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

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0
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2answers
33 views

Establish identitiy $(\cos\theta - \sin\theta)^2 + (\cos\theta + \sin\theta)^2 = 2$

Hey guys this is the question: $$(\cos\theta - \sin\theta)^2 + (\cos\theta + \sin\theta)^2 = 2$$ I did this before but I forgot how I did it.. I tried to do this but don't know how to continue.. ...
6
votes
4answers
669 views

Integration of arctan(x) is itself?

Sorry if this seems like a useless question, but I can't seem to work out what's wrong with the following reasoning, which leads to what is definitely an incorrect answer. so, $$ \int \arctan(x) dx = ...
2
votes
2answers
52 views

prove $\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$

Today I found the identity : $$\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$$. How to prove or disprove this? Thank you.
1
vote
1answer
64 views

How do I show that $\frac {\cos^2 A}{\cos^2 B} + \frac {\cos^2 B}{\cos^2 C} + \frac {\cos^2 C}{\cos^2 A} \ge 4(\cos^2 A + \cos^2 B + \cos^2 C)$?

Let $A, B, C$ be the angles of an acute triangle. Show that $$\frac {\cos^2 A}{\cos^2 B} + \frac {\cos^2 B}{\cos^2 C} + \frac {\cos^2 C}{\cos^2 A} \ge 4(\cos^2 A + \cos^2 B + \cos^2 C).$$ How should ...
0
votes
0answers
54 views

Show that triangle $ABC$ is isosceles if $\sin A=2\sin B\sin C$

The exercise is that $$\sin A=2\sin B\sin C$$ I need to show that $B=C$. Then I try to use $\sin(\frac A2)=\sin B$ and $\cos(\frac A2)= \sin C$
0
votes
2answers
49 views

Solve $2\cos x^{\circ} + 3\sin x^{\circ} = -1$ where $0 \le x \le 360$

Solve $2\cos x^{\circ} + 3\sin x^{\circ} = -1$ where $0 \le x \le 360$ I am not asked to use any form so I am going to use $k\cos(x-alpha)$ $2\cos x^{\circ} + 3\sin x^{\circ} = k\cos(x-alpha)$ $$= ...
0
votes
1answer
32 views

$2\cos B=\cos C+\cos A $ then the area of triangle ABC can be expressed as $p\sqrt q/r \ $ & g.c.d of p,r =1 & ..

Triangle ABC has an in-radius of 5 and a circum-radius of 16. If $2\cos B=\cos C+\cos A$ then the area of triangle ABC can be expressed as $p\sqrt q/r$, where $p,q,r$ are Natural numbers and g.c.d of ...
0
votes
1answer
36 views

Find the inclination of the legs to the horizontal, and the height of the apex.

The legs of a tripod are each $10$ cm. in length,and their points of contact with a horizontal table on which the tripod stands from a triangle whose sides are $7,8,9$ cm in length.Find the ...
3
votes
2answers
47 views

Properties of sin(x) and cos(x) from definition as solution to differential equation y''=-y

I recently came across the interesting definition of the sine function as the unique solution to the Initial Value Problem $$y'' = -y$$ $$y(0) = 0, y'(0) = 1$$ (My first question would be why this ...
3
votes
1answer
62 views

Given that $\cos A + \cos B + \cos C = 0$ and $\sin A + \sin B + \sin C = 0$.

If $\cos A + \cos B + \cos C = 0$ and $\sin A + \sin B + \sin C = 0$. The value of $ \sin^3A+\sin^3B+\sin^3C$ What I can see here is that as $\sin A + \sin B + \sin C = 0$ hence $ ...
0
votes
1answer
13 views

Find the triangle with the greatest area using trigonometric ratios

The hypotenuse, c, of right $\triangle$ABC is $7.0$cm long. A trigonometric ratio for angle $A$ is given for four different triangles. Which of these triangles has the greatest area? a) sec $A$ = ...
4
votes
3answers
80 views

How to proof $\sum^{n}_{k=1}\sin^2\left(x+ \frac{\pi(k-1)}{n} \right)=\frac{n}{2}$ [duplicate]

I would appreciate if somebody could help me with the following problem: Q: How to proof? $(x\in \mathbb{R})$ $$\sum^{n}_{k=1}\sin^2\left(x+ \frac{\pi(k-1)}{n} \right)=\frac{n}{2}$$
0
votes
1answer
29 views

Linear algebra - Proof of a thesis concerning the height in triangles!

