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

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3
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2answers
327 views

Proving $\tan (\frac{\pi }{4} - x) = \frac{{1 - \sin 2x}}{{\cos 2x}}$

How do I prove the identity: $$\tan \left(\frac{\pi }{4} - x\right) = \frac{{1 - \sin 2x}}{{\cos 2x}}$$ Any common strategies on solving other identities would also be appreciated. I chose to ...
0
votes
2answers
225 views

Finding all rational $p,q,r$ satisfying $p\cos{\frac{\pi}{7}}+q\cos{\frac{2\pi}{7}}+r\cos{\frac{3\pi}{7}}=1$

Find all rational numbers $p,q,r$ such that $$p\cos{\dfrac{\pi}{7}}+q\cos{\dfrac{2\pi}{7}}+r\cos{\dfrac{3\pi}{7}}=1.$$ My idea: we can find $x=\cos{\dfrac{\pi}{7}}$ the equation root? Because ...
9
votes
3answers
272 views

How to prove $\cos\left(\pi\over7\right)-\cos\left({2\pi}\over7\right)+\cos\left({3\pi}\over7\right)=\cos\left({\pi}\over3 \right)$

Is there an easy way to prove the identity? $$\cos \left ( \frac{\pi}{7} \right ) - \cos \left ( \frac{2\pi}{7} \right ) + \cos \left ( \frac{3\pi}{7} \right ) = \cos \left (\frac{\pi}{3} \right ...
0
votes
3answers
80 views

How to find remain factor of this trigonometic equation?

The equation $$3\sin^2 x - 3\cos x -6\sin x + 2\sin 2x + 3=0$$ has a solution $x = 0$. That is mean it has a factor $\cos x - 1$. I tried write the given equation has the form $$(\cos x - 1)P(x)=0.$$ ...
5
votes
1answer
243 views

Expressing $\sin\pi/n$ in terms of radicals of integers

Are values of $$\sin \frac{\pi}{n}$$ where $n$ is a positive integer all expressible in terms of radicals of integers? If not, what is the first $n$ for which it is not?
19
votes
2answers
455 views

How to find the value of $\sin{\dfrac{\pi}{14}}+6\sin^2{\dfrac{\pi}{14}}-8\sin^4{\dfrac{\pi}{14}}$

Determine $$ \sin\left(\pi \over 14\right) + 6\sin^{2}\left(\pi \over 14\right) -8\sin^{4}\left(\pi \over 14\right) $$ My idea: Let $\displaystyle{\sin\left(\pi \over 14\right)} = x$.
6
votes
8answers
1k views

Solving $\sin \theta + \cos \theta=1$ in the interval $0^\circ\leq \theta\leq 360^\circ$

Solve in the interval $0^\circ\leq \theta\leq 360^\circ$ the equation $\sin \theta + \cos \theta=1$. I've got the two angles in the interval to be $0^\circ$ and $90^\circ$, it's not an answer I'm ...
1
vote
3answers
68 views

Ease numbers in a certain range

I'm trying to find an easing function taking in values from 0 to π/4 and outputting values in the same range, which starts ...
1
vote
1answer
102 views

Determine the central angle theta

Suppose the ends of the cylindrical storage tank in the figure arc circles of radius $3$ ft and the cylinder is $20$ ft long. Determine the volume of the oil in the tank to the nearest cubic foot if ...
3
votes
3answers
182 views

Trying to solve equation $\sin^2 x = 1$

The assignment is that I'm suppose to correct a flawed solution to the equation $\sin^2 x = 1$. The flawed solution is: $\sin^2 x = 1$ $\sin x = 1$ $x = 90^\circ + 2n\pi$ I thought I was simply ...
3
votes
5answers
1k views

Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$

$$\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$$ I have attempted this question by expanding the left side using the cosine sum and difference formulas and then multiplying, and then ...
9
votes
3answers
488 views

Being ready to study calculus

Some background: I have a degree in computer science, but the math was limited and this was 10 years ago. High school was way before that. A year ago I relearnt algebra (factoring, solving linear ...
3
votes
2answers
202 views

Solving $a \sin(\alpha) - c \sin^2(\alpha) = b \cos(\alpha) - c \cos^2(\alpha)$

$a, b, c$ are given positive integers. I need $\sin(\alpha)$ or $\cos$ or anything simple with $\alpha$ from the equation: $$a \sin(\alpha) - c \sin^2(\alpha) = b \cos(\alpha) - c \cos^2(\alpha)$$
3
votes
1answer
1k views

How is the formula for the focal point of a ball lens derived?

How can the focal point of a ball lens be found?
1
vote
1answer
63 views

Can the distance between $3$ points be worked out given their magnetic bearings?

I have 3 moving points which I want to find the distance between. Each point transmits radio waves which the others receive. At each point the field strength of the transmitted wave from the other ...
-1
votes
5answers
11k views

$\cos(\arcsin(x)) = \sqrt{1 - x^2}$. How?

