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
votes
0answers
21 views

When $\cos(\theta) = 1/8$ it's easy to show $\theta$ is an irrational angle. Is it algebraic?

Along the lines of my lines of my previous question about irrational angles "$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?", I was working on ...
-3
votes
0answers
26 views

solve another system of three equations [on hold]

I have: $x=\dfrac{-.5b-.5c+.25d}{b+c+d}$ $y=\dfrac{.5b\sqrt{3}+.5c\sqrt{3}+.25d\sqrt{3}}{b+c+d}$ $z=b+c+2d$ I need help moving the $b$, $c$, and $d$ to the Left-hand-side; and moving the x, y, and ...
1
vote
2answers
29k views

Calculate width and height of a rectangle, given its diagonal and ratio

Well, I know, it's easy. We did it in class some time ago and I forgot it, I'm stupid because I can't figure it out: E.g. I have a 32" TV with 16:9 ratio and I want to know its width and height. I'd ...
2
votes
0answers
27 views

How to find unkown height of triangle without hyptenuse

I been trying to solve this question and have tried to solve it for many days, but do not know how, any help would be much oblidged. A cable company owns the roads marked with the dotted lines in ...
5
votes
7answers
1k views

How many ways are there to define sine and cosine?

Sometimes there are many ways to define a mathematical concept, for example the natural base logarithm. How about sine and cosine? Thanks.
-1
votes
0answers
37 views

Solve a system of three equations [on hold]

$x=\dfrac{a-.5c+.25d}{a+c+d}$ $y=\dfrac{.5c\sqrt{3}+.25d\sqrt{3}}{a+c+d}$ $z=a+c+d2$ How do I make it so that only $x$, $y$, and $z$ are on the Right-Hand-Side of the equation while only $a$, $c$, ...
0
votes
2answers
41 views

Solve the equation $(tan θ − 2)(9 sin^2 θ − 1) = 0$

Solve the given equation. (Enter your answers as a comma-separated list. Let $k$ be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) ...
1
vote
3answers
21 views

Write the trigonometric expression as an algebraic expression

Write the trigonometric expression as an algebraic expression. $6 \cos(2 \cos^{-1} x)$ Can someone explain to me how to do this? I tried it on my own after watching a youtube video from patrickjmt ...
3
votes
5answers
52 views

Trigonometry identity $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$

How to prove that $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$? I tried manupulating the left hand side but ended up in $\frac{\cos x}{\sin^2 x}$. Can someone show me? Thanks in advance.
0
votes
2answers
46 views

Finding the Zeroes of a Second Derivative to Determine Points of Inflection

I have been told to graph the function $y=\cos \left(x^2\right)$ for $-2π ≤ x ≤ 2π$. I have determined key features of the graph but need help when it comes to determining the points of inflection for ...
-7
votes
0answers
39 views

How to solve equation like this [on hold]

A trigonometric equation has the following solution: $ x=20°+n*15°$ where n is an integer . which answer(s) of the of the following answers give(s) the exact same result, note that m is also an ...
3
votes
1answer
56 views

When is the cosine of $\pi/n$ of a certain form?

I have a few questions concerning $\cos(\frac{\pi}{n})$. Are there characterizations for the values $n \in \mathbb{N}$, such that $\cos(\frac{\pi}{n})$ ... is an algebraic number? ... can be written ...
17
votes
6answers
384 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
2
votes
2answers
23 views

Show that any 2D vectors can be expressed in the form…

(a) Show that any 2D vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let $u$ and $v$ be ...
0
votes
2answers
48 views

Trigonometry problem

Okay..this one simple problem but I am really stuck and have no idea how to start.. $\cos(a-b)+\cos(b-c)+\cos(c-a)=-\frac32$ we need to prove $\cos(a)+\cos(b)+\cos(c)=\sin(a)+\sin(b)+\sin(c)=0 $
2
votes
3answers
80 views

Find max of $S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$

Let $x$, $y$, $z$ are positive constants. $A$, $B$, $C$ are three angles of the triangle. Find max of $$S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$$
0
votes
3answers
34 views

If $a \sin \alpha = b \sin \beta$, then show that $ b \cot\alpha + a \cot \beta = (a+b)\cot \frac{\alpha +\beta}{2}$

The question is: If $a \sin \alpha = b \sin \beta$, then show that $$ b \cot\alpha + a \cot \beta = (a+b)\cot \frac{\alpha+\beta}{2} $$ Could I get any hints to the problem? When do I need to ...
2
votes
1answer
34 views

Find $Z$ transform of given signal

Given the discrete signal $h(n)=r^n\frac{\sin{[(n+1)\theta]}}{\sin{\theta}}$ if $n \geq 0$ and $h(n)=0$ otherwise, find the $Z$ transform of $h(n)$. What I did: We know that ...
-1
votes
4answers
30 views

prove that $\sin^2 b +\sin^2 c - \sin^2 a= -2\cos a \sin b\sin c$. [on hold]

