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

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0
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4answers
1k views

Find all solutions of $4\cos^2(x)-4\sin(x)-5=0$

Find all solutions of $4\cos^2(x)-4\sin(x)-5=0$ in the interval $(6\pi, 8\pi)$. I tried to work it out and got: $4y^2-4y -9 = 0$, but I can't figure out what $\cos x = $from there to finish the ...
1
vote
1answer
395 views

Higher Derivatives of trigonometric functions

The position of a particle is given by $s = 5 \cos (2t+ (\pi/4))$ at time $t$ . What are the maximum values of the displacement,the velocity,and the acceleration? The answers are displacement: $5$ ...
-1
votes
2answers
266 views

Find all solutions of $\cos (x) + 1/2 \sec (x) = -3/2$ in the interval $(2\pi, 4\pi)$

Find all solutions of $\cos (x) + 1/2 \sec (x) = -3/2$ in the interval $(2\pi, 4\pi)$ (Leave your answers in exact form and enter them as a comma-separated list.)
7
votes
5answers
442 views

What's the importance of the trig angle formulas?

What's the importance of the trig angle formulas, like the sum and difference formulas, the double angle formula, and the half angle formula? I understand that they help us calculate some trig ratios ...
6
votes
2answers
127 views

Sequence and Series - If $a_n =\int^{\frac{\pi}{2}}_0 \frac{\sin^2nx}{\sin^2x}dx,$…

If $\displaystyle a_n =\int^{\frac{\pi}{2}}_0 \frac{\sin^2nx}{\sin^2x}dx, $ then find the value of $$\begin{vmatrix} a_1 & a_{51} & a_{101} \\ a_2 &a_{52} & a_{102}\\ a_3 & ...
8
votes
3answers
843 views

Finding the limit $\lim \limits_{n \to \infty}\ (\cos \frac x 2 \cdot\cos \frac x 4\cdot \cos \frac x 8\cdots \cos \frac x {2^n}) $

This limit seemed quite unusual to me as there aren't any intermediate forms or series expansions which are generally used in limits. Stuck on this for a while now .Here's how it goes : $$ \lim ...
0
votes
2answers
3k views

How would you calculate the Tangent without a calculator? [duplicate]

I was just curious as to how you would calculate it without a calculator. I don't care if it's in radians or degrees, but I just would like it to be specified.
1
vote
3answers
488 views

Show that $\sin3\alpha \sin^3\alpha + \cos3\alpha \cos^3\alpha = \cos^32\alpha$

Show that $\sin3\alpha \sin^3\alpha + \cos3\alpha \cos^3\alpha = \cos^32\alpha$ I have tried $\sin^3\alpha(3\sin\alpha - 4 \sin^3\alpha) = 3\sin^4\alpha - 4\sin^6\alpha$ and ...
10
votes
1answer
3k views

How were Hyperbolic functions derived/discovered?

Trig functions are simple ratios, but what does Cosh, Sinh and Tanh compute? How are they related to euler's number anyway?
3
votes
2answers
130 views

Showing a particular function is onto

Consider $f: \mathbb{R} \rightarrow \mathbb{R}$ is given by $f(x) = x\cos(x)$, we want to show $f$ surjective. Usually these type of proofs aren't very difficult as you can isolate x in terms of y. In ...
2
votes
2answers
76 views

Sine Over Cosine Limit Doubt

Intuitively, I suppose that: $\displaystyle \lim_{x\to \infty} \dfrac{x + 5 \sin x}{x-\cos x} = 1$ Analytically, though, I get stuck at $\cos / \sin$ limits... Thanks.
3
votes
2answers
244 views

Proving a fact: $\tan(6^{\circ}) \tan(42^{\circ})= \tan(12^{\circ})\tan(24^{\circ})$

Prove that $\tan(6^\circ)\tan(42^\circ) = \tan(12^\circ) \tan(24^\circ)$. I don't know how to approach this problem. One approach might be to note that $42-6= 24+12$, and then apply the identities ...
0
votes
3answers
77 views

