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

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-1
votes
4answers
42 views

How to calculate the tangent of x?

I've looked it up of course and got $\tan(x) = \cos(x)/\sin(x)$. For example $\tan(60) = \cos(60)/\sin(60)$ I get $0.32004$ but when I use a calculator I get $1.7320508075688772935274463415059$? Is ...
2
votes
2answers
28 views

Reduction formulae in definite integration

$$I_n = \int_0^{\pi}\frac{\sin^2(nx)}{\sin^2(x)}dx $$ Find relation between $I_n$, $I_{n+1}$ and $I_{n+2}$ I tried integration by parts by taking $\sin^2(nx)$ as the first function, but reached ...
2
votes
4answers
60 views

Prove that $\cos \arctan 1/2 = 2/\sqrt{5}$

How can we prove the following? $$\cos \left( \arctan \left( \frac{1}{2}\right) \right) =\frac{2}{\sqrt{5}}$$
0
votes
2answers
39 views

Trying to solve $\sqrt{2\cos^2(x)-\sqrt{3}}+\sqrt2 \sin(x)=0$

The equation is $$\sqrt{2\cos^2(x)-\sqrt{3}}+\sqrt2 \sin(x)=0$$ I solve it thus: $$ \begin{cases} 2\cos^2(x)-\sqrt3=2\sin^2(x) \\ -\sqrt2 \sin(x)\ge 0 \iff \sin(x)\le 0 \end{cases} $$ The first ...
0
votes
1answer
20 views

Right triangle trigonometry help?

I've got a right triangle where I know the slope of side $c$ based on the two points $(-150,200)$ and $(0,0)$. Also I know the length of side $a$. I was wondering based on these two known factors how ...
-5
votes
0answers
16 views

prove the given question [on hold]

Prove that $\sec(2 \alpha)\cos(45^{\circ}-\alpha)\sin(45^{\circ}+\alpha) = \dfrac{1}{2}$.
1
vote
0answers
8 views

trying to grasp disphenoid tetrahedral honeycomb, what are the dihedral angles?

what are the dihedral angles in a disphenoid with 4 identical triangles of 1 edge length '2' and 2 edges of length 'sqrt(3)'? Tried to look it up, but couldn't find it... perhaps there is no such ...
2
votes
2answers
49 views

Find min of $M=\frac{1}{2+\cos2A}+\frac{1}{2+\cos2B}+\frac{1}{2-\cos2C}$

Find min of $$M=\frac{1}{2+\cos2A}+\frac{1}{2+\cos2B}+\frac{1}{2-\cos2C}$$, where $A, B, C$ are three angle of triangle $ABC$ Using Cauchy-Schwarz, we obtain: \begin{align*} M &= ...
0
votes
1answer
17 views

Intersection of angular ray with circle

I have a geometric/trigonometric problem. I will include a diagram but I know images are not ideal so I will do my best to describe the figure as well. Sorry for the Paint diagram. The angle corner ...
-2
votes
2answers
63 views

Resolving $x^5=i$ using algebra and trigonometry, prove that [on hold]

Resolving $x^5=i$ using algebra and trigonometry, prove that $\cos( 18^{\circ})=\frac{\sqrt{5+2\sqrt{5}}}{\sqrt[5]{176+80\sqrt{5}}})$ $\sin( 18^{\circ})=\frac{1}{\sqrt[5]{176+80\sqrt{5}}})$
-1
votes
0answers
44 views

Can $ \tan^2 \theta \sin^2 \theta$ be written as $ \sin^2 \theta \tan^2 \theta$? [on hold]

Is the following expression valid? $$ \tan^2 \theta \sin^2 \theta = \sin^2 \theta \tan^2 \theta$$
2
votes
0answers
40 views

