Questions tagged [trigonometry]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
5 views

Calculating pitch and roll from an IMU relative to a direction

I have an IMU sensor that reports its orientation in the form of a unit quaternion. The sensor resides in a device that will be placed in a moving vehicle. In the ideal case, the sensor is properly ...
1
vote
3answers
28 views

confusion about negative trigonometric identities

Given that $\cos A=1/2$ and $\cos A$ and $\sin A$ have the same sign, find the value of $\sin(-A)$. If the question is referring to the first quadrant, where all trigonometric identities are positive, ...
0
votes
2answers
34 views

If in a triangle $ABC$, $b = ( a + c ) \cos \theta,$ find $\sin \theta$

If in a triangle $ABC$, $b = ( a + c ) \cos \theta$, find $\sin \theta$ Please help, I wasn't able to figure out how to solve this.
-3
votes
0answers
16 views

Prove Closed square theta- cot square theta=1 [on hold]

Trigonometric identities of mathematics class 10
0
votes
1answer
16 views

Determine potential coordinates for point $A$ on the terminal arm if angle $\theta$ lies in Quadrant $2$ with $\sin\theta = 3/\sqrt{45}$

I want to learn how to complete questions like these but when I look at the equation all I think I can do is simplify $\sqrt{45}$ into $3\sqrt{5}$. What do I do next? It says quadrant $2$, so the sin ...
0
votes
0answers
23 views

Geometric Sequence Complex Numbers

Can someone evaluate this? I'm having trouble as I could not get that answer I get $$-i(\frac{z(z^{44}-1)}{z-1} - \frac{z^{-1}(z^{44}-1)}{z^{-1}-1}) + 1 $$ $$-i(z^{44}-1)\frac{z+1}{z-1} +1$$ since ...
0
votes
1answer
37 views

What is the solution to arctan(i) [on hold]

I have no idea how to anywhere with this problem, can somebody please help me answer this question, I am a beginner at trig and am looking for some help
0
votes
1answer
24 views

General solution of intersection of sines

Is there a general way to solve this equation for $x$: $$ A_1 \sin ( B_1 x + C_1) = A_2 \sin ( B_2 x + C_2) $$ where $A_1, B_1, C_1, A_2, B_2, C_2$ and $x$ are real, and $A_1, B_1, A_2$ and $B_2$ ...
3
votes
3answers
80 views

Why cant I prove this trigonometric equation straight down?

The question is as follows: Given that $\tan^2a - 2 \tan^2b = 1$. Show that $\cos2a + \sin^2b = 0$. After a few attempts, I successfully came up with a solution as follows: $$\tan^2a - 2 \tan^2b = ...
-3
votes
0answers
29 views

Find earth radius [on hold]

Given two equations: 1) A=inverse sine(21/R). 2) 0.0008=R-R*cos(A) . A Laser beam points horizontally at height of 1.5 meter above sea level to distance of 21 km. The drop on the the other end is ...
1
vote
1answer
48 views

What connection do the hyperbolic trig functions have to the actual trig functions?

As far as my understanding goes, trigonometry is the math of right triangles. Sine is the opposite side over the hypotenuse, cosine is the adjacent side over the hypotenuse, etc. The unit circle ...
0
votes
1answer
43 views

Given the equations $y=3\sin x+2$ and $y=x+c$, which statements are true?

$y=3\sin x+2$ $y=x+c$ where c is a constant Which of the following statements is/are true? For some value of c: there is exactly one solution with $0\leq x\leq \pi$ and there is at ...
-1
votes
1answer
59 views

Proof of $\sin(x) > \frac{2}{\pi} \cdot x $ [duplicate]

I want to prove that $\left(\forall x \in \left( 0; \frac{\pi}{2} \right)\right) \left[ \sin(x) > \frac{2}{\pi} \cdot x \right]$. This is quite easy to see when drawing the functions, but I wonder ...
1
vote
3answers
67 views

For non-negative $a$ and $b$ with $a+b \leq c$ for a small constant $c$, what is the minimum of $\cos a + \cos b$?