How does that bit work? How is $$\cos(\arcsin(x)) = \sin(\arccos(x)) = \sqrt{1 - x^2}$$
2
votes
2answers
168 views

How to get angle bewteen two vectors in range -1 to 1 without using arc cosine?

Given two normalized vectors in 3d space, how can I get a value from $-1$ to $1$ based on their angle without using arc cosine? With use of arc cosine, I think this would give me the correct result. ...
1
vote
0answers
127 views

Shortest time taken for a targeted object with a set speed to meet a body orbiting in a circle

I'm trying to figure out how to find the optimum point that a ship in 2D space would meet a planet which was orbiting in a perfect circle. The orbit is at a constant rate, and the ship can only move ...
1
vote
2answers
121 views

Proving that $\sin(a)\cos(b)$ and $\cos(a)\sin(b)$ identities are identical using $\sin(-x)=-\sin(x)$

This website states the two trig identities below are identical: [\begin{array}{l} \sin (a)\cos(b) = \frac{1}{2}\left[ {\sin (a + b) + \sin (a - b)} \right] \Rightarrow 1\\ \cos (a)\sin (b) = ...
2
votes
0answers
501 views

Solving the equation $\cos(x) \cdot \cosh (x) + 1 = 0$

$$\cos(x) \cdot \cosh (x) + 1 = 0$$ Sorry I am a software developer and I have forgotten this part of mathematics! What is the value of $x$ in the above equation? I need the steps to solve the ...
6
votes
2answers
129 views

Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$

Show that $$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$$
2
votes
1answer
274 views

A Trigonometric Sum Related to Gauss Sums

This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it. Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} ...
3
votes
1answer
68 views

$\tan B\cdot \frac{BM}{MA}+\tan C\cdot \frac{CN}{NA}=\tan A. $

Let $\triangle ABC$ be a triangle and $H$ be the orthocenter of the triangle. If $M\in AB$ and $N \in AC$ such that $M,N,H$ are collinear prove that : $$\tan B\cdot \frac{BM}{MA}+\tan C\cdot ...
4
votes
3answers
272 views

Can't find solutions for $\tan{2x} = \tan{x}$

Solving $\tan{2x}=\tan{x}$ Reducing the left side: $$\frac{\sin{2x}}{\cos{2x}} = \frac{2\sin{x}\cos{x}}{2\cos^{2}(x)-1}.$$ Reducing the right side: $$\tan{x} = \frac{\sin{x}}{\cos{x}}.$$ therefore: ...
3
votes
2answers
91 views

$\sin{\frac{A+B}{2}}+\sin{\frac{B+C}{2}}+\sin{\frac{C+A}{2}} > \sin{A}+\sin{B}+\sin{C}. $

Help me please to prove that: for any $\triangle ABC$ we have the following inequality: $$\sin{\frac{A+B}{2}}+\sin{\frac{B+C}{2}}+\sin{\frac{C+A}{2}} > \sin{A}+\sin{B}+\sin{C}. $$ It's about ...
0
votes
2answers
315 views

How to count angle between vector and horizontally oriented vector?

I need to calculate in my Java application an angle between my line and horizontal line that has the same beginning. I have a line described by its equation: $$f(x) = ax + b.$$ I would like to know ...
3
votes
2answers
480 views

How to get the minimum angle between two crossing lines?

I'm not a student, I'm just a programmer trying to solve a problem ... I just need the practical way to calculate the smallest angle between two lines that intersect. The value, of course, must always ...
13
votes
3answers
337 views

Can $\sin n$ get arbitrarily close to $1$ for $n\in\mathbb{N}?$

Or put differently, does $$\lim_{n \to \infty}\big(\max \{\sin 1, \sin 2, \ldots ,\sin n\}\big) = 1?$$ My intuition says yes, but how can one prove this?
2
votes
1answer
64 views

Trigonometric Sums - URSS

Calculate the value of the sums: (a) $\cos x+\binom{n}{1}\cos 2x +\cdots+\binom{n}{n} \cos (n+1)x $; (b) $\sin x+\binom{n}{1}\sin 2x +\cdots+\binom{n}{n} \sin (n+1)x $.
1
vote
1answer
46 views

approximation of law sines from spherical case to planar case

we know for plane triangle cosine rule is $\cos C=\frac{a^+b^2-c^2}{2ab}$ and on spherical triangle is $ \cos C=\frac{\cos c - \cos a \cos b} {\sin a\sin b}$ suppose $a,b,c<\epsilon$ which are ...
1
vote
3answers
170 views

Why does tan(t) touch the unit circle at (1,0)?