Solve following trigonometric equation

How to solve $$2\sin(x) / (1+2\cos(x)) = \sqrt{3}/2$$ where $0< x <180$. (Final answer may be in inverse form)
0
votes
1answer
34 views

determine the bearing of the plane

A jet takes off bearing N28∘E and flies 5 miles and then makes a left (90∘) turn and flies 12 miles further. If the control tower operator wanted to locate the plane, what bearing would she use?
0
votes
1answer
645 views

Solving negative domain trigonometric equations with unit circle

How would you solve a trigonometric equation (using the unit circle), which includes a negative domain, such as: $$\sin(x) = 1/2, \text{ for } -4\pi < x < 4\pi$$ I understand, that the sine ...
-1
votes
1answer
118 views

linear velocity conversion problem

A ford F-$150$ comes standard with tires that have a diameter of $25.7$ inches. If the owner decided to upgrade to tires with a diameter of $28.2$ inches without having onboard computer updated, how ...
0
votes
1answer
109 views

Consider the function $f(x)=\tan(\frac{\pi x}{2})$ and $g(x)=\frac12\sec(\frac{\pi x}{2})$

I need an explanation on the following problem Consider the function $$f(x)=\tan(\frac{\pi x}{2})$$ and $$g(x)=\frac12\sec(\frac{\pi x}{2})$$ on the interval $[-1,1]$: a: Approximate the interval ...
1
vote
2answers
355 views

Express $f(\theta)=6\cos2\theta+5(\sin2\theta)^2+3$ as a sum of powers of $\sin\theta$

Please explain how to express the below formula as a sum of powers of $\sin \theta$ $$f(\theta)=6\cos2\theta+5(\sin2\theta)^2+3$$
11
votes
2answers
487 views

Find the sum : $\sin^{-1}\frac{1}{\sqrt{2}}+\sin^{-1}\frac{\sqrt{2}-1}{\sqrt{6}}+\sin^{-1}\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}+\cdots$

Problem : Find the sum of : $$\sin^{-1}\frac{1}{\sqrt{2}}+\sin^{-1}\frac{\sqrt{2}-1}{\sqrt{6}}+\sin^{-1}\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}+\cdots$$ My approach : Here the $n$'th term is given ...
2
votes
2answers
535 views

What are the steps involved in reducing cos(arctan(2/5))?

I was completely stumped on this problem during a test. I have googled the answer and came across this link: http://answers.yahoo.com/question/index?qid=20110310212347AAJ21sr However, I still do not ...
2
votes
1answer
263 views

Tangent half-angles and linear fractional transformations

Suppose $z=x+iy$, $x$ and $y$ are real, and $|z|=x^2+y^2=1$ so that $z=e^{i\alpha}$ for some real $\alpha$. Then for some real $\gamma$, $$ \begin{align} e^{i\gamma} = f(e^{i\alpha}) = f(z) & ...
4
votes
2answers
101 views

What this sine function equation means?

Apostol's book "Calculus" asks to prove that $$\sin\frac{\pi }{6}=\frac{1}{2}$$ using the fact that $$\sin 3x=3\sin x-4\sin^3 x$$ and $$\sin \frac{\pi}{2}=1$$ So, we take $x=\frac{\pi}{6}$ and ...
6
votes
2answers
203 views

$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$ without using trigonometry?

$$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$$ Is there any way to find the answer without using trigonometry, like this? Hint by Parth Thakkat: $$\int \dfrac {\sqrt{x+1}} {x^{7/2}} dx$$ $$ = \int ...
4
votes
0answers
102 views

Generalized sine integral $ \int_0^\infty \sin^m(x^n)/x^p\,\mathrm dx$

I have seen that both the integrals $$ \color{black}{ \int_0^\infty \operatorname{sinc}(x^n)\,\mathrm{d}x = \frac{1}{n-1} \cos\left( \frac{\pi}{2n}\right)\Gamma ...
0
votes
3answers
147 views

Can you get the exact real value of $ \left((-1)^{\frac{1}{180}}\right)^{89}-\left((-1)^{\frac{1}{180}}\right)^{91}$?

