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

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2
votes
1answer
38 views

Are there complex numbers whose sines are zero?

I recently learned that $\sin(z)$ has an extension into the complex plane, namely: $$\frac{e^{iz}-e^{-iz}}{2i}$$ Is there any complex number $z=a+bi$, with $b≠0$ such that $\sin(z)=0$ ? I am ...
0
votes
0answers
16 views

Translate Pitch and Roll Angles of Object to those at different Yaw

I have been trying to find a method to translate the pitch and roll angles of one object to those of another connected object at a different yaw - i.e I have an IMU mounted on a quadcopter frame and a ...
0
votes
2answers
35 views

Finding the solutions of $\sin\left( x - \frac{\pi}{4} \right) = \sin\left( 3x + \frac{\pi}{4} \right)$

Find all the solutions for $x$ in the following equality. $$ \sin\left( x - \frac{\pi}{4} \right) = \sin\left( 3x + \frac{\pi}{4} \right) $$ I tried using the following formulas for both ...
-6
votes
0answers
54 views

Find $a$, $b$ and $c$. [on hold]

As title said, How to find angles $a$, $b$ and $c$? Thanks in advance!
7
votes
2answers
56 views

Is there no formula for $\cos(x^2)$?

I was wondering if there was a "formula" or an "identity" for $\cos(x^2)$, as there is for $\cos(2x)$. My question is closely related to this one, which was only asking for $\cos(ab)$. For instance ...
0
votes
1answer
45 views

Using derivatives to get some trigonometric identities

Is there a way of using derivatives to get some trigonometric identities in a straight-forward fashion? I use to forget them, so that would help me a lot... For example, since when we get the ...
0
votes
3answers
53 views

how **(1)** $(2n-1)\pi/2 + (-1)^n\pi/3$ and **(2)** $2n\pi±\pi/6$ indicates the same angle?

I'm learning Trigonometry right now with myself and at current about General solution. I have a question in my book which I don't understand how to proof. The question is Show that the two angles are ...
-3
votes
2answers
51 views

Equetion of difference sinus and cosinus functions

Hi i have question i have something like this: $2 \sin 5t - 5 \cos 5t = A \sin(5t + \varphi)$ The main question is : Is there any formulas or something to change left side of "$=$" to look similar ...
0
votes
0answers
13 views

what functions/formula gives me linear progression from 90 -> 180/-180 -> -90

in my programming code the angles of a camera goes from TOP 90 to MIDDLE 180 to -180 to BOTTOM -90 what formula would give me a linear progression between those values? i need to move the camera up ...
2
votes
3answers
68 views

Trigonometric Equation Simplification

$$3\sin x + 4\cos x = 2$$ To solve an equation like the one above, we were taught to use the double angle identity formula to get two equations in the form of $R\cos\alpha = y$ where $R$ is a ...
0
votes
3answers
32 views

Find the ratios of the sides of a triangle

If the perimeter of a the right-angle triangle is six times its smallest side, find the ratios of the three sides. I tried solving it by using the normal area and volume.
0
votes
1answer
31 views

What is the third positive number starting from zero that will satisfy $\cos 6x = \cos x$? [on hold]

What is the third positive number starting from zero that will satisfy $\cos 6x = \cos x$? I need help on this one. I don't know how to approach this problem.
1
vote
1answer
23 views

Compound angle formula

I understand how to use the compound angle formula when solving $\sin(\pi/12)$. However I dont understand how I can use a compound angle formula to show $$\arctan(3)-\arctan(1/2)=\pi/4$$ Thankyou Any ...
1
vote
0answers
32 views

Mentally approximating an inverse sine?

Is there a method to approximate the inverse of a sine function in ones head? I know one can approximate the inverse of a cosine with the following equation: $\cos^{-1}(x) = ...
2
votes
3answers
28 views

how to find $\lim_{x\to 0}\sin^2(\frac{1}{x})\sin^2 x$

How to find $\lim_{x\to 0}\sin^2(\frac{1}{x})\sin^2 x$ ? I tried using taylor expansion: $$((x-\frac{x^3}{6}+\frac{x^5}{120}+O(x^5))(\frac{1}{x}-\frac{1}{6x^3}+\frac{1}{120x^5}+O(x^{-5})))^2$$ but ...
2
votes
2answers
69 views

Compute $(\sin4^\circ)^2 +(\sin8^\circ)^2+(\sin12^\circ)^2+\cdots+(\sin176^\circ)^2$

Angle of sine is in degrees, can anyone show me an easy soln to this? This was question was given to us for 1minute without calcu. I know that $\sin4^\circ=\sin176^\circ$, ...
0
votes
0answers
13 views

Bearing of a line

Please help me to find the bearing. I've attached the image.I've tried by drawing a North a B and C, and D but couldn't figure out the angle that give me bearing of C from D. Thanks for all your ...
6
votes
4answers
895 views

What is meant by a 'pure' wave?