Let $a,b \geq 0$ with $a+b \leq c$ for a small constant $c$ between $0$ and $1$. What is the minimum of $\cos(a) + \cos(b)$? I conjecture it is $\cos(0)+\cos(c) = 1 + \cos(c)$ but I have no ...
0
votes
1answer
35 views

How does the unit circle relate to triangles with a hypotenuse larger than 1?

Why is a unit circle representative of all triangles, even triangles larger than the circle itself? I understand that $\sin\theta= Y$ and $\cos\theta = X$, but that is only because the hypotenuse is $...
3
votes
2answers
56 views

Calculate arctangent expression without using a calculator

I need help calculating the following:$$\arctan \frac{21 \pi}{\pi^2-54}+\arctan \frac{\pi}{18} + \arctan \frac{\pi}{3}$$ I don't know how to start, can anyone give me any information what should I do? ...
2
votes
4answers
121 views

How to solve for $~ 2x - \tan(x)=0~$

I need to find the roots for this function $$~ 2x - \tan(x)=0~$$ in order to graph it. I have found the one root $~(x=0)~$ but there are two more $~(x= -1.164 ,~ x= 1.164)~$. How can I find these ...
2
votes
1answer
34 views

Question Regarding Solving for Theta within a Trigonometric Equation

I am currently puzzled over how to deal with the following exercise: $$2\cos(\theta)-1=0$$ Here is the work I have done so far concerning the listed equation: $$2\cos(\theta)-1=0 \\2\cos(\theta)=1 \...
-2
votes
1answer
36 views

Seperate the following into real and imaginary parts: [on hold]

$$\frac{e^{ix}}{1-ce^{iy}}$$ solve this Trigonometry question of bsc. 1 semester
3
votes
2answers
65 views

How are Modular Forms used in number theory

I've been reading up on Fermat's last theorem and the Beal conjecture and in that context watched some of Edward Frenkel's lectures on Youtube. I understand how periodic trigonometric functions like $\...
0
votes
1answer
39 views

What is the total length of any interval for which $-1 \leq \tan(x) \leq1$ and $\sin(2x)\geq0.5$?

What is the total length of any interval for which $-1 \leq \tan(x) \leq 1$ and $\sin(2x)\geq0.5?$ We know that $0\leq x\leq \pi$, $\sin(2x)\geq 0.5$ can be rewritten as $\sin(x) \cos(x)\geq \frac{1}...
2
votes
3answers
63 views

Using de Moivre's formula to find an expression for $\sin 3x$ in terms of $\sin x$ and $\cos x$

I was asked to use de Moivre's formula to find an expression for $\sin 3x$ in terms of $\sin x$ and $\cos x$. De Moivre's formula is this: $$\cos nx+i\sin nx=(\cos x+i\sin x)^n$$ I plugged $3$ in ...
2
votes
3answers
73 views

How to calculate a point on a hypotenuse given two angles

I would like to have a formula to calculate the $x,y$ coordinates of a point "$B_2$" on the hypotenuse of a right triangle, given the angles "$b$" and "$a$" or the length of line $A-B_2$: see the ...
4
votes
4answers
229 views

tangent inequality in triangle

Let $a$, $b$ and $c$ be the measures of angles of a triangle (in radians). It is asked to prove that $$\tan^2\left(\dfrac{\pi-a}{4}\right)+\tan^2\left(\dfrac{\pi-b}{4}\right)+\tan^2\left(\dfrac{\pi-...
0
votes
0answers
12 views

Singularity conditions

$f(x)=\cot(k(x-c))$ for $[0,a]$ Now I have singularities at 0 and π So I choose the end points to be points of singularities since it is allowed and have the whole f(x) continuous in the interval (0,...
0
votes
2answers
44 views

Can anyone explain proving trig identites and finding angles (using unit circle)? [on hold]

I get the concept, but I'm often confused about which identities to use. Does anyone have any conceptual tips to make it easier to figure out? Thanks!! E.g. Solve $\sin(2x) + \cos(2x)^2 =2$, for $0&...
1
vote
0answers
30 views

Cotangent property

If I have $\cot(k(x-a+π+π/k))$ Then, is $\cot(kx-ka+kπ+π)=\cot(kx-ka+kπ)=\cot(k(x-a+π))=\cot(k(x-a))$ correct if I use $\cot(a+π)=\cot(a)$ property?
3
votes
3answers
61 views