By using euler formula,one can obtain: $$ 2\sin\left(\frac{\pi}{180}\right)=\left((-1)^{\frac{1}{180}}\right)^{89}-\left((-1)^{\frac{1}{180}}\right)^{91}. $$ In order to get the exact real value of ...
2
votes
1answer
4k views

Bearing and Course Trigonometry Question

This is the question: A jet flew 140 mi on a course of 196 degrees and then 120 mi on a course of 106 degrees then the jet return to its starting point via the shortest route possible. Find the total ...
3
votes
4answers
538 views

Show that $\sin2\alpha\cos\alpha+\cos2\alpha\sin\alpha = \sin4\alpha\cos\alpha - \cos4\alpha\sin\alpha$

Show that $\sin2\alpha\cos\alpha+\cos2\alpha\sin\alpha = \sin4\alpha\cos\alpha - \cos4\alpha\sin\alpha$ I know that $\sin2\alpha = 2\sin\alpha\cos\alpha$ so ...
16
votes
2answers
372 views

$\tan (n) > n$ for infinitely many positive integers

I heard the following problem is open: $ \tan(n ) > n $ for infinitely many positive integers in radians. Does anyone know if it is still open or if any progress has been made on this ...
3
votes
1answer
121 views

Dot Product/ Cross Product Proof

Let $\hat{a}$, $\hat{b}$, and $\hat{c}$ $\in \mathbb{R}^3$, using the properties of vectors, prove $$ (\hat{a} \times \hat{b}) \cdot [(\hat{b} \times \hat{c}) \times (\hat{c} \times \hat{a})] = ...
2
votes
2answers
661 views

If $\sin\theta+\sin\phi=a$ and $\cos\theta+ \cos\phi=b$, then find $\tan \frac{\theta-\phi}2$.

I'm trying to solve this problem: If $\sin\theta+\sin\phi=a$ and $\cos\theta+ \cos\phi=b$, then find $\tan \dfrac{\theta-\phi}2$. So seeing $\dfrac{\theta-\phi}2$ in the argument of the tangent ...
1
vote
1answer
118 views

parenthesis ,or without parenthesis?

I have the following function: $$f(x)=\sin2x;-\pi\leq x<-\frac{\pi}{2}$$ if i consider 2 within parenthesis $\sin[2(-\pi)]$, it is equal to $0$. if i do not consider 2 within parenthesis ...
2
votes
1answer
248 views

Calculating % Grade from latitude and longitude

I am trying to calculate % Grade from GPS points. I am using the basic equation of $$\dfrac{\text{Rise}}{\text{Run}} \times 100$$ The problem is I am getting spikes in my GPS data (example $1$ ...
3
votes
1answer
55 views

Parametrization of unit sphere in $\mathbb{R}^3$

I would like to show (I'm not yet sure if it's true, though), that any vector $v\in \mathbb{R}^3$ with $\|v\| = 1$ can be written as $\left(\cos(\beta)\sin(\alpha),\; \sin(\alpha)\sin(\beta), \; ...
0
votes
1answer
77 views

Integrating trig functions with $R(\frac {z+1/z} {2}, \frac {z - 1/z} {2i} )$

Someone told me that there is a method for integrating rational functions $R(\cos{\theta}, \sin { \theta})$ by doing contour integration of the complex function $$\frac {R \left( \frac {z + \frac1z} ...
2
votes
1answer
548 views

Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder?

I have a solid that is a combined shape of a cylinder and a concentric cone (a round sharpened pencil would be a good example) Know values are: Total Volume = 46,000 Height to Base Ratio = 2/1 ...
4
votes
2answers
161 views

Integrating $e^{ \cos(x)\sin(x)}$ from $0$ to $2\pi$

So i'm trying to find out how, or if its even possible, to integrate $e^{\sin(x)\cos(x)}$ analytically from $0$ to $2\pi$. I know that i can integrate $e^{\cos(x)+\sin(x)}$ or $e^{\cos(x)^2}$ and ...
3
votes
6answers
289 views

If $\sin\alpha + \cos\alpha = 0.2$, find the numerical value of $\sin2\alpha$.