My Math teacher gave us some tasks we should work on. I solved most of them already, however I still could not manage to figure out the solution for this one! I would really appreciate, if someone of ...
-2
votes
1answer
39 views

Find all possible solutions between 0 and 2π [closed]

I really could use help here, i am trying to find all possible solutions between 0 and 2 pi.
1
vote
0answers
38 views

Is there a general rule to find period of multiplied functions?

We know that $g(x)$ and $f(x)$ are both periodic and trigonometric functions and we also know its period interval. How can we find the period of the function $f(x)g(x)$?
0
votes
0answers
15 views

Proof of alternate cartesian to polar transformation of theta

My vector calculus lecturer has claimed that rather than the angle $\theta$ in the transformation from cartesian coordinates $(x,y)$ to polar coordinates $(r,\theta)$ can not only be given by: $$ ...
0
votes
2answers
28 views

$f(x) = x^2 - \sin2x$ function, slope and degrees

I'm new here and sorry for my bad English, not my first language. Anyway, I have this function: $f(x) = x^2 - \sin2x\;,\;\;\left[-\frac\pi2< x <0\right]$ And I've been asked to find what is: ...
0
votes
4answers
55 views

In an triangle the least angle is $45^\circ$ and the tangents of the angles are in $A.P.$If its area be $27$ sq.cm.Find the lengths of its sides.

In an triangle the least angle is $45^\circ$ and the tangents of the angles are in $A.P.$If its area be $27$ sq.cm.Find the lengths of its sides. Let $A$ be the smallest angle.$\angle ...
1
vote
1answer
16 views

If one angle of a triangle be $60^\circ,$ the area $10\sqrt3$ sq cm.,and the perimeter $20$ cm,find the lengths of the sides.

If one angle of a triangle be $60^\circ$, the area $10\sqrt3\ \mbox{cm}^2$, and the perimeter $20\ \mbox{cm}$, find the lengths of the sides. Let $\angle A=60^\circ$ and $\frac{1}{2}bc\sin ...
1
vote
1answer
33 views

Prove that $\sin^2\frac{A}{2}\csc2A$, $\sin^2\frac{B}{2}\csc2B$, $\sin^2\frac{C}{2}\csc2C$ are in harmonic progression

If sides $a,b,c$ of $\triangle ABC$ are in arithmetic progression (AP), then prove that $$\sin^2\frac{A}{2}\csc2A, \quad\sin^2\frac{B}{2}\csc2B, \quad \sin^2\frac{C}{2}\csc2C$$ are in harmonic ...
1
vote
1answer
31 views

Prove that there are two values to the third side,one of which is $m$ times the other.

Let $1<m<3$. In $\triangle ABC$, if $2b=(m+1)a$ and $\cos A=\frac{1}{2}\sqrt{\frac{(m-1)(m+3)}{m}}$, prove that there are two values to the third side, one of which is $m$ times the other. ...
2
votes
4answers
62 views

Solve equation $8\cos x - 6\sin x = 5$ where $0 \le x \le 360$

Solve equation $8\cos x - 6\sin x = 5$ where $0 \le x \le 360$. I am not asked to use any form, so I am going to use $k\cos(x-\alpha)$. $8\cos x - 6\sin x = k\cos(x-\alpha)$ $$=k(\cos x\cos\alpha + ...
1
vote
0answers
25 views

Prove a harmonic range from a familiar picture

During solving some simple problem (10th grade), I found this interesting problem, which I got no clue to solve it clean and properly. Hope someone can give me some hint to solve it. Thanks. Given ...
0
votes
1answer
66 views

Trigonometric equation: $\sin^3x+\cos^3x+\sin^2x+\sin x+\cos x=2$.

Well, the title says it all. I've tried utilizing the fact that $\sin^3x+\cos^3x=(\sin x+\cos x)(1-\sin x\cos x)$ and then the equation becomes $(\sin x+\cos x)(2-\sin x\cos x)=1+\cos^2x$. Squaring ...
1
vote
1answer
45 views

Estimate trigonometric functions with complex argument

I would like to prove the following estimates $\vert \sin(z)\vert\leq \sinh(s)$ and $\vert \cos(z)\vert\leq \cosh ( s )$ ,where $z\in D_s(0)\subset\mathbb{C}$ and $D_s(0)$ denotes the disc with ...
0
votes
2answers
48 views

Why does $\cos(\pi x)$ approach $0^-$, as $x\to{\frac{1}2}^-$

In problem 220, I understand the rest of the solution except that I don't know why $\cos(\pi x)$ approaches to $0$ from the left, making the solution $-\infty$. I have seen a few problems like this, ...
0
votes
1answer
23 views