I tried nearly but couldn't prove it afterwards. Please tell me the the way to prove this. The conditions of the problem is that $a+b+c=0$.
4
votes
0answers
31 views

Generalized Trigonometric Functions in terms of exponentials and roots of unity

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x ...
2
votes
2answers
114 views

Finding $4$ variables using $3$.

if I have: $ x=\dfrac{a-.5b-.5c+.25d}{a+b+c+d}$ $ y=\dfrac{\dfrac{b\sqrt{3}}{2}+\dfrac{c\sqrt{3}}{2}+\dfrac{d\sqrt{3}}{4}}{a+b+c+d}$ $ z=a+b+c+2d $ Then how do I get back to: $ a= $ , $ b= $ , $ ...
3
votes
2answers
368 views

What is the actual geometric meaning of trigonometric operations such as adding cos,sine,tan

$$\sin(\pi/4)+\cos(\pi/4)=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}= \frac{2\sqrt{2}}{2}=\sqrt{2}$$ Thinking of trig components (cosine, sine) that I used to produce the result using the mechanics of ...
1
vote
2answers
99 views

Find the value of $h$ from a Kepler-type equation

$$V = \frac{0.5r^{2}\cdot \cos^{-1}(\frac{r-h}{r})\cdot 2-\sin\big(\cos^{-1}(\frac{r-h}{r})\cdot 2\big)}{10^{6}}\tag1$$ This is the equation to find the volume of liquid in a tank in the shape of a ...
1
vote
2answers
59 views

Solve $2+ \cos{\frac{3x}{2}} + \sqrt{3} \sin{\frac{3x}{2}} = 4\sin^2{\frac{x}{4}}$

$$2+ \cos{\frac{3x}{2}} + \sqrt{3} \sin{\frac{3x}{2}} = 4\sin^2{\frac{x}{4}}$$ My try: $$ \cos{\frac{3x}{2}} + \sqrt{3} \sin{\frac{3x}{2}} = \sqrt{4}\left(\frac{\sqrt{3}}{2} \sin{\frac{3x}{2}} + ...
0
votes
0answers
21 views

Smoothly interpolating between functions to create a bouncing wave

How can I create a function which allows me to control the roundness of a wave so I can transition between an Round Wave -> Linear Wave -> Inverted Round wave? I've made a function which creates a ...
3
votes
2answers
83 views

How to prove an identity (Trigonometry Angles--Pi/13)

In this page http://mathworld.wolfram.com/TrigonometryAnglesPi13.html I found equation (11) and (12). $$\cos^2\frac{\pi}{13}+\cos^2\frac{3\pi}{13}+\cos^2\frac{4\pi}{13}=\frac{11+\sqrt{13}}{8}$$ ...
0
votes
2answers
36 views

Trigonometric Form of Complex Numbers question. [on hold]

What is the following quotient expressed in polar form: $$\frac{10(\cos(35^{\circ})+ isin(35^{\circ}))}{5(\cos(100^{\circ}) +i\sin(100^{\circ}))}?$$ Please enter your answer in cis notation and ...
3
votes
2answers
59 views

Easy question Find $\sin 2x$, $\cos 2x$, and $\tan 2x$

Ok so I was absent from school yesterday because long story short I had no way to get to class b/c something happened last minute. I'm pretty sure this is easy but I keep getting the wrong answer for ...
3
votes
1answer
59 views

$D, E, F$ are respectively projection of $O$ on $BC, CA, AB$. Prove that $\cot{\angle ADB} + \cot{\angle BEC} + \cot{\angle CFA} =0$

Let $O$ be an arbitrary point located inside the triangle $ABC$. Let $D, E, F$ be (respectively) the projections of $O$ on $BC, CA, AB$. Prove that $$\cot{\angle ADB} + \cot{\angle BEC} + ...
2
votes
4answers
87 views

Find the area of a triangle given the radius of its incircle and a tangential point

A friend gave me recently the following interesting problem and I would like to share a couple of solutions. Any additional contributions are welcome. A triangle $\vartriangle ABC$ is given and we ...
0
votes
2answers
14 views

In the sin of two angles are equal, then proving that two angles are equal - w.r.t different traingles

From the text book: What do they mean by: Therefore, AC/PR = AB/PQ ? Is the / division or ratio? What rule says that AC/PR = AB/PQ in this example? What do ...
1
vote
2answers
143 views

Find the values of the positive constants $k$ and $c$ such that $-37\le k(3\sin\theta + 4\cos\theta) +c\le 43$ for all values of $\theta$

Hi how do i go about solving this? Find the values of the positive constants $k$ and $c$ such that $$-37\le k(3\sin\theta + 4\cos\theta) +c\le 43 $$for all values of $\theta$ $$\rightarrow-37\le ...
0
votes
2answers
56 views

Find the smallest value of $x$ such that $10\cos\left (\frac{x+1}{2}\right)=3$ [on hold]