Sum of arctans of trignometric expressions

Let $s_k=\sin\frac{2\pi(4k+1)}{4n}$ and $c_k=\cos\frac{2\pi(4k+1)}{4n}$ for some positive integer $n$. If $n=2007$ and $x=3$ , find $\tan \sum_{k=0}^{n-1} \arctan(\frac{s_k}{x-c_k})$ I tried using ...
0
votes
1answer
19 views

Rotated parabola 2d vertex

I'm implementing an application where I need to get the vertex of a parabola, the parabola might be tilted; so it can have an angle with the x-axis not necessarily vertical or horizontal. Can I get ...
2
votes
0answers
51 views

Will $x=0$ satisfy the equation $\sqrt{\tan(3x)}=\sqrt{-\tan(x)}$?

The equation is $$\sqrt{\tan(3x)}=\sqrt{-\tan(x)}$$ And the one condition set for the solution is that $x$ should fall within this range: $0\le x < \pi$ The solution process boils down to $$ ...
12
votes
3answers
131 views

Prove that $\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $

I've found the following identity. $$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ I could verify it by using CAS, and calculate the integrals in term of ...
0
votes
2answers
40 views

Maximum of $\cos \alpha_{1}\cdot \cos \alpha_{2}\cdot \cos \alpha_{3}…\cos \alpha_{n}.$

Maximum value of $\cos \alpha_{1}\cdot \cos \alpha_{2}\cdot \cos \alpha_{3}\cdot \cos \alpha_{4}....\cos \alpha_{n}.$ If it is given that $\cot \alpha_{1}\cdot \cot \alpha_{2}\cdot \cot ...
1
vote
4answers
35 views

Epsilon-Delta Limit Proof: Arccos(x) Inequalitiy

I'm studying a Calculus proof using notes (proving that $\lim_{x \to 1} \cos(x) = \cos(1)$ from the definition of limit). The text says that we get from: $\cos(1) −\epsilon < \cos(x) < ...
0
votes
4answers
659 views

Limits of cosine and sine [duplicate]

When $\theta$ is very small why $\sin \theta$ is similar to $\theta$ and $\cos\theta$ similar to $1$? Is it related to limits or we can prove it simply by using diagrams?
0
votes
1answer
42 views

Length of all sides of a triangle, knowing one angle one length and the perimeter of the triangle.

i am sure this question is answered in a round about way, but my math is not strong enough to put it all together so i need a direct answer for my direct question if you don't mind (: Now i did draw ...
3
votes
3answers
93 views

Proving uniqueness of solutions to $\sin^2A + \sin^2B = \sin (A+B)$ without using multivariable calculus

In the course of solving a trigonometric problem (see $a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle), in one approach the following equation ...
1
vote
2answers
64 views

Trig equation: $a \sin \frac{a \pi}{2} = 1$

How do I solve the following? I am having a bit of a slow moment. $$a \sin \frac{a \pi}{2} = 1$$
0
votes
4answers
47 views

Calculate area of a triangle with just one length and a tangent-relation(?)

I am looking through some old mathematics that I did 5 years ago and don't remember 100%. Right now I am learning about trigonometry and have some problem with a question. "The triangle ABC is ...
0
votes
5answers
204 views

can a real number be added to a complex number [on hold]

does it make sense to add a real to a complex given that addition binary operation is only defined for set of complex numbers OR real numbers also a related question: how can exponential $e^x$ which ...
0
votes
1answer
41 views

Calculating the resultant of two forces and angle? [on hold]

A force of $256 N$ and a vertical load of $537 N$. Trying to work out the resultant of the two forces and the angle at which it acts to the horizontal?
1
vote
1answer
52 views

a simple question: whence the $\pi$ symbols in the solution of a trig equation?

There's a step-by-step discussion of an example irrational trig equation in my textbook. $$\sqrt{3\sin(2x)}=\sqrt{-5\cos(x)\cot(x)}$$ One of the solutions is $$\cos(x)=-\frac23$$ The solution to ...
1
vote
2answers
53 views

Does $\sin^2(-x)$ simplify?

