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
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2answers
117 views

Trigonometry, Using sine rule and area formula.

Two ships P and Q are observed to be NW and NE respectively of a port A. From a second port B, which is 1km due east of A, the ships P and Q are observed to WNW and NNE respectively. Show that the two ...
1
vote
2answers
237 views

Why is $\sin^2x + \cos^2x = 1$ important?

To start off, I understand the proof behind this identity, and I can visualize it in my head with the unit circle. But I read this quote: They only need to remember three facts – that $\sin ...
2
votes
4answers
132 views

Trigonometry Equations.

Solve for $0 \leq X \leq 360$, giving solutions correct to the nearest minute where necessary, a) $\cos^2 A -8\sin A \cos A +3=0$ Can someone please explain how to solve this, ive tried myself and no ...
2
votes
1answer
345 views

Whats the formula for the amount to scale up an image during rotation to not see the edges

I'm trying to figure out a formula... for how much a picture (rectangle) would have to be scaled up during a rotation (at any rotation amount) so that you don't see the edge of the picture in the ...
10
votes
6answers
446 views

How to calculate $\cos(6^\circ)$?

Do you know any method to calculate $\cos(6^\circ)$ ? I tried lots of trigonometric equations, but not found any suitable one for this problem.
10
votes
2answers
1k views

Evaluate $\cos 18^\circ$ without using the calculator

I only know $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$ as standard angles but how can I prove that $$\cos 18^\circ=\frac{1}{4}\sqrt{10+2\sqrt{5}}$$
0
votes
1answer
34 views

Will this Trigonometric give the following answer?

If $n$ is an integer, can $$\cos[(2n-1)\pi/2]-\cos[(2n-1)\pi/4]$$ be equal to $\;\;\cos[(2n-1)\pi/4],\;\;?$ I have tried the formula for $\cos A-\cos B$ but that would give result in $sine$
2
votes
2answers
82 views

General solution for trigonometry equation

How should I state the general solution for the equation $\sin(4\phi)=\cos(2\phi)$. The angles are $15$, $45$, $75$ and $135$ if I restrict myself within the range $[0,360]$
1
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1answer
116 views

Trigonometric equations with more than one function

This is a general question about how to solve trigonometric equations which involve different functions. I have been multiplying and dividing the functions but have not been able to attain an ...
1
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2answers
115 views

The $\cos(\sin 60^\circ)$

I stumbled across this question and I cannot figure out how to use the value of $\cos(\sin 60^\circ)$ which would be $\sin 0.5$ and $\cos 0.5$ seems to be a value that you can only calculate using a ...
0
votes
1answer
144 views

Express $[\cos(x) + \sqrt3 \sin(x)]$ in the form $[r\cos(x-a)]$

Express $[\cos(x) + \sqrt3\sin(x)] $ in the form $[r\cos(x-a)]$, where $r>0$ and $ 0\leq360$, hence solve the equation $[\cos(x) + \sqrt3\sin(x)= \sqrt2]$ This is as far as i have completed. I ...
1
vote
1answer
114 views

Trigonometric manipulation of complex number, how does this step occur?

I was reading the section about DeMoivre, and my book showed how to derive his formulas. The next part is supposed to be about finding roots of complex and real numbers. Roughly, it says: "Let $z$ be ...
4
votes
2answers
951 views

Minimum and maximum of $ \sin^2(\sin x) + \cos^2(\cos x) $

I want to find the maximum and minimum value of this expression: $$ \sin^2(\sin x) + \cos^2(\cos x) $$
3
votes
1answer
435 views

Find $\cos(2\alpha)$ given $\cos(\theta -\alpha)$ and $\sin(\theta +\alpha)$

My question is: If $\cos(\theta -\alpha) = \frac{3}{5}$ and $\sin(\theta +\alpha) =\frac{12}{13}$, find $\cos(2\alpha)$. Attempt I: \begin{align*} &\cos^2(\theta -\alpha)+\sin^2(\theta ...
2
votes
1answer
158 views

Homework Help: Prove Pythagorean identities

The original question is boxed in red. The question asks to "Prove the identities:" I have tried one method as shown below the box then another method under the squiggly but both come to an answer ...
0
votes
2answers
40 views

Consider: $\tan x^3=-\frac{3}{2x^3}$ problem with the logic of finding two values for $\tan$?

$\tan x^3=-\dfrac{3}{2x^3}$ $\tan x^3=\dfrac{2}{3x^3}$ $3\tan x=\dfrac{2}{x^3}$ $3\tan x^3=2$ $\tan x^3=\dfrac{2}{3}$ Hang on!! Now $\tan x^3 =\dfrac{2}{3x^3}$ but also: $\tan ...
4
votes
1answer
116 views

Solve $x\sqrt{10} = \prod\limits_{k = 1}^{90} \sin(k), x\in \mathbb Q$.

