101} \\ a_2 &a_{52} & a_{102}\\ a_3 & ...
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2answers
2k views

How would you calculate the Tangent without a calculator? [duplicate]

I was just curious as to how you would calculate it without a calculator. I don't care if it's in radians or degrees, but I just would like it to be specified.
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3answers
400 views

Show that $\sin3\alpha \sin^3\alpha + \cos3\alpha \cos^3\alpha = \cos^32\alpha$

Show that $\sin3\alpha \sin^3\alpha + \cos3\alpha \cos^3\alpha = \cos^32\alpha$ I have tried $\sin^3\alpha(3\sin\alpha - 4 \sin^3\alpha) = 3\sin^4\alpha - 4\sin^6\alpha$ and ...
7
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1answer
3k views

How were Hyperbolic functions derived/discovered?

Trig functions are simple ratios, but what does Cosh, Sinh and Tanh compute? How are they related to euler's number anyway?
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2answers
126 views

Showing a particular function is onto

Consider $f: \mathbb{R} \rightarrow \mathbb{R}$ is given by $f(x) = x\cos(x)$, we want to show $f$ surjective. Usually these type of proofs aren't very difficult as you can isolate x in terms of y. In ...
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2answers
73 views

Sine Over Cosine Limit Doubt

Intuitively, I suppose that: $\displaystyle \lim_{x\to \infty} \dfrac{x + 5 \sin x}{x-\cos x} = 1$ Analytically, though, I get stuck at $\cos / \sin$ limits... Thanks.
3
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2answers
220 views

Proving a fact: $\tan(6^{\circ}) \tan(42^{\circ})= \tan(12^{\circ})\tan(24^{\circ})$

Prove that $\tan(6^\circ)\tan(42^\circ) = \tan(12^\circ) \tan(24^\circ)$. I don't know how to approach this problem. One approach might be to note that $42-6= 24+12$, and then apply the identities ...