Rearrangement of Complex Sin and Cos

From my complex numbers course notes, there is the following derivation: The definitions of sin and cos I'm very comfortable with, but I cannot see how we get from the definition to the given ...
3
votes
2answers
94 views

Prove that $6(\sin^{10}A+\cos^{10}A) – 15(\sin^8A+\cos^8A) + 10(\sin^6A+\cos^6A) – 1 = 0​$ [closed]

Prove that: ​$$6(\sin^{10}A+\cos^{10}A) – 15(\sin^8A+\cos^8A) + 10(\sin^6A+\cos^6A) – 1 = 0​$$ Expression can be verified for different values of $A$ such as $\frac{\pi}{4},\frac{\pi}{2}$ etc. But ...
2
votes
0answers
50 views

Integral with irrational functions and polynomials

I thought this integral was simple, but it turns out it's not. $$\int \frac{xdx}{\left(1-x^3\right)\sqrt{1-x^2}}$$ I tried the substitution $1-x^3=\frac{1}{t}$, but that leaves me again with $x = ...
0
votes
3answers
48 views

Prove that $ \tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{7} + … + \tan^{-1}\frac{1}{n^2+n+1} = \tan^{-1}\frac{n}{n+2}$

Prove that $$ \tan^{-1}\frac{1}{3} + \tan^{-1}\frac{1}{7} + ... + \tan^{-1}\frac{1}{n^2+n+1} = \tan^{-1}\frac{n}{n+2}$$ I have been trying to solve it step by like $ \tan^{-1}\frac{1}{3} + ...
0
votes
1answer
58 views

How to solve $\cos3\theta + 7\cos\theta=4$

If we expand $\cos 3\theta$ in the equation $\cos3\theta + 7\cos\theta=4$ then we end up with $\cos^3\theta+\cos\theta-1=0$ but is it possible to solve the first one directly?
0
votes
2answers
40 views

Find $\frac{\cos A}{p_1}+\frac{\cos B}{p_2}+\frac{\cos C}{p_3}=$

Let $p_1,p_2,p_3$ be the altitudes of $\triangle ABC$ from vertices $A,B,C$ respectively, $\Delta$ is the area of the triangle,$R$ is the circumradius of the triangle,then$\frac{\cos ...
0
votes
1answer
57 views

Does there exists an addition formula for $\arctan(2a)$? [closed]

I want to find an expression for $\arctan(2a)$ in terms of a sum of $\arctan$s, so $\arctan(2a) = c_1\cdot \arctan(b_1)+ c_2\cdot \arctan(b_2)$. Does there exists such a formula? I tried to derive it ...
1
vote
3answers
39 views

Solve limit $\lim_{x\to 0} \frac{x - \sin (x)}{(x \sin (x))^{3/2}}$

Hi I must solve the next limit $\infty - $\infty$ usibg L'Hopital and Taylor series. $\lim\limits_{x \to 0} \frac{x - \sin (x)}{(x \sin (x))^{3/2} }$ I tried to eliminate the root with ...
0
votes
0answers
30 views

If $\frac{1-3\epsilon}{2(1+\epsilon)} \le \cos\theta \le \frac{1+3\epsilon}{2(1-\epsilon)}$, then how does $\theta$ compare to $\frac{\pi}{3}$?

I have $$\frac{1-3\epsilon}{2(1+\epsilon)} \le \cos\theta \le \frac{1+3\epsilon}{2(1-\epsilon)}\qquad\qquad\text{where}\quad 0 < \epsilon < 1$$ What can be said about the value of $\theta$ ...
0
votes
1answer
30 views

Operator confusion when writing trigonometric expression in $k\sin(x-\alpha)$

If I want to rewrite the following 2 expressions in the form $k\sin(x-\alpha)$ a) $4\cos x - 3\sin x$ b) $\cos x + \sin x$ I am confused what happens to the operator in each case. In case a ...
1
vote
1answer
39 views

Representing a number as $a^2+db^2$ given $d$

Given integers $n$ and $d$, how can I find integers $a$ and $b$ (or show that they do not exist) such that $n=a^2+db^2$? If it helps, in my present application I know the factorizations of $n$ and ...
0
votes
2answers
37 views

How to find $\sin\left(-\frac{11\pi}{12}\right)$ from the unit circle?