Can someone help me with this question? I've found a solution but it's not a very nice one. I used 6 times the relation $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. There's got to be a better way. ...
1
vote
2answers
214 views

solving an equation of the type: $t \sin (2t)=2$ where $0<t< 3 \pi$

Need to solve: How many solutions are there to the equation, $t\sin (2t)=2$ where $0<t<3 \pi$ I am currently studying calc 3 and came across this and realized i dont have a clue as to how to ...
3
votes
1answer
79 views

Testing $\sin\theta$ and $\cos\theta$ without referring to the trigonometric functions

This is very much not my area so apologies if this is an obvious no. Suppose values have been calculated for $\sin\theta$ and $\cos\theta$. Is it possible to test their correctness, without referring ...
2
votes
2answers
91 views

Trigonometry - Finding the range of the function

Problem : $$f(\theta)=(2\sqrt{3}+4)\sin\theta +4\cos \theta $$ I have studied if the function is in the form : $f(\theta)=a\cos\theta + b\sin\theta$ then the range of this function can be given as ...
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vote
2answers
69 views

Finding the relation between function x,y,z - trigo problem

Problem : For $\displaystyle 0 < \theta < \frac{\pi}{2}$ if $$\begin{align}x &= \sum^{\infty}_{n =0} \cos^{2n}\theta \\ y &= \sum^{\infty}_{n =0} \sin^{2n}\theta\\ z &= ...
2
votes
2answers
179 views

If $\sin a+\sin b=2$, then show that $\sin(a+b)=0$

If $\sin a+\sin b=2$, then show that $\sin(a+b)=0$. I have tried to solve this problem in the following way : \begin{align}&\sin a + \sin b=2 \\ \Rightarrow ...
1
vote
1answer
53 views

Sectors of a Circle

I am programatically drawing sectors of a circle with radius 55 on a cartesian plane which runs from -55 to 55 on the x and y axes. I would like the first sector to be drawn at 0,55. I know I can ...
2
votes
2answers
362 views

The golden ratio and a right triangle

Assume the square of the hypotenuse of a right triangle is equal to its perimeter and one of its legs is $1$ plus its inradius(the radius inside the circle inscribed inside the triangle.) Find an ...
4
votes
1answer
66 views

Polynomials and Trig

Question: The equation $x^{2}-x+1=0$ has roots $\alpha$ and $\beta$. Show that $\alpha ^{n}+\beta ^{n}=2\cos\frac{n\pi }{3}$ for $n=1, 2, 3...$ Attempt: $x^{2}=x-1 \Rightarrow ...
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vote
2answers
88 views

How to find what point a wave is reflected off

If a wave is reflected off a surface, the angle of reflection is equal to the angle of incidence. But, how can we use this to find the actual path of the incident and reflected waves if we only know ...
0
votes
1answer
16k views

Finding functions for an angle whose terminal side passes through x,y

How would I "Find the six trigonometric functions for the angle theta whose terminal side passes through the point (-8,-5)"?. I learned this material over 2 years ago and since then have forgotten. I ...
1
vote
1answer
744 views

Derivative of tan(x) with product and chain rules instead of quotient rule

So I usually just use the product and chain rules for quotient functions, because I can never remember which product to substract from which in the numerator. But somehow I'm doing it wrong for ...
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vote
5answers
355 views

How to integrate these integrals

$$\int^{\frac {\pi}2}_0 \frac {dx}{1+ \cos x}$$ $$\int^{\frac {\pi}2}_0 \frac {dx}{1+ \sin x}$$ It seems that substitutions make things worse: $$\int \frac {dx}{1+ \cos x} ; t = 1 + \cos x; dt = ...
5
votes
2answers
163 views

Transforming trigonometric identities

The problem goes like this: If $$N=2\sec^4x-3\sec^2x+2=\frac{\cos^2x}{\cos^2y}$$ Calculate the equivalent of $$M=2\tan^4x+3\tan^2x+2$$ The alternaties I have are: $$\frac{\tan^2x}{\tan^2y},\mbox{ ...
0
votes
1answer
70 views

Angle limit problem

I have been trying to interpret orientation angle data retrieved from a sensor device. It returns the angle in Radian units towards North that the device is measuring at the moment. The problem I am ...
5
votes
1answer
434 views

Continued fraction for $\tan(nx)$

I found this beautiful continued fraction expansion of $\tan(nx)$, $n$ being a positive integer, online but I don't remember the source now: $\displaystyle \tan(nx) = \cfrac{n\tan x}{1 -\cfrac{(n^{2} ...
2
votes
3answers
85 views