What is meant by a 'pure' wave? I know it might sound like a basic question, but I've never been taught this. I saw that a sine wave is a pure wave. I tried Googling what a pure wave is, but all I ...
0
votes
4answers
63 views

Using trig identities to evaluate $\int_{0}^{\pi/2} \sqrt{1-\sin x} \, dx$

Use the identities $$\cos 2x=2\cos^2 x -1=1-2\sin^2 x$$ $$\sin x=\cos \left(\frac{\pi}{2}-x\right)$$ to help evaluate $$\int_{0}^{\pi/2} \sqrt{1-\sin x} \; dx$$ I've already done ...
2
votes
4answers
180 views

Proving algebraic equations with circle theorems

I got as far as stating that OBP=90˚ (as angle between tangent and radius is always 90˚), and thus CBO=90˚- 2x. CBO=OCB as they are bases in a isosceles. COB=180-90-2x-90-2x. But after this, i am ...
2
votes
1answer
42 views

If $A,B>0$ and $A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$.

If $A,B>0$ and $\displaystyle A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$. $\bf{My\; Try::}$ Given $$\displaystyle A+ B = \frac{\pi}{3}$$ and $A,B>0$. So we can say ...
24
votes
1answer
724 views

Moriarty's calculator: some bizarre and deceptive graphical anomalies

Background: This is a problem I first came across a few years ago in a calculus textbook (a James Stewart one), where it addressed some of the pitfalls of using graphing calculators. The original ...
8
votes
3answers
147 views

Differential Equation of the Form $\frac{dy}{dx}=\sin(x+y)$ [duplicate]

I have been attempting to solve the above differential equation for some time now, and I remain stuck on one step. After substituting $u=x+y$, separating the variables, and integrating both sides, I ...
3
votes
0answers
58 views

PDF of Random Variable $\sin\alpha \cdot \cos\beta$ with $\alpha,\beta$ uniform

As part of a bigger problem, I want to compute the probability density $f_Z(z)$ of $$Z = \sin\alpha \cdot \cos\beta$$ where $\alpha, \beta$ are random variables, independently and uniformly ...
4
votes
4answers
395 views

Partial fractions and trig functions

A long time ago I wrote down a silly problem. It starts with Attempt to write $$\frac{1}{\sin(x)\cos(x)}$$ using partial fractions. and then goes on to prove a trig identity. I was wondering if ...
12
votes
1answer
207 views

Inequality with summation of cosine terms $\left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| \leq 1 + 2\sum_{j=1}^k \cos (\frac{2\pi }{q}j)$

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
122
votes
3answers
7k views

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
31
votes
2answers
650 views

Is Pythagoras the only relation to hold between $\cos$ and $\sin$?

Pythagoras says that $\cos^2 \theta + \mathrm{sin}^2\theta = 1$ for all real $\theta$. (Vague) Question. Is this the only relationship between the functions $\cos$ and $\sin$? More precisely: Let ...
8
votes
3answers
514 views

Prove $\alpha \in\mathbb R$ is irrational, when $\cos(\alpha \pi) = \frac{1}{3}$

I am trying to prove: If $\cos(\pi\alpha) = \frac{1}{3}$ then $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ So far, I've tried making it into an exponential, since exponentials are easier to ...
25
votes
2answers
399 views

Disproving an “almost true” trigonometric identity

The plausible looking "identity" $$\sin(\frac{\pi}{51})+\cos(\frac{\pi}{74})=\frac{3}{2\sqrt 2}$$ is not true, but it is close indeed: $$LHS=1.0606\color{blue}{598...}$$ ...
6
votes
5answers
360 views

How to find the polynomial such that …

Let $P(x)$ be the polynomial of degree 4 and $\sin\dfrac{\pi}{24}$, $\sin\dfrac{7\pi}{24}$, $\sin\dfrac{13\pi}{24}$, $\sin\dfrac{19\pi}{24}$ are roots of $P(x)$ . How to find $P(x)$? Thank you very ...
2
votes
2answers
63 views

Taylor series convergence for sin x

a. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}<\sin x<x-\frac{x^3}{3!}+\frac{x^5}{5!},$ b. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots-\frac{x^{4k-1}}{(4k-1)!}<\sin ...
8
votes
4answers
131 views