If $\sin\alpha + \cos\alpha = 0.2$, find the numerical value of $\sin2\alpha$. How do I find a value for $\sin\alpha$ or $\cos\alpha$ so I can use a double angle formula? I know how to solve a ...
5
votes
1answer
1k views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
1
vote
2answers
115 views

How to prove two trigonometric identities

I want to show that $${\sin}^2 \alpha + 4{\sin}^4\frac{\alpha}{2} = 4{\sin}^2 \frac{\alpha}{2}$$ and $${\sin}^2 \alpha + 4{\cos}^4\frac{\alpha}{2} = 4{\cos}^2 \frac{\alpha}{2}$$ They should be true, ...
5
votes
1answer
629 views

Finding the derivative of $\sin \sqrt {x^2+1}$ from the definition?

This means finding $\lim_{h \to 0} \large \large \frac{\sin \sqrt {(x+h)^2+1}-\sin \sqrt {x^2+1}}{h}$ . The only way I could think of to do this is to replace $h$ by some function $f(h)$ such that ...
4
votes
3answers
256 views

Differentiate $\sin \sqrt{x^2+1} $with respect to $x$?

Differentiate $$ \sin \sqrt{x^2+1} $$ with respect to $x$? Can someone please help me with question, im very lost.
5
votes
3answers
539 views

How to be good at angles and trigonometry

I am Computer Science Engineer and loved algebra side of Mathematics. But when it comes to trigonometry and angles and triangles, I do not understand anything since college time. And till now also ...
1
vote
0answers
126 views

Bound on the angle between a vector and a subspace

Suppose you have three complex vectors $x_1$, $x_2$, and $x_3$. Define $a = \angle(x_1,x_2)$, $b = \angle(x_1,x_3)$. My question is about $c = \angle(x_1, span(x_2,x_3))$, the angle between the vector ...
3
votes
2answers
577 views

Properties of Triangle - Trigo Problem : In $\triangle $ABC prove that $a\cos(C+\theta) +c\cos(A-\theta) = b\cos\theta$

Problem : In $\triangle $ABC prove that $a\cos(C+\theta) +\cos(A-\theta) = b\cos\theta$ My approach : Using $\cos(A+B) =\cos A\cos B -\sin A\sin B and \cos(A-B) = \cos A\cos B +\sin A\sin B$, we ...
0
votes
5answers
514 views

Trigonometry question: Find in simplest surd form: $\cos 195^{\circ}$

Find in simplest surd form: $\cos 195^{\circ}$. Ive recently been doing the trigonometry topic form textbook and have oftenly come across these questions. Can someone please justify how you do this ...
6
votes
3answers
395 views

Calculate trig limit of type $\frac{0}{0}$ without L'Hopital

I am trying to figure out the solution to this Limit without using L'Hopital. $$ \lim \limits_{x \to \pi} \frac {(\tan (4x))^2 } {(x - \pi )^2} $$ Any help would be greatly appreciated.
3
votes
2answers
227 views

How prove this $\frac{1}{\cos{A}}-\frac{\sin{\frac{A}{2}}}{\sin{\frac{B}{2}}\sin{\frac{C}{2}}}=4$

in $\Delta ABC $ not An equilateral triangle, let $$\begin{vmatrix} 2\sin{A}\sin{C}-\cos{B}&2\sin{A}\sin{B}-\cos{C}\\ ...
2
votes
2answers
169 views

Proving (or disproving) that the sine and cosine of integers are always unique

Can it be proven that $ \forall x, y \in \mathbb{Z} \left( \sin(x) = \sin(y) \iff x = y\right)$ ? Or disproven, of course. And likewise with cosine? Since sine and cosine have periods of $2\pi$, ...
0
votes
3answers
85 views

Showing that $A = \{\tan x \mid x \in (-\frac{\pi}{2}, \frac{\pi}{2})\}$ is not bounded, without calculus.

I have to show that $A = \{\tan x \mid x \in (-\frac{\pi}{2}, \frac{\pi}{2})\}$ is not bounded. Since $\tan\colon (-\frac{\pi}{2}, \frac{\pi}{2}) \to \mathbb R$ is surjective $\Rightarrow ...
1
vote
6answers
699 views

Show that $\frac {\sin(3x)}{ \sin x} + \frac {\cos(3x)}{ \cos x} = 4\cos(2x)$

Show that $$\frac{\sin(3x)}{\sin x} + \frac{\cos(3x)}{\cos x} = 4\cos(2x).$$