Trigonometric equations: cotangent

If I have $cot(x-a)=cot(x-b)$ Where x is in radians and equal on both the sides and not equal to $0$ or $π$ Also for a and b, they are not equal to $0$ or $π$ Does the above equality mean $a=b$? ...
0
votes
2answers
41 views

Given that $\sin θ = 5/6$, what is the exact value of $\sin(2θ)$? ($0^\circ < θ < 90^\circ$) [on hold]

Can anyone please explain this to me? Thank you!
1
vote
2answers
30 views

Find distance AB given bearings and angles of elevation

I'm having issues with the following problem, which I think are due to how I am modelling the problem. Seems like I have to think in 2D and 3D which is where I get confused. Could someone show me how ...
7
votes
2answers
57 views

Show that the general value of $\theta$ satisfying $\sin\theta=\sin\alpha$ and $\cos\theta = \cos\alpha$ is given by $\theta = 2n\pi + \alpha$ [duplicate]

The general value of $\theta$ simultaneously satisfying equations $$\sin\theta = \sin\alpha \quad\text{and}\quad \cos\theta = \cos\alpha$$ is given by $\theta = 2n\pi + \alpha$, where $n\in\mathbb{Z}$ ...
3
votes
2answers
142 views

value of $(\cos\frac{2\pi}{7})^{{1}/{3}}+ (\cos\frac{4\pi}{7})^{{1}/{3}} + (\cos\frac{8\pi}{7})^{{1}/{3}}$

This question was on my list. I was trying to apply the $n$-th roots of unity, but other ideas are welcome. I also tried Newton's sums, but it's not working. I searched around here and I didn't find ...
4
votes
1answer
76 views

Finding global extrema of $a\left(\frac{1}{2}-b\frac{\sin(cx)}{x} \right) - b(1-\cos(cx))$, for $x\geq 0$

I have the following equation: \begin{equation} f(x) = a\left(\frac{1}{2}-b\frac{\sin(cx)}{x} \right) - b(1-\cos(cx)), \quad x\geq0, \end{equation} where a, b and c are strictly positive constants....
4
votes
3answers
60 views

If $u=\sqrt {a\cos^{2}x+b\sin^{2}x} + \sqrt {b\cos^{2}x+a\sin^{2}x}$ , find the maximum and minimum value of $u^2$.

If $u=\sqrt {a\cos^{2}x+b\sin^{2}x} + \sqrt {b\cos^{2}x+a\sin^{2}x}$ , find the maximum and minimum value of $u^2$. This problem was bothering me for a while. The minimum value of $u $ seemed ...
3
votes
4answers
169 views

Prove $x^2-2x+\sin(\frac{\pi}{2}x) \le 0$

I have been searching non-trigonometric approximations for some trigonometric functions and have found myself in need of showing that, $$x^2-2x+\sin\left(\frac{\pi}{2}x\right) \le 0,$$ in the range $...
2
votes
0answers
25 views

Different approaches to calculating angle between two vectors

I can calculate angle θ between two dimensional vectors a and b as the inverse cosine of their dot product divided by their magnitude: ...
4
votes
1answer
39 views

Finding $\phi$ so that we have $3$ equal sets

Find all angles $\phi$, if any, for which the set $S=\left \{ \sin(\phi),\sin(2\phi),\sin(4\phi) \right \}$, the set $C=\left \{ \cos(\phi),\cos(3\phi),\cos(9\phi) \right \}$, and the ...
0
votes
0answers
24 views

Trigonometric equations for three different regions on number line

I have $$f(x)=\begin{cases}k\cot(sx),&x<pa<qa\\ k\cot(s[x-b]),&x>qa>pa\end{cases}$$ I have to find $f(x)$ for $pa<x<qa$, given $s(a-b)=\pi$. $p,q$ are any real nos. Do ...
2
votes
0answers
108 views
+50

Prime number and Relationship of Sequences of period 4,5,and 6

Let $p$ be a prime number.($p \neq 2,3,5$) Let $t^+,t^-,a$ be sequences. $t^+_{k+5}=t^+_k,t^+_1=0,t^+_2=-1,t^+_3=-1,t^+_4=0,t^+_5=2$ $t^-_{k+5}=t^-_k,t^-_1=-1,t^-_2=0,t^-_3=0,t^-_4=-1,t^-_5=2$ $a_k=...
1
vote
1answer
54 views

Why do the sines of the numerators of $\pi$’s continued fraction convergents approach zero?