Given that $x$ is measured in radians and $x > 10$, find the smallest value of $x$ such that $$10\cos \left(\frac{x+1}{2}\right)=3$$ How to solve this question? I've no idea.
-3
votes
0answers
25 views

why $\cos 2x$ is positive for $0 < k < 1$? [on hold]

The reflex angle x is such that $\sin x = –k$, where $0 < k < 1$ How to explain why $\cos 2x$ is positive for $0 < k < 1$?
-3
votes
1answer
31 views

find the exact value of $\cos^2x$ and $\csc x$. [on hold]

Given that $x=\tan ^{-1}(\frac{1}{3})$, find the exact value of $\cos^2x$ and $\csc x$. How to find this without using calculator?
1
vote
1answer
27 views

Express various trig functions in terms of the sine.

The acute angle $x$ radians is such that $\sin x = k$, where $k$ is a positive constant. Express, in terms of $k$. i) $\sin (2\pi-x)$ ii) $\tan(\frac{1}{2}\pi-x)$ iii) $\cos (\pi+x)$ My attempt: ...
1
vote
1answer
134 views

What is the value of $\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$? [duplicate]

How to compute $$S=\csc^2\frac{\pi}{14}+\csc^2\frac{3\pi}{14}+\csc^2\frac{5\pi}{14}$$ I tried to rewrite it in terms of $\sin$ $$ ...
1
vote
1answer
35 views

Show that $x^2+y^2$ is constant for all values of $\theta$.

Given that $x=3\sin \theta-2 \cos \theta$ and $y=3\cos \theta+2 \sin \theta$ i)Find the value of the acute angle $\theta$ for which $x=y$ ii)Show that $x^2+y^2$ is constant for all values of ...
1
vote
1answer
33 views

Show that the angle between $OP$ and the normal to the curve at $P$ satisfies the following

I'm struggling to answer the following question below I've already worked out the gradient to the curve at $P$, but I'm having difficulty answering the second part of the question. MY attempt is as ...
0
votes
2answers
68 views

Trigonometry - how to find angles of triangle within another triangle?

What are the angles of angle 1 and 2? I don't see how any of them could be corresponding angles... The adjacent side of angle 2 is parallel to the hypotenuse of the bigger triangle, just to make ...
2
votes
1answer
415 views

How to model an aviation holding pattern mathematically?

A standard aviation holding pattern has four sections, each of which, in windless conditions, takes one minute to fly. The first is the inbound leg, which is the only one for which precise ...
1
vote
2answers
878 views

Calculate Triangle Ground using Height and Top Angle

Is it possible to calculate the ground of a triangle only using the height and top angle. Click here to see a poorly draw sketch of what I'm trying to calculate. So is it possible and how, to ...
4
votes
4answers
309 views

What is the limit of $x/(x+\sin x)$ as $x$ approaches infinity?

I am trying to determine $$\lim_{x \to \infty} \frac{x}{x+ \sin x} $$ I can't use here the remarkable limit (I don't know if I translated that correctly) $ \lim (\sin x)/x=1$ because $x$ approaches ...
18
votes
1answer
3k views

Infinite product of sine function

How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) ...
-1
votes
3answers
31 views

Value of Sine Function from data given [on hold]

If $0 \le \alpha , \beta \ge 90\ $ and $ \tan(\alpha - \beta) = 2$ and $\tan (\alpha + \beta) = 3 $, Then what is the value of $\sin 2\beta$ ?
-1
votes
1answer
52 views

Maximize the trigonometric expression

Find the maximum value of $$4\sin^2 x+3\cos^2 x+\sin(x/2)+\cos(x/2)$$ Please give some hints. I tried writing the angles in half-angles but it didn't help. Thanks.
3
votes
1answer
43 views

My attempt regarding finding critical ponts of $(\cos x)(\cos y)(\cos(x+y))$

Given this problem Restrictions on $x$ any are that $x\in[0,\pi]$ , $y\in[0,\pi]$ I have $f_x=-(\cos y)({\sin(2x+y))}--------*$ $f_y=-(\cos x)(\sin x+2y)-----------**$ So from $*$ I get either ...
4
votes
1answer
64 views

Calculation of integral using two different methods? [on hold]

Find $$\int \dfrac{x^3}{(x^2+1)^3}dx$$ in two different ways, first using the substitution $u=x^2+1$ and then using the substitution $x=\tan \theta$. I managed to do both of these but the answer is ...
-4
votes
2answers
50 views

How to calculate an elementary integral

How do you calculate $$\int\dfrac{2 du}{(u^2+1)^2}$$ It does not seem too difficult but I do not know which method to use.
2
votes
7answers
43 views

Some help on trigonometric equation

So I have $\sin^3x = \frac 34 \sin x$. Can you expand so the answer is either $\sin x(\sin^2x +\frac 34)$ which leads to the answer $\frac 12 + 2n\pi$ or that $\sin^3x = \frac 14(3\sin x-\sin^3x) - ...