Does $\sin^2(-x)=-\sin^2(x)$, if not, does it simplify to something else?
0
votes
7answers
71 views

Why is $1+\cos(\theta)=2\cos^2(\frac{\theta}{2})$

Why is $1+\cos(\theta)=2\cos^2(\frac{\theta}{2})$? Where this comes from? I don't get it. From $\sin^2\theta+\cos^2\theta=1$?? I search everything but I really don't find that.
0
votes
2answers
31 views

how to find the value of the trigonometric function in the question

$$\text{if }\frac{\sin\theta}{\sin\phi}=\frac12 \text{ , }\frac{\cos\theta}{\cos\phi}=\frac32 \text{ ; if both the angles are the acute angle, then find } \tan\theta \text{ and } \tan\phi.$$ this ...
2
votes
8answers
82 views

Prove the trigonometric identity $\cos(x) + \sin(x)\tan(\frac{x}{2}) = 1$

While solving an equation i came up with the identity $\cos(x) + \sin(x)\tan(\frac{x}{2}) = 1$. Prove whether this is really true or not. I can add that $$\tan\left(\frac{x}{2}\right) = ...
1
vote
1answer
37 views

Dividing a trigonometric expression

Given: $$\sin {x} ⋅ \cos {3x} = \sin {x} ⋅ 2\sin {3x} ⋅ \cos {3x}$$ Can I divide by $\sin {x} ⋅ \cos {3x}$ ? If I check $\sin {x} ⋅ \cos {3x} = 0$ I get 2 more answers that are correct to the ...
0
votes
1answer
42 views

I apply the sum-to-product identity for $\sin$, but my result differs from the textbook's

I don't understand the last transformation here: $$\sin x - \cos 3x = 0\iff \sin x -\sin\left(\frac\pi2 - 3x\right) =0\iff 2\sin\left(\frac\pi4-x\right)\cos\left(2x-\frac\pi4\right)=0$$ When I apply ...
-1
votes
2answers
42 views

Simplify $\tan3x/\tan x$. Answer given is $(2\sin 2x +1)/(2\sin 2x-1)$ [on hold]

The question is to simplify $\displaystyle \frac{\tan{3x}}{\tan x}$. The answer given in my book is $\displaystyle \frac{2\sin 2x+1}{2\sin 2x- 1}$ but I am not getting this answer by solving it. Can ...
1
vote
1answer
33 views

Inverse sum representation of sine

The other day I was playing with functions of the form $$ f(x) = \frac{1}{\frac{1}{a_0(x-b_0)} + \frac{1}{a_1(x-b_1)} + \cdots + \frac{1}{a_n(x-b_n)}} $$ and I found particularly that $$ ...
1
vote
4answers
63 views

Trignometric Identities and Equations

For the following problem(s) I cannot get any answer(s). I would appreciate your help very much. $$\tan { \theta -\sec { \theta } =\sqrt { 3 } } $$ TI get 30 degrees as the reference angle. What ...
0
votes
2answers
42 views

Infinite series of trigonometric ratios

The question is to compute: $$(1+\cos A)+2(1+\cos A)^2 + 3(1+\cos A)^3+\ldots = \sum_{k=1}^{\infty}k(1+\cos A)^k.$$ I tried by setting $1+\cos A=y$, then the serie becomes $$y+2y^2+3y^3+\ldots = ...
2
votes
1answer
42 views

Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$.

Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$. Can I use Taylor's polynomial?
2
votes
1answer
46 views

Gaussian function in the limit of trigonometric functions

I've noticed that $$ (\sin\theta \cos\phi)^{2n} + (\sin\theta \cos\phi)^{2n-1} $$ increasingly resembles a Gaussian function of $(\theta, \phi)$ as $n$ goes to infinity. In particular, when I take ...
0
votes
0answers
15 views

Trigonometric identities for Bessel Functions?

