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

learn more… | top users | synonyms (1)

0
votes
2answers
35 views

Trigonometric identity on $\cos \pi/7$

I found this in a book I used to train myself for the Math Olympics a bunch of years ago: Prove that $$\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}=\frac{1}{2} $$ I couldn't solve it ...
2
votes
3answers
37 views

What are the roots of $\sin(ax) + \sin((a + 2)x)$?

I was playing around with $\sin(5x) + \sin(7x)$, wondering where the roots of the function are. I graphed it on wolframalpha and from the list of solutions I guessed that the solutions to $\sin(5x) + ...
0
votes
2answers
53 views

Trigonometric identity for $\sin 6x$

I got this question from my teacher: $\sin {6x}=\dots$ Try to make this one from this: $\sin(3x+3x)$, then according to the formula ended up like this: $$2\sin{(2x+x)}\cos{(2x+x)}$$ ...
0
votes
3answers
44 views

Formula of Trigonometry

I have a question like this: If $A + B + C = 90°$ then find $\sin \frac{B + C}{2}$ I try to make it start from this part: $B = 90° - A - C$, $C = 90° - B - A$ Then substitute both of it, and ...
3
votes
0answers
49 views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
0
votes
1answer
30 views

Trigonometry about formula of trigonometry [on hold]

if $\tan 2x = 2$ then find $\tan x$ if $A + B + C = 90$ then $\sin \left(\frac{B + C}{2}\right)$ $\tan 9 - \tan 27 - \tan 63 + \tan 81$ $\sin 6x$ $\sin 18 \cos 36$
-2
votes
0answers
40 views

Trigonometric equality to prove [on hold]

I am trying to prove that $\frac{1}{32}(\cos(6\theta)-2\cos(4\theta) -\cos(2\theta) + 2)= \frac{1}{64}((\cos(4\theta)4\cos(2\theta)-4)-4\cos(2\theta)+4)$ I have already tried use various trig ...
1
vote
2answers
19 views

Rewriting a trigonometric inequality (including a parameter)

How is it possible to rewrite these equations? $\sin{x}- \cos{x} ≤ \mu(\cos{x} + \sin{x}) \implies \tan{}x ≤ \frac{1 + \mu}{1 - \mu}$ and $\cos{x}- \sin{x} ≤ \mu(\cos{x} + \sin{x}) \implies \tan{}x ...
0
votes
1answer
20 views

Trigonometric AP relation on sides of a triangle

The sides of a triangle are in AP (Arithmetic Progression) and the greatest angle exceeds the least angle by $90$ degrees prove that the sides are proportional to $7^{\frac{1}{2}}+1$ , ...
1
vote
0answers
23 views

Calculate a point on a geodesic line on an ellipsoid

I have a problem which i don't understand how to achieve. Maybe someone could sheed some light on it. Have a look at this picture: What I try to achieve is to determine the point D on the geodesic ...
0
votes
4answers
53 views

$\tan \theta =b/a$ then find the value of $a\cos2\theta+b\sin2\theta$

Given $\tan \theta =b/a$, then find $a\cos2\theta+b\sin2\theta$ in terms of $a$ and $b$. I tried to solve the problem by first converting $\sin2\theta$ and $\cos2\theta$ in the $\tan$ terms ...
0
votes
2answers
48 views

Prove that the expression is Independent of theta

Prove that $2\sin^2 \theta +4 \cos(\theta + \alpha ) \sin \alpha \sin \theta +\cos(2(\theta + \alpha))$ is independent of $\theta$. How do we solve such problems ?
0
votes
2answers
54 views

Trignometric problem

Show that: $$\large2^{\sin{x}} + 2^{\cos{x}} \ge 2^\left({1-\frac{1}{\sqrt{2}}}\right)$$ This looks like an am gm problem to me where we should be using the fact that am is more that or equal to gm ...
3
votes
3answers
55 views

Evaluate the limit of $\ln(\cos 2x)/\ln (\cos 3x)$ as $x\to 0$

Evaluate Limits $$\lim_{x\to 0}\frac{\ln(\cos(2x))}{\ln(\cos(3x))}$$ Method 1 :Using L'Hopital's Rule to Evaluate Limits (indicated by $\stackrel{LHR}{=}$. LHR stands for L'Hôpital Rule) ...
2
votes
0answers
23 views

Decomposition of $a\sin(\varphi t)+b\sin(\vartheta t)$ into AM and carrier

I feel like this should not be so hard, but I am somehow stuck. I would like to decompose the signal $$a\sin(\varphi t)+b\sin(\vartheta t)$$ into an amplitude modulation and a periodic carrier ...
1
vote
1answer
46 views

Hard Trigonometric Equation

its possible to solve a equation like $$\prod^{45}_{k = 0} \left( 1 + \tan \frac{k \pi }{180} \right) = \left[ \log_{\frac{\sqrt{6}}{3}} \big| \sin(2x)\big| \right]^{\frac{9}{2}\sin(2x) + 20}$$ ...
0
votes
1answer
24 views

Trigonometric Partial Derivative

I need to find $$\frac{\partial Z}{\partial U} \text{ and } \frac{\partial Z}{\partial V}$$ for a $z=f(x,y) = \cos(xy) + y\cos(x)$. After a bit of an internet search, I think I have found the ...
4
votes
4answers
62 views

How does $x^3 - \sin^3 x$ become $x^3 + \frac{1}{4}\sin{3x}-\frac{3}{4}\sin x$?

