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

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4
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1answer
110 views

Calculate depth using triginometry

I was asked a question like this on an exam today and I'm wondering if I got it right or not. A boat is laying a pipe on the ocean floor using two cables. The angle of depression for one of the ...
-1
votes
1answer
64 views

An error in Wikipedia? (trigonometry)

https://en.wikipedia.org/wiki/N-sphere In "Spherical coordinates" section the hyperspherical coordinates are results of arccosinus function. In some other sources there is arccotangent instead: ...
5
votes
1answer
180 views

How to find inverse of $\sin(x) + \sin(2x) = y$?

I was wondering if there were any way to solve the equation $$\sin(x) + \sin(2x) = y$$ in terms of $x$. This of course would allow us to express the inverse for this function on $-\frac{\pi}{4}$ to ...
0
votes
0answers
41 views

Find all the angles between $0^\circ$ and $360^\circ$ which satisfy the equation.

Find all the angles between $0^\circ$ and $360^\circ$ which satisfy the equation $$\tan 3x-3 \sin 30^\circ=0$$ I tried searching for examples but didn't get any. Please teach me how to solve such ...
1
vote
1answer
50 views

Question regarding trigonometry

I've got this thing on my mind : we know that $cos(x)$ is a periodic function , hence integral from $2(k-1) \pi$ to $2k \pi$ will yield the same value for any $k \geq1$. My question is , why is ...
0
votes
1answer
161 views

Solve an Angle-Side-Angle special case triangle if it has an obtuse angle?

I've seen this type of problem multiple times on homework, and it's confusing me like mad. The scenario: We have a triangle. It is a special case triangle, with one angle, one side, and another ...
2
votes
2answers
62 views

Trigonometric substitution

Been out of touch with trigonometry for some time now. Need help proving this expression. $$\sin^{2}\left(\frac{x}{2}\right) = \frac{1}{2}(1-\cos\left(x\right))$$ Any help will be appreciated. ...
1
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2answers
43 views

Trigonometric equation problem

This is the following equation: $$\arccos x= \arctan x$$ Could someone give me at least a tip how to begin with?
0
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3answers
82 views

Solving a simple trigonometric equation $\sin x = -\sin y$

What is the solution set of the following trigonometric equation? $$\sin x = -\sin y$$
0
votes
2answers
50 views

Doubt regarding trigonometric equation

In a book of mine it says solution of $\sin^2(x) = \sin^2(y)$ is $x = n\pi \pm y$ But if we take sq root on both sides we get sinx = siny for which the solution is $x = n\pi + (-1)^ny$ Which is ...
22
votes
13answers
6k views

How to solve $4\sin \theta +3\cos \theta = 5$

Another problem that I already wasted hours on. Given $$4\sinθ +3\cosθ = 5$$ Find $$4\cosθ -3\sinθ$$ Help me guys (PS:I'm not that good in maths)
4
votes
1answer
68 views

Solving a trigonometric equation

Can someone help me to solve this problem? Find all number pairs $x,y$ that satisfy the equation: $$\tan^4(x) + \tan^4(y) + 2\cot^2(x)\cot^2(y) = 3 + \sin^2(x+y)$$
1
vote
1answer
352 views

Calculus - Trig Maximum Value Problem

When the rules of hockey were developed, Canada did not use the metric system. Thus, the distance between the goal posts was designated to be six feet. If Sidney Crosby is on the goal line, three feet ...
2
votes
3answers
336 views

Alternative definition of hyperbolic cosine without relying on exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula ...
0
votes
2answers
46 views

Verify :$\cos^2x=\cot^2x-\frac{\cos^2x}{\tan^2x}$

$$\cos^2x=\cot^2x-\frac{\cos^2x}{\tan^2x}$$ How can I solve it?
0
votes
3answers
33 views

Sketch the graph for $0^\circ \leqslant x \leqslant 360^\circ$.