Need to find function related to Knoedel numbers that satisfies these conditions

I need to find the continuous function $f(x)$ that satisfies $f(0)=0$ and: $$\frac{f(\sin(\pi/6))^2}{\sin^4(\pi/6)}=135$$ $$\frac{f(\sin(\pi/4))^2}{\sin^4(\pi/4)}=63$$ ...
6
votes
3answers
222 views

If $A = \tan6^{\circ} \tan42^{\circ},~~B = \cot 66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$

My trigonometric problem is: If $A = \tan6^{\circ} \tan42^{\circ}$ B = cot$66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$. Working : $$B = \cot 66^{\circ} \cot78^{\circ} = ...
2
votes
3answers
2k views

Express $\cos 6\theta $ in terms of $\cos \theta$

I think I'm supposed to use the chebyshev polynomials, as in $$ \cos n \theta = T_n(x) = \cos(n \arccos x)$$ But no idea what now?
0
votes
1answer
52 views

Trigonometric Anti-derivative

What is $$\int \frac{\sin(x)^2}{\cos(x) + 1}dx\;?$$ I've tried everything I can think of, but I can't get it into a form that I can solve.
3
votes
1answer
268 views

Solving $(Ax + B)\sin x = C$ for $x$

I have a pretty messy equation that I'm trying to solve for x. I've been able to get it down to: $$(Ax + B)\sin x = C$$ Where $A,B,C$ are all constants. Is there an analytical solution to this?
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vote
3answers
500 views

Identity as lower bound of sine

I'm struggling to rigorously proof $$ \sin(2x) \geq x \qquad (0 \leq x \leq \pi/4) $$ Any ideas?
1
vote
1answer
142 views

Problem with radian and degrees resulting in error

Hey all I'm trying to calculate the declination of a star using the following formula. The problem is I keep getting the wrong answer. I believe radians and degrees might be an issue. I'm expecting ...
1
vote
0answers
138 views

Roots of linear combinations of trigonometric functions

How many roots does $a\cos x + b \sin x = 0$ have, if $x \in [0, \pi]$, and $a$ and $b$ are nonzero? I deduced that it has at most one, but I am quite aware of my carelessness in trigonometry, so ...
2
votes
1answer
149 views

Trigonometry and algebra question

Given: The total length of ad + dc The lengths of each ab, bc and ...
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0answers
28 views

Trigonometry and algebra question [duplicate]

Given: The total length of ad + dc The lengths of each ab, bc and ...
2
votes
1answer
81 views

Solve for the domain of x

Solve for the domain of x $$y=\sqrt{\text{Cos}\left[x^2\right]}\tag1$$ my answer:where is wrong? $\text{Cos}\left[x^2\right]\geq 0$, so $x^2\in [2k \pi -\pi /2,2k \pi +\pi /2]\Rightarrow x\in ...
2
votes
1answer
74 views

shape regular triangulations and Zlamal's condition

I'm trying to show that a triangulation $\tau_h$ is regular if and only if there exists $\theta_0>0$ such that for all $T\in\tau_h$ we have $\theta_T\geq \theta_0>0$, whereas $\theta_T$ is the ...
5
votes
1answer
228 views

Size of new box rotated and the rescaled

I have a box of height h and width w. I rotate it to r degrees. Now I resize it so that it can original box in it. What will be the size of newly box. Original Box: Box after rotating some ...
1
vote
1answer
122 views

Domain and range of a cyclometric function

My textbooks tells me: $$f(x) = \arctan(x)$$ $D_f = \mathbb{R}$ and $B_f = \left\langle -\dfrac{1}{2} \pi, \dfrac{1}{2} \pi \right\rangle$ $$g(x) = \arcsin(x)$$ $D_g = [-1, 1]$ and $ B_g = [ ...
1
vote
2answers
112 views

Trigonometry: Law of Cosines

How to solve using rule of cosines? I can solve using law of sines but trying to check using rule of cosines is tripping me up, can anyone help clear things up?
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3answers
78 views

Trigonometry: Law of Sines

How can this be solved using law of sines? I get a different answer then when I solve it using law of cosines..
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3answers
120 views

Does $a_n = \cos\left(n\ln \left(1+\frac{\pi}{n}\right)\right)$ converge?

I want to check if a sequence converges or diverges. The sequence is the following: $$a_n = \cos\left(n\ln \left(1+\frac{\pi}{n}\right)\right)$$ I though of maybe using sandwich theorem, but can I ...
5
votes
1answer
92 views

Obtaining function's extreme values without derivate

What is other method to obtain a function min/max value without any use of derivative? For example in this function: $f(x) = 4x + \dfrac{9\pi^2}{x} + \sin(x)$ My teacher used a method that goes ...