I was going through answers on this question and came across this answer and I was wondering how the user arrived at the first line where they state: $$f(x) \equiv x^3 - \sin^3 x = x^3 + {1 \over 4} ...
0
votes
1answer
37 views

What is the graph of $y = \sin n$ and why is it different from the graph of $y = \sin x$?

I have downloaded a book about Calculus from MIT OCW. In that book, there is a section "A Thousand points of Light". (You can download the relevant section from here.) In that section, it is written ...
1
vote
9answers
137 views

Find $\lim_{x\to0}\frac{\sin5x}{\sin4x}$ using $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$.

I am trying to find $$\lim_{x\to0}\frac{\sin5x}{\sin4x}$$ My approach is to break up the numerator into $4x+x$. So, $$\begin{equation*} ...
0
votes
0answers
20 views

How to calculate the coordinate of a point which depends on other points on a plane with specific distances

I have $8$ points on a plane $(x_1,y_1)....(x_8,y_8)$ among these $8$ points I know the coordinates for $7$ points and I have to find the $8^{th}$ point. Each points has the difference between all ...
2
votes
2answers
36 views

Find the smallest positive number $p$ for which the equation $\cos(p\sin x)=\sin(p \cos x)$ has a solution $x\in[0,2\pi].$

Find the smallest positive number $p$ for which the equation $\cos(p\sin{x})=\sin(p\cos{x})$ has a solution $x$ belonging $[0,2\pi]$. I am not able to solve this problem. Please help me.
-5
votes
0answers
34 views

Trigonometric AP [on hold]

If $a, b, c $ are in $AP$ prove that $$\cos(A)\cot\left(\frac{A}{2}\right),\quad\cos(B)\cot\left(\frac{B}{2}\right),\quad\cos(C)\cot\left(\frac{C}{2}\right)$$ are in $AP$ too. Where $a, b, c$ ...
2
votes
6answers
213 views

Algebraic proof of $\tan x>x$

I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 ...
2
votes
2answers
34 views

How to find perpendicular point of a vector to another vector 2d

Given the axis x-y and some random points to the vectors AB and CD, how can i find out where will the point D lie when the vector CD(dashed line) is perpendicular to AB. For example if point A has ...
1
vote
1answer
68 views

Why was $\tan\theta$ used?

In reference to $(a)$, I do not understand why the tan ratio was used? Tan is $\frac{Opposite} {Adjacent}$ . Where should I be looking at to be able to determine the $\frac{Opposite} {Adjacent}$? ...
-1
votes
4answers
72 views

Why is $\sin 30^\circ=\frac{1}{2}$

Take half a square with side length $1$. The resulting right-angled triangle ABC has two angles of $45^\circ$. By Pythagoras’ theorem, the hypotenuse AC has length $\sqrt{2}$. Applying the definitions ...
0
votes
1answer
48 views

Find $\sin(x+y)$, given $\tan x$ and $\cos y$

Given that $\tan x= -2$ and $\cos y= 1/2$ where $x$ and $y$ are in the 4th and 1st quadrants respectively. Find, without evaluating angles $x$ and $y$, a) $\sin (x+y)$ Here is what i have done so ...
1
vote
1answer
32 views

For all $x \in [0,90]$ show that $\cos(\sin(x))>\sin(\cos(x))$ diff from the one which I posted earlier

For all $x \in [0,90]$ show that $\cos(\sin(x))>\sin(\cos(x))$ I understood the solution given in my book which said  $\cos(x)+\sin(x)≤\sqrt{2}<pi/2$ $\cos(x)<pi/2−\sin(x)$. Over here if ...
2
votes
0answers
52 views

Eliminate variable in trigonometry equations

Say you have the equations: \begin{align} -S_1\sin\left(2\psi+\theta\right)+S_2\cos\left(\psi\right)&=S_3\\ S_1\cos\left(2\psi+\theta\right)+S_2\sin\left(\psi\right)&=S_4 \end{align} or ...
0
votes
1answer
35 views

Complex trigonometry problem

The value of $$\large \displaystyle e^{\log(\tan 1^\circ) + \log(\tan 2^\circ)+ \cdots+\log(\tan 89^\circ)}$$ Base is $10$. I guess it should simplify to $\large\displaystyle e^{89 \log(\tan ...
0
votes
1answer
16 views

Eliminating theta problem

Eliminate theta if $\tan(\theta - \alpha)=a$ and $\tan(\theta + \alpha)=b$. I tried using the $\tan (a+b)$ and $\tan(a-b)$ identity and adding the two equations but $\theta$ failed to get eliminated. ...
1
vote
4answers
55 views

Find 3rd side, given two sides and bearings

The bearing from A to B is N $42^\circ$ E. The bearing from B to C is S $44^\circ$ E. A small plane traveling $65$ miles per hour, takes $1$ hour to go from A to B and $2$ hours to go from B to C. ...
2
votes
3answers
28 views

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$?