I can't get my head around this, any help would be very much appreciated. Thanks EDIT: t is an angle, where 0 < t < 90, angle t is in degrees EDIT: Added a picture I lifted from google
5
votes
3answers
211 views

Show that for all $(x,y)$ there exists $(r,\theta)$…

Given a problem wherein $(x,y) \in \mathbb{R}^2$, I often transform to polar coordinates by introducing the assumption that $(r,\theta)$ satisfy $x = r\cos \theta, y = r\sin \theta.$ Of course, it's ...
1
vote
2answers
325 views

A question on Trigonometry (bisector)

If two bisector of a triangular is equal, then it is Isosceles triangular.
0
votes
1answer
86 views

The law of cosines for a sphere

$\cos(c) = \cos(a)\cos(b) + \sin(a)\sin(b)\cos(C)$ Prove that if $a$, $b$, and $c$ is approximately $0$, then $c^2 = a^2 + b^2 - 2ab~\cos(C)$. I wasn't sure how to prove this. One thought I had was ...
1
vote
5answers
2k 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 ...
1
vote
0answers
126 views

Differential geometry textbook or lecture notes on the riccati equation and riccati inequality

I took a course on differential geometry and didn't get one specific topic well, so I am searching on some additional metrial to understand it in a better way. This wasn't a course about classical ...
1
vote
2answers
151 views

trigonometry equations

Take this question: "We follow the tips of the hands of an old fashioned analog clock (360 degrees is 12 hours) . We take the clock and put it into an axis system. The origin (0,0) of the axis ...
4
votes
1answer
138 views

finding $\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$

I want to find $$L=\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$$ I already know that I need to split the expression $\frac{2r}{1-r^2+r^4}$ of the form ...
1
vote
1answer
51 views

Uncertainty in measurements: if $x$ has uncertainty $\pm\epsilon$, what is the uncertainty in $\sin x$?

I have two questions regarding uncertainties in measurements. First, if I have some measured value for $x$ with an uncertainty $\pm e$, what would be the uncertainty in $\sin x$, $\pm\sin e$? ...
5
votes
5answers
520 views

Complex numbers and trig identities. I've heard this question is easy but I don't know how. Help?

Using the equally rule $a + bi = c + di$ and trigonometric identities how do I make... $$\cos^3(\theta) - 3\sin^2(\theta)\ \cos(\theta) + 3i\ \sin(\theta)\ \cos^2(\theta) - i\ \sin^3(\theta)= ...
3
votes
5answers
259 views

elegant proof that $\sin(x)\cdot\cos(x)=\sin(2x)/2$

I tried for a few days to prove the identity $\sin(x)\cos(x)=\frac{\sin(2x)}{2}$ and finally got the following proof. I wanted to know if someone knew a simpler or more elegant way to proof it. ...
2
votes
1answer
36 views

Missing rule for this simple identity?

Is there a simple rule I am missing to help prove the identity below? Of course I can find the common denominator, do multiplication, summarize, apply trigonometric product-to-sum identities, and then ...
4
votes
1answer
2k views

Where are secant, cosecant, and cotangent applied?

The textbook we use says: The cosecant function and the secant function are the reciprocal functions of the sine function and the cosine function, respectively, and thus are also periodic ...
1
vote
1answer
2k views

Solving the inverse of cos^2

The following equation provides the inclination ($i$) of a galaxy, using the ratio of its two axes: $$ \cos^2 i = {(b/a)^2 − (b/a)^2_{eos} \over 1 − (b/a)^2_{eos}} $$ All I need however is to ...
3
votes
3answers
317 views

Find minimum of $\sin^4 x + \cos^4 x + \sec^4 x$

I tried manipulating the terms but I couldn't get anywhere. The only other thing I can observe is that the minimum must be greater than $1$ since all the terms are non-negative and the range of ...
0
votes
2answers
128 views

Can you express $\cos(\frac{x}{2})\sin(x)$ as a linear combination of even multiples of $\frac{x}{2}$ in sin and odd ones in cos?

I was going through some of my notes and found they said we can express $\cos(\frac{x}{2})\sin(x)$ as a linear combination of even multiples of $\frac{x}{2}$ in sin and odd ones in cos. However, I ...
3
votes
3answers
75 views

Trigonometric Functions And Identities Question .

If $A$ , $B$ and $C$ are the angles of a triangle , I have to show that : $ \tan^2 \cfrac{A}{2} + \tan^2 \cfrac{B}{2} + \tan^2 \cfrac{C}{2} \ge 1 $ . I had only arrived at $A+B+C = \pi $ , thus ...
3
votes
3answers
2k views

What are some useful tricks/shortcuts for verifying trigonometric identities?

What "tricks" are there that could help verify trigonometric identities? For example one is: $$a\cos\theta+b\sin\theta = \sqrt{a^2+b^2}\,\cos(\theta-\phi)$$
0
votes
2answers
71 views

Area of a rectangular triangle

We need to calculate the area of the triangle shown in figure: The text of the problem also says that: $\sin \alpha =2 \sin \beta$. What is the area of ​​the triangle?
0
votes
2answers
34 views

For the angles a and b it is given that:-

For the angles $a$ and $b$ it is given that: $\sin a = 0.6$, $\sin b = 0.8$, $\cos a = 0.8$ and $\cos b = 0.6$ Find the value of $\sin( a + b)$ with out calculator :- thanks all - happy mother day ...