Sketch the graph $y= cos \frac{3}{4}x-2$ for $0^\circ \leqslant x \leqslant 360^\circ$. Please help me draw this. I found out that $y= cos \frac{3}{4}x-2$ has a period of ...
0
votes
1answer
184 views

Find the resulting speed and direction

A barge is pulled by two tugboats. The first tugboat is traveling at a speed of 15 knots with heading 130°, and the second tugboat is traveling at a speed of 16 knots with heading 190°. Find the ...
0
votes
2answers
33 views

Solving for x on unit circle equation

I have been given the equation $$\cos^2{x} + 2\sin{x}=2.$$ I have factored it, and the only answer I got was $x=\frac{\pi}{2}$. Is this correct or is there more than one answer? The interval is $0 ...
0
votes
1answer
40 views

Simplifying Trig Identity

I have an equation I have been given to solve, I know how to start but I do not know what to do after I use the Trig Identities. Any help? Here is what I was given $$ \frac{\cos(A + B) + \cos(A - ...
1
vote
2answers
180 views

Inequality: $\tan(x) > 1$

So far, I've not come very... far. It ends up with me trying to solve it more intuitively than mathematically. I figured, first I'll find the place of equality, which is at $x = \arctan 1 = ...
1
vote
2answers
325 views

Finding (sin(A+B))^2 given roots of a quadratic equation.

If tan A and tan B are the roots of the equation x^2 -ax + b = 0, then the value of sin(A+B)^2 is? Options are: ((a^2)/((a^2)+(1-b)^2), (a^2)/(a^2+b^2), a^2/(b+a)^2, a^2/(b^2*(1-a)^2) The value ...
4
votes
2answers
166 views

Showing $\sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64}$

I would like to show that $$ \sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64} $$ I've been working on this for a few ...
0
votes
2answers
307 views

“Which is equivalent for restricted x values to”

I've been checking my homework via Wolfram Alpha, and for several questions (example below) in this section (trigonometric integrals). I'd be correct up until the last step, in which Wolfram Alpha ...
1
vote
1answer
100 views

Find the solutions of: $\sin x+\cos x=\sin^2 x+0.5\sin{2x}$

Find the solutions of: $\sin x+\cos x=\sin ^2 x+0.5\sin{(2x)}$ How can I find the solutions ?
0
votes
1answer
31 views

Law of Sine Problem

I know the law of Sine. SinA/a=SinB/b=SinC/c I think I'm missing something here... I am given ∠A=68.41°,∠B=54.23° and a=12.75ft. I found b with no trouble which is 11.119ft. I used SinA/a=SinB/b... ...
-1
votes
1answer
170 views

How to draw arrows by rotating lines in 3d space?

I am trying to figure out direction vectors of the arrowheads of an arrow. Basically I'm given a normalized direction vector ...
3
votes
2answers
3k views

How to find opposite and adjacent lengths of a right triangle given the hypotenuse and angle?

I'm writing a few functions for a JavaScript game engine. Is it possible to calculate the length of the legs of a right triangle given ONLY the length of the hypotenuse and an angle?
0
votes
1answer
40 views

Finding the value of trigonometric functions

This is probably one of the easiest concepts but I do not get it, so I am going to give the two problems that are giving me the most trouble on my very long worksheet I have to do, maybe you guys can ...
1
vote
1answer
41 views

Integrating an equation with both cos and tan

$$\int2\cos^5x\cdot\tan^6x\cdot dx$$ $$2\int\cos^5x\cdot\frac{\sin^6x}{\cos^6x}\cdot dx$$ $$2\int \frac{\sin^6x}{\cos{x}} dx$$ $$2\int\cos^{-2}x\cdot \sin^6x\cdot \cos{x}\cdot dx$$ ...
1
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3answers
70 views

I have problem with Trigonometry

Tomorrow I have a test and there is one exercise in my textbook that isn't explained. Here is the exercise. ...
0
votes
2answers
4k views

Real world tangent functions

I am a high school math teacher and one of my students asked me for examples of real world tangent functions. Not using tangent to find a side length but a relationship that can be represented by a ...
0
votes
1answer
44 views

When to use what inverse trig?

When do I use $\arcsin$ and when do I need to include all of the outcomes? My gut feeling is if you have an equation like $\sin(x)=0$, then $x=0,\pi,2\pi...$ whilst if you are using it in integration ...
1
vote
1answer
83 views

When is $ 4 ab \sin^2 θ = (a+b)^2 $ ?

