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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3answers
103 views

How do I write a sum of cosines as a product of sines?

I am trying to prove that $$\cos A+\cos B+\cos C=4\sin\frac A2\sin\frac B2\sin\frac C2$$ for ABC is a triangle. I tried up to the stage of $$-2\sin^2 C+2\cos\frac{180-C}2 \cos\frac{A+B}2$$ but how ...
1
vote
4answers
534 views

Prove that $\cot(A+B)=\frac{\cot A\cot B-1}{\cot A+\cot B}$

The question is: Prove that: $$ \cot(A+B)=\frac{\cot A\cot B-1}{\cot A+\cot B} $$ I have tried expanding it as $\dfrac{\cos(A+B)}{\sin(A+B)}$ and $\dfrac{1}{\tan(A+B)}$.
0
votes
2answers
71 views

If $\tan A=2$, find the possible values of $\csc A$

If $\tan A=2$, find the possible values of $\csc A$. Can someone please show me thorough steps as to how to do this question?
0
votes
3answers
131 views

Given that $\csc A=-\frac{5}{2}$ and $\sin A< 0$, find $\cot A$

Trig question, Given that $\csc A=-\frac{5}{2}$ and $\sin A< 0$, find $\cot A$ Ive done this so far, although my answer is abit off, where the correct answer is $-\frac{1}{2}\cdot\sqrt{21}$ while ...
5
votes
4answers
626 views

Given that $\cos x =-3/4$ and $90^\circ<x<180^\circ$, find $\tan x$ and $\csc x$

Given that $\;\cos x =-\frac{3}{4}\,$ and $\,90^\circ<x<180^\circ,\,$ find $\,\tan x\,$ and $\,\csc x.$ This question is quite unusual from the rest of the questions in the chapter, can someone ...
2
votes
0answers
101 views

Trigonometric functions of rational fractions of pi

Consider rational numbers $\frac{m}{n}$ and $\frac{m'}{n'}$, where $0<\frac{m}{n}, \frac{m'}{n'} <1$. Then $$\sin^2 (\tfrac{m}{n} \pi) = 2 \sin^2 (\tfrac{m'}{n'} \pi)$$ When $\frac{m}{n} = ...
0
votes
2answers
31 views

Trigonometry Functions/equations

Given that $\sin A=\dfrac{8}{17},\;$ find the possible values of $\cos A$ and $\cot A$. Can someone please explain this question? I' new to the topic, and I'm very unsure as to how its done.thanks
5
votes
1answer
140 views

Sum of the arctangents of roots of a cubic equation (multiple choice)

If $A$, $B$, and $C$ are the roots of the equation $x^3+mx^2+3x+m=0$, then the value of $$\arctan(A)+\arctan(B)+\arctan(C)$$is given by A. $n\pi$ B. $n\pi/2$ C. $\pi(2n+1)/2$ D. none ...
3
votes
3answers
653 views

What is a rational trigonometric function? Is $\cos x$ rational?

I am reading Trigonometry by Gelfand and Saul. On p.140 they discuss rational trigonometric functions and define one as: A rational trigonometric function is a function you can get by taking the ...
2
votes
3answers
224 views

Tirgo problem knowing roots are given find the value of : $\sin^2(\alpha +\beta) +p\sin(\alpha +\beta) \cos(\alpha +\beta)+q\cos^2(\alpha +\beta)$

Question : Knowing that $\tan\alpha$ , $\tan\beta$ are roots of the quadratic equation $x^2+px+q=0$ ; Compute the expression $\sin^2(\alpha +\beta) +p\sin(\alpha +\beta) \cos(\alpha ...
-1
votes
1answer
56 views

Trigonometric Equations

Eliminate B from each pair of equations: x=sinB -3cosB and y=sinB+2cosB I've tried solving this simultaneously just as the textbook has guided me through, but it still doesn't work. My ...
3
votes
1answer
591 views

Solving a system of non-linear (trig) equations:

I am having trouble trying to solve the following equations: $\sin(\alpha)+\sin(\beta)=\dfrac {1000} A$ $\sin(\alpha)+\sin(\gamma)=\dfrac {800} A$ $\dfrac {20(1+\cos(\alpha-\beta))} {\cos(\beta)} ...
2
votes
1answer
429 views

Trigonometric problem : Eliminate $\theta$ and $\phi$ from the relation and find relation between p and q

Question : Eliminate $\theta$ and $\phi$ from the relation $$\begin{align} p \cot^2\theta + q \cot^2\phi &= 1 &(1)\\ p \cos^2\theta + q \cos^2\phi &= 1 &(2)\\ p \sin\theta &= ...
2
votes
2answers
114 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
236 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
336 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
444 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 $[0,360]$
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1answer
112 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
110 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
913 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
434 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
156 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
115 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
211 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
52 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
357 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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2answers
84 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
15k 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
708 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 ...
1
vote
5answers
353 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
418 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
211 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
260 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?