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$? I am learning trigonometric identities one identity I have to proof is the next: $ (1- \sin \alpha + \cos \alpha)^2 = ...
1
vote
1answer
34 views

Eliminate $A$ from these two equations.

$$x = \cot A + \tan A$$ $$y = \cos A + \sec A$$ Eliminate $A$ from these two equations. We tried doing $y^2 - x^2$ but it did not eliminate $A$.
0
votes
3answers
66 views

Trigonometry - Find the exact value of $\sin15^\circ$

I am having problems understanding how to solve $\frac{1}{4}(\sqrt6 - \sqrt2)$, find the exact value of $\sin15^\circ$. I have the answer, but I need help understanding the methods to achieve the ...
0
votes
3answers
36 views

Derivation of sine and cosine case

I am struggling to see this. I know that we can factor out $ a$, but I don't see how we can end up with the right hand side. $$a \cos ^2(a t)-a \sin ^2(a t)=a \cos (2 a t)$$
1
vote
0answers
14 views

Trig equation that fits the plot points (octagonal pyramid)

I'm looking for an equation that satisfies these conditions: Input 90 degrees, result is 90 degrees Input 45 degrees, result is 60 degrees Input 0 degrees, result is 45 degrees For an input value ...
1
vote
1answer
23 views

How could I calculate the local size of an object given its distance and actual size?

Lets say I take a picture of a sign. I know that sign is 20ft (width), 10ft height. I'm standing 40 feet away. If I were to take a picture, how could I calculate how wide and how high the sign is in ...
2
votes
4answers
56 views

Show that there is an angle $\theta$ such that $a=\cos\theta$ and $b=\sin\theta$

My problem is from Israel Gelfand's Trigonometry textbook. Page 50. Exercise 3: Suppose that $\alpha$ is some angle. If $a=4\cos^3\alpha-3\cos\alpha$ and $b=3\sin\alpha-4\sin^3\alpha$, show that ...
1
vote
4answers
61 views

Trignometry-Prove that $(\csc\theta - \sec\theta )(\cot \theta -\tan\theta )=(\csc\theta +\sec\theta )(\sec\theta ·\csc\theta -2)$

Prove that $$(\csc\theta - \sec\theta )(\cot \theta -\tan\theta )=(\csc\theta +\sec\theta )(\sec\theta ·\csc\theta -2)$$ I tried solving the LHS and RHS seperately but they were not coming out to be ...
-2
votes
1answer
69 views

For all $x$ in $[0,90]$ show that $\cos(\sin x ) >\sin(\cos x )$

For all $x$ in $[0,90]$ show that $\cos(\sin x ) >\sin(\cos x )$ I understood the solution given in my book which said  $$\cos(x)+\sin(x)\leq\sqrt{2}<90$$ $$\cos(x)<90-\sin(x)$$ But if ...
0
votes
4answers
41 views

How to find an angle (in degrees) in a right triangle, given its sides?

I need to find out a degree of an angle. Pretty simple, or so I thought. I remember doing a crap-ton of these in high-school, sadly the details did not remain. Anyway, let's take a look at this ...
1
vote
3answers
30 views

Basic question about angles

Why is the answer a)? Why can't it be d)? Why are the choices listed in this format, i.e., $(x \pm \theta^{\circ})$, and why is angle C $(x+30^{\circ})$ and not just $30^{\circ}$? Thanks.
6
votes
5answers
564 views

A strange trigonometric equation

Today,in our class, we received a trigonometric equation $$\sin^{10}{x}+\cos^{10}{x}=\frac{29}{16}\cos^4{2x}$$ and the question was to find the general solution of this equation. My approach was, at ...
2
votes
3answers
179 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
5
votes
4answers
81 views

Simplify $2 \sin(x) \cos(7x) + \sin(6x)$

I was doing a problem and in my chain of computations I arrived at a seemingly complicated function $$2 \sin(x) \cos(7x) + \sin(6x)$$ However, I typed it into Wolfram and was surprised to find $$2 ...
3
votes
3answers
49 views

The average value of the function $y=\tan(2x)$ over the interval $[0,\frac{\pi}{8}]$

I was given the following question in a technology free exam. How would one go about solving this without the use of a calculator? NB. I am currently in my last year of high school so please don't ...
-1
votes
1answer
43 views

How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
8
votes
2answers
152 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...