I Want Above To Be In Unit Circle.... unit circle. like convert $\frac{\pi}{3}$ to $\frac{4\pi}{12}$ to make it same denom... and then do stuff.... i tried to do with $\frac{\pi}{2}$ (convert to ...
2
votes
2answers
22 views

Understanding simplifications of complex terms $\exp(-ik\pi/4)$

I read that $1\over{2}$$\pi$$i$($\exp[-3i\pi/4]+\exp[-9i\pi/4])$ = $1\over{2}$$\pi$$i$($-\exp[i\pi/4]+\exp[-i\pi/4])$ = $\pi$$\sin(\pi/4)$ = $\pi\sqrt{2}$ Can you help me to understand how we move ...
0
votes
1answer
27 views

Equating coefficients explanation for trigonometric identities

Equating coefficients for the trigonometric identities does not make sense for me and I will explain why: If I use the example Write $5\cos x - 3 \sin x$ in the form $k\sin(x - \alpha)$, where $0 ...
10
votes
4answers
216 views
+100

How do I transpose an ellipse function to stretch the ellipse into curved space?

I'm working on an engineering project, using CAD software. I can write simple parametric functions to draw an ellipse, with $\theta$ ranging from $0$ to $2\pi$ ...
0
votes
2answers
21 views

Simplifying Inverse Trig Function

I'm trying to figure out how to simplify this expression but I'm not quite sure on how to approach this question. How should I approach this question? Any help is greatly appreciated! ...
3
votes
1answer
67 views

proof of a definite integral

Is there a proof of the definite integral function given below. I found the result by trial and error method and is stumped by the simplicity of the result. It seems to work for all the values for n > ...
0
votes
1answer
77 views

Prove that $R_1+R_2+R_3=R+r$,where $R$ is the circumradius and $r$ is the inradius of $\triangle ABC.$

Consider a triangle $DEF$,the pedal triangle of the triangle $ABC$ such that $A-F-B$ and $B-D-C$ are collinear.If $H$ is the incenter of $\triangle DEF$ and $R_1,R_2,R_3$ are the circumradii of the ...
0
votes
1answer
55 views

Prove that $2A+A_0=A_1+A_2+A_3$,where $A$ is the area of the triangle $ABC.$

If $A_0$ denotes the area of the triangle formed by joining the points of contact of the inscribed circle of the triangle $ABC$ and the sides of the triangle;$A_1,A_2,A_3$ are the corresponding areas ...
-4
votes
0answers
75 views

I asked somewhere about the relation between angles and pi and I was told incoherent.simple aritmetics [closed]

==consecutive numbers new way without the need of pi== $a,b,c$ $a<b<c$ $\frac{a}{2}+\frac{c}{2}=b$ $a+b+c=3b$ $a+c=2b$ $\frac{a+b+c}{a+c}=\frac{3}{2}$ $(1-\frac{a}{c})\times c=c-a$ ...
1
vote
3answers
41 views

Write $4\cos x - 3 \sin x$ in the form $k\sin(x - \alpha)$ where $0 \le \alpha \le 360$

Write $4\cos x - 3 \sin x$ in the form $k\sin(x - \alpha)$ where $0 \le \alpha \le 360$. $4\cos x - 3 \sin x = k\sin(x - \alpha)$ => $k(\sin x \cos \alpha - \cos x \sin \alpha)$ => $k\cos ...
0
votes
0answers
18 views

write $-10\cos x -7\sin x$ in the form $k\cos(x - \alpha)$ where $0 \le \alpha \le 360$.

write $-10\cos x -7\sin x$ in the form $k\cos(x - \alpha)$ where $0 \le \alpha \le 360$. $-10\cos x -7\sin x= k\cos(x - \alpha)$ => $k(\cos x \cos \alpha + \sin x \sin \alpha)$ => $k\cos \alpha ...
3
votes
3answers
87 views

How to evaluate this limit using Taylor expansions?

I am trying to evaluate this limit: $\lim_{x \to 0} \dfrac{(\sin x -x)(\cos x -1)}{x(e^x-1)^4}$ I know that I need to use Taylor expansions for $\sin x -x$, $\cos x -1$ and $e^x-1$. I also realise ...
1
vote
3answers
67 views

How to simplify $y = \sin(\frac{\arcsin(x)}{n}), n≥1$?

$y = \sin(\frac{\arcsin(x)}{n}), n≥1$ I know that: $\lim \limits_{x \to 0} \frac{x}{y} = n$ But I can't figure out what the curve of $x/y$ practically represents. Is there an obvious simple ...