Relation between $\sin(\cos(\alpha))$ and $\cos(\sin(\alpha))$

If $0\le\alpha\le\frac{\pi}{2}$, then which of the following is true? A) $\sin(\cos(\alpha))<\cos(\sin(\alpha))$ B) $\sin(\cos(\alpha))\le \cos(\sin(\alpha))$ and equality holds for some ...
20
votes
4answers
917 views

Elementary Proof of Euler's Sine Expansion $\prod_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$

I've been looking at proofs of Euler's sine expansion, that is $$\frac{\sin x}{x}=\prod_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$$ All the proofs seem to rely on complex analysis and ...
3
votes
1answer
40 views

Another Verify the identity: $\sec^2 \frac{x}{2} = \frac{2}{1+\cos x}$

Another Verify the identity that I can't get: $$\sec^2 \frac{x}{2} = \frac{2}{1+\cos x}$$ $$ = \frac{1 + \left(\frac{1}{\cos x}\right)}{2}$$ $$ = \frac{\cos x + 1}{2 \cos x}$$
30
votes
4answers
940 views

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
4
votes
2answers
465 views

Fixed-Point Iteration method unable to converge to any of a function's infinte roots

An equation is given to me which has to be solved by direct iteration method: $$\sin(x) = {x+1 \over x-1}$$ or $$f(x)=\sin(x)-{x+1 \over x-1} = 0$$ I follow the following procedure with reasons ...
3
votes
2answers
330 views

Prove that $x^2<\sin x \tan x$ as $x \to 0$ [duplicate]

$$x^2<\sin x \tan x \quad as \; x \to 0$$ I made the substitution $x \to \arctan x$ . $\arctan^2 x<x\sin (\arctan x)$ $\arctan x < \large \frac{x}{(x^2+1)^{\frac 14}}$ There are two ...
6
votes
6answers
4k views

Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$

We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$ I have tried and it gets confusing.
1
vote
5answers
3k views

In the context of the Unit Circle why is tan$(\theta)$ defined as $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$?

I understand why the circular functions $\sin(\theta)=y$ and $\cos(\theta)=x$, but why does $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$? Is there any particular reason why $\tan ...
43
votes
3answers
1k views

Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$

Prove that $$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
7
votes
6answers
3k views

Integral of $\int_0^{\pi/2} \ (\sin x)^7\ (\cos x)^5 \mathrm{d} x$

I am trying to find this by using integration by parts but I am not sure how to do it. $$\int_0^{\pi/2} (\sin x)^7 (\cos x)^5 \mathrm{d} x$$ I tried rewriting as $$\int_0^{\pi/2} \sin x\cdot\ ...
18
votes
5answers
5k views

Do “imaginary” and “complex” angles exist?

During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting: $ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $ $ \sin ...
161
votes
14answers
22k views

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
6
votes
6answers
584 views

Proving a trigonometric identity

How can one prove the validity of this trigonometric identity? \begin{equation} 2\arccos\sqrt{x} = \frac{\pi }{2}-\arcsin(2x-1) \end{equation}
38
votes
3answers
1k views

Making trigonometric substitutions rigorous

I've been tutoring some basic calculus, and it made me think about something pretty basic. Let me explain the problem by example: Say we are given the integral $\int \frac{x^2}{\sqrt{1-x^2}}\ ...
8
votes
3answers
470 views

$\int \cos^2 x$ - Where did I go wrong?

So when looking on the question: $$\int_{0}^{\pi} \cos^2 x \ \text{d}x$$ I would just subtract $\cos^2(0)$ from $\cos^2(\pi)$, but doing so would get me 1 - 1 = 0. When the answer is $\pi/2$. Where ...
0
votes
2answers
77 views

Is there a sequence of $v_{n}$ such that $G=\prod_{n=1}^{\infty} \frac{2\tan^{-1}(v_{n})}{\pi}> 1$?

Let $G=\prod_{n=1}^{\infty} \frac{2\tan^{-1}(v_{n})}{\pi}$, where $v_{n}$ is an increasing, monotonic sequence of natural numbers: is it true that there is no sequence of $v_{n}\in\mathbb{N}$ such ...
2
votes
4answers
17k views

Finding parametric equations for the tangent line at a point on a curve

Find parametric equations for the tangent line at the point $(\cos(-\frac{4 \pi}{6}), \sin(-\frac{4 \pi}{6}), -\frac{4 \pi}{6}))$ on the curve $x = \cos(t), y = \sin(t), z=t$ I understand that in ...
20
votes
10answers
22k views

Real world uses of hyperbolic trigonometric functions

I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful. Is there any good examples of their uses ...