I was messing with the sine function and tried getting values close to zero with integer inputs. I found a peculiar pattern. If you take pi’s continued fraction and write them out as one whole ...
1
vote
1answer
50 views

Twice angle conditions with a point in a triangle

Let $X$ be a point lying in the interior of the acute triangle $ABC$ such that $\angle BAX = 2\angle XBA$ and $\angle XAC = 2\angle ACX$. Denote by $M$ the midpoint of the arc $BC$ of the ...
0
votes
2answers
40 views

Is there an easy way to quickly prove (or memorize) inverse trig formulas such as $ \arcsin(a) = \arctan(\frac{a}{\sqrt{1-a^2}}) $?

Is there an easy way to quickly prove these formulas? If not, is there any easy mnemonic way to memorize them fast? $$\begin{align} \arcsin(a) &= \arctan\left(\frac{a}{\sqrt{1-a^2}}\right) \...
-1
votes
1answer
22 views

How to find the angle of a vector based on origin? [closed]

How can I find the angle between a vector and unit vector [1, 0], but I don't want the shortest angle, but always the right side angle like this: ...
1
vote
2answers
59 views

How to form functions similar to $\frac 12 - \frac12 \cos \pi x$

I wanted to know how to form curves similar to that of $y = \frac12 - \frac12\cos(\pi x)$. What I need is a function that forms a smooth curve been $x = 0$ to $x = 1$. As you see with the $y = \frac12 ...
1
vote
1answer
62 views

$ \cos(2 \pi x)$, $ \cos(4 \pi x)$, $ \cos(8 \pi x)$, $ \cos(16 \pi x)$, and $ \cos(32 \pi x)$ are all nonpositive

$ \cos(2 \pi x)$, $ \cos(4 \pi x)$, $ \cos(8 \pi x)$, $ \cos(16 \pi x)$, and $ \cos(32 \pi x)$ are all nonpositive. What is the smallest positive $ x$? I tried ranges for each of cos(?)..that did not ...
0
votes
2answers
35 views

Proving $ \sin{x} - \sin{y} = \sin{(x - y)} \cdot \sqrt{\frac{1 + \cos{(x + y)}} {1 + \cos{(x - y)}}} $

I want to prove following trig identity: $$ \sin{x} - \sin{y} = \sin{(x - y)} \cdot \sqrt{\frac{1 + \cos{(x + y)}} {1 + \cos{(x - y)}}} $$ for $$ 0 < x < \pi, 0 < y < \frac{\pi}{2}$$ ...
0
votes
1answer
39 views

Can this system of trigonometric equations be solved? How to solve it?

Given an odd integer number n, and x is an unknown odd integer number and $ 1 < x \leq n $ Can i solve the following system of equations to find $x$? If i can, how to solve it?: $$\begin{cases} y ...
0
votes
4answers
32 views

Question About Trig Substitution

I was asked to evaluate $\displaystyle\int\frac{1}{x^2\sqrt{1-x^2}}\text{d}x$ Here's my attempt: $\text{Let $x=\sin(\theta)$}$ $\text{Then }\text{d}x=\cos(\theta)\text{d}\theta$ $$\int\frac{1}{\sin^...
1
vote
2answers
53 views

How do I show that the partial sums of the sequence $a_n=\cos( \log n)$ are bounded?

I am just stuck. How do I show that the partial sums of the sequence $a_n=\cos( \log n)$ are bounded? Log n makes everything twisted...Could anyone please help me?
-1
votes
0answers
34 views

State the number of the solution in this domain of the equation $x^2=3x+\sin(x)$ [on hold]

Sketch on the same diagram the graph of $y=3x-x^2$ and $y=-\sin(x)$..Hence state the number of the solution in this domain of the equation $x^2=3x+\sin(x)$.