I know that by trial and error it is only possible when $ a=b $, but what is the actual solution process?
2
votes
1answer
121 views

Weierstrass function

I got stuck on this exercise from Prof. Tao's real analysis notes. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic ...
1
vote
3answers
57 views

Problem with trigonometric equation

I am having trouble solving this equation $$4\cdot \sin \theta + 2 \cdot \sin 2\theta =5$$ Thank you for your help.
0
votes
2answers
38 views

Trig Identity Proofs

I'm having a really hard time understanding how to do these. The directions are to verify that each of the following is an identity: $$\dfrac{\csc x}{\cot x+\tan x}=\cos x$$ I have to get the left ...
0
votes
2answers
30 views

Express the following in terms of $q$.

Given that $\cos{160^{\circ}} = -q$, express $\cos70^{\circ}$ in terms of $q$. No example in the book, don't know how to do it?? I need a complete explanation.
2
votes
1answer
32 views

Can I find this trigonometric expression without a calculator?

I know that $\sin A= 0.75$ will give me the answer of $A= 48.6^\circ$ or $\ 131^\circ$. Is there a way to find what $A$ equals manually. Thank you.
0
votes
3answers
39 views

Problem with this Trigonometric Equation

I am having trouble figuring out how to solve such an equation can anyone please tell me the steps to solve it as I have been solving a lot of trigonometric equations but I am stuck in this one: ...
0
votes
1answer
40 views

Trigonometric problem

I am having trouble solving simple trigonometric equations without a calculator which I am required to be doing in my course since I cant get to understand how to get for example sin x=-1/2 I know sin ...
2
votes
1answer
68 views

Sum of fractions of squared sines

I'm trying to prove the following approximate identity for $p$ integer: $$ \sum_{l=1}^m\frac{\sin^2\left(\frac{\pi l}{p}\right)}{\sin^2\left(\frac{\pi l}{mp}\right)}\sim \frac{m^2(p-1)}{2}+O(m) $$ ...
0
votes
1answer
31 views

Simple algebraic question mixed up

I know it is very simple but do not know why I am mixed up in it $(.5)(r^2)\cfrac{20-2r}r$ how is this equal to $10r-r^2$ Sorry if it is too easy, thanks for the help.
0
votes
1answer
28 views

Trigonometric equation problem.

Simply and shortly how do I show that this $33 = 33 + 5 \cos(720\cdot t)$ is equal to this $720 \cdot t = 90.$ Thank you for your help.
2
votes
0answers
51 views

How find the range value $a^2+b^2$ if $\cos{(a\sin{x})}=\sin{(b\cos{x})}$ have no solution

if the equation $$\cos{(a\sin{x})}=\sin{(b\cos{x})}$$ have no zero solution,then $a^2+b^2$ range of value $A:[0,\dfrac{\pi}{4})$,$B: [0,\dfrac{\pi^2}{2})$,$C: ...
2
votes
4answers
70 views

Minimum value Of trigonometry expression

FIND THE MIN VALUE OF 4 cosec^2 x + 9 sin^2 x ? Please explain by both calculus and non-calculus methods ?
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2answers
213 views

What is the minimum value of $\csc x - \sin x$?

What is the minimum value of $\csc x - \sin x$? Differentiating and setting it to zero yields nothing meaningful. How can I find the minimum value?
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0answers
41 views

Finding the inverse of trig functions

I'm supposed to find the inverse of $$f(x) = \cos(x)+x$$ I usually just substitute $x$ for $y$ and then re-arrange. What do I do in this scenario?
2
votes
6answers
119 views

explicit expression sought

Consider the equation $$ \cos^2\phi + \alpha\sin\phi\cos\phi-\beta=0\;, $$ where $\alpha,\beta\in\mathbb{R}$. I need to find an explicit expression for $\phi$. I have tried completing the square, but ...
3
votes
2answers
81 views

Finite-case symmetry leads to infinite-case asymmetry

Formulas for sines or cosines of sums superficially appear to have a certain symmetry, specifically it looks as if sine and cosine play something like symmetrical roles: $$ \begin{align} & ...
2
votes
2answers
77 views

Complex Numbers and Hyperbolic Functions

How would you evaluate: $\mathfrak{R}\left[(1+i)\sin\left(\dfrac{(2+i)\pi}{4}\right)\right]$? I know that $\cos x = \dfrac{e^{ix}+e^{-ix}}{2}$ and $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$. I have also ...