I'm wondering if there exists extensions of trigonometric identities to special functions like Bessel? For example, is there an alternative way to express the following? $J_0((a+b)x) = ?$ Thanks
3
votes
1answer
45 views

Should cosecant be defined as $\csc \theta = \frac{1}{\sin \theta}$, specifying the constraint: $\sin \theta \neq 0$?

I'm studying trigonometry on my own, and I keep noticing that the trigonometric functions are never defined with constraints to deal with divide-by-zero issues. As an example, I've seen cosecant ...
2
votes
2answers
71 views

Prove $\frac{\sin (2016x)}{2016x}<\frac{\sin x}{x}$

How can we prove that $$\frac{\sin (2016x)}{2016x}<\frac{\sin x}{x}$$ with $x$ close to/near $0$ I don't know where to start from, but I think that we need to examine distinct cases of $x>0$ ...
1
vote
3answers
38 views

If $f’(x) = \sin x + (\sin4x)(\cos x)$, then $f’(2x^2 + \pi/2) $is?

If $$f'(x) = \sin x + \sin4x \cdot \cos x,$$ then $$f'(2x^2 + \pi/2)$$ is? Given answer: $$4x\cos(2x^2) – 4x\sin(8x^2) \sin(2x^2)$$ I tried and I'm getting the answer as $\cos(2x^2) - ...
2
votes
2answers
70 views

Solving $\cos^2{\theta}-\sin{\theta} = 1$

Can someone please help me solve this? $$\cos^2{\theta}-\sin{\theta} = 1, \quad\theta\in[0^\circ, 360^\circ]$$
0
votes
2answers
95 views

How to go about proving that $\cos(\frac{\pi}{2}-x) = \sin(x)$?

I have very little experience writing proofs so I don't know how to begin. I recognize that the statement is always true, but I can't go about proving it without using circular reasoning. How could ...
1
vote
2answers
35 views

Working out the length of the 3rd side of an isosceles triangle- Pythagoras' theorem

I have been revising some maths equations and see that you can work out the third side of an isosceles triangle using the formula $\sqrt2 x$ $x$ being one of the equal sides. Could someone explain ...
0
votes
2answers
48 views

find the coefficient

If $n$ is an odd natural number, and $\sin(n\theta) = \Sigma_{r=0}^{n} b_r \sin^r\theta$, then find $b_r$ in terms of $n$. I have tried this using trigonometric expansion but unable to find solution ...
3
votes
2answers
55 views

If $f(x) = \cos x\cos2x\cos4x\cos8x\cos16x$, then $f’(\pi/4)= ?$

If $f(x) = \cos x\cos2x\cos4x\cos8x\cos16x$, then $f’(\pi/4)= ?$ Ans: $\sqrt{2}$
0
votes
1answer
22 views

converting cos to sin and tan in specific quadrants

I'm having issues understanding as to how to go about doing this. I cant seem to figure out how to find the values of sin and tan in terms of the given cos value in the 3rd quadrant. Thanks with any ...
6
votes
4answers
530 views

How can I simplify this complex number to get a real number?

$$\large \frac {e^{i \frac{\pi a}{2}}[1-e^{i\pi a}]} {[1-e^{i2\pi a}]}$$ I am trying to arrive at $$\frac {1}{2\cos\left(\frac{\pi a}{2}\right)}$$ I've tried dividing top and bottom by one of the ...
0
votes
1answer
28 views

Get Distance Between Point and Side of Ellipse

So I have an ellipse where I know the two foci, the length and the width and all the relevant information. I then have a point somewhere in the ellipse. This point is known and an arbitrary angle ...
3
votes
1answer
45 views

Prove that $IL,JK$ and angle bisector of angle $BCD$ are concurrent

Given a convex quadrilateral $ABCD$. In $\Delta ABC$, $I$ is the incentre and $J$ is the excentre opposite to vertex $A$. Similarly, $K$ is the incentre and $L$ is the excentre opposite to vertex $A$ ...