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

learn more… | top users | synonyms (2)

0
votes
1answer
176 views

If $\cos\pi x=x^2-x-5/4$ then the value of $x$ is

If $\cos\pi x=x^2-x-5/4 \,$ then the value of $x$ is? I have completed: $\cos \pi x=(4x^2-4x-5)/4$ $\cos \pi x=3/8 \,$ or $1/8$
0
votes
2answers
30 views

$x/|x|$ question about division

What is $\frac{x}{|x|}$ can it be simplified? Because look at this. $\frac{r\cosh(x)}{\sqrt{\cosh^2(x)}} = \frac{r\cosh(x)}{|\cosh(x)|}$ How do you do this?
1
vote
1answer
72 views

Amplitude of a Product of Trigonometric Functions

We know that $|a|$ is called the amplitude of $a \sin(bx\pm \delta)$ and $a \cos(bx\pm \delta)$. But what is the amplitude of a product of trigonometric functions like: $a \cos(bx\pm \delta_1) \cos(...
1
vote
2answers
24 views

Confused About Trigonometric Substitution

I'm learning Trigonometric Substitutions, they gave us the following example in the book: I'm confused about how exactly we make the substitution $x= a\sin(\theta)$ In regular substitution we have ...
0
votes
2answers
46 views

Finding the norm of a complex trigonometric function?

Given that the complex norm $|z| = 1$, how would I go about proving that $|cos(z)| \leq e$? Just a hint would be helpful.
0
votes
2answers
41 views

Simplify a trigonometric equation in quadratic form

I have a computer problem that I was able to reduce to an equation in quadratic form, and thus I can solve the problem, but it's a little messy. I was just wondering if anybody sees any tricks to ...
0
votes
2answers
28 views

How can I determine if this draws a triangle?

Given a compass and these instructions 1. Go 50 meters 7 degrees 2. Go 50 meters 127 degrees 3. Go 50 meters 247 degrees Questions How can I use ...
0
votes
1answer
81 views

Inequality method to solve trigonometric equation.

Ok, while seeing examples in my book concerning with the general solution of trigonometric equations, I saw they used this inequality method . What is this??? The book didn't give any precise ...
2
votes
1answer
259 views

Proving $\frac{\sin x + \sin 2x + \sin3x}{\cos x + \cos 2x + \cos 3x} = \tan2x$

I need to prove: $$ \frac{\sin x + \sin 2x + \sin3x}{\cos x + \cos 2x + \cos 3x} = \tan2x $$ The sum and product formulae are relevant: $$ \sin(A + B) + \sin (A-B) = 2 \sin A \cos B \\ \sin(A + B) -...
0
votes
2answers
44 views

Find the value of $1-2\sin^2\theta+\sin^4\theta$

Find the value of $1-2\sin^2\theta+\sin^4\theta$ I have done so far: $1-2\sin^2\theta+\sin^2\theta\cdot \sin^2\theta$ $1-2\sin^2\theta+\sin^2\theta\cdot sin^2\theta$ $1-\sin^2\theta\cdot (2+\sin^2\...
3
votes
2answers
77 views

Why is the derivative of the arccos the negative derivative of arcsin?

$$ \dfrac{d}{dx} \sin^{-1}x = \dfrac{1}{\sqrt{1-x^2}}$$ $$\dfrac{d}{dx} \cos^{-1}x = - \dfrac{d}{dx} \sin^{-1}x$$ What is the reason for this?
4
votes
3answers
203 views

Proving Identities.

I tried to solve it but I cant get the answer. How to prove this by using a hand? $$ \sec^2x + \csc^2x = \sec^2x \csc^2x $$ $$ \frac{\sec\theta + 1}{\sec\theta - 1} = \frac{1 + \cos\theta}{1 - \cos\...
0
votes
1answer
36 views

Find the nearest regular polygon, given a side length and an approximate radius

I want to create a regular polygon with a given side length, s, and an maximum radius, r1. The radius value needs to be decreased (or increased if it simplifies things) to the closest length, r2, ...
0
votes
2answers
30 views

Trigonometry finding constant with angle

$\cos(x)=P$ (i) Find $\sin(x)$ I try $\sin(x)=\frac{\sqrt{(1-p^2)}}{1}$ (ii) Find $\sin(90+x)$ (iii) Find $\sin(180-x)$ (iv) Find $\sin(360-x)$ How to solve for part(ii)(iii)(iv)
0
votes
2answers
53 views

What is $2\sec^2(x)$ evaluated at $5\pi/6$?

What is $2\sec^2(x)$ evaluated at $5\pi/6$? I don't know when to apply the squared part of the secant identity. Now that I know when to apply the square.... doing this part of the equation I get -2....
1
vote
1answer
51 views

Indefinite integral with trig components

The following integral has me stumped. Any help on how to go about solving it would be great. $\int\frac{\cos\theta}{\sin2\theta - 1}d\theta$
1
vote
2answers
29 views

How would you solve this limit?

How can you solve this limit without using the aid of a graphing calculator? lim x-> 7 (x^2−15x+56)/ sin(x-7) I can figure it out using a graphing calculator, or by inputting numbers really close ...
0
votes
1answer
44 views

Can every smooth sine function be given a smooth argument?

Here's a conjecture that I believe to be true, but I couldn't find a proof: Let $\alpha: \mathbb{R}\longrightarrow\mathbb{R}$ be a function such that $\sin\alpha$ is smooth. Then there is a smooth ...
2
votes
2answers
41 views

$-1.4\sin 3x - 0.2 \cos 3x$ in the form $R \sin (3x+\alpha)$ such that $R>0$ and $0<\alpha<2\pi$

Write $-1.4 \sin 3x - 0.2 \cos 3x$ in the form $R \sin (3x+\alpha)$ such that $R>0$ and $0<\alpha<2\pi$ I found $R= \sqrt{(-1.4)^2+(-0.2)^2}= \sqrt{2}$ And $\alpha= \arctan \frac{0.2}{1.4}...
1
vote
1answer
40 views

How do you call this kind of functions in english?

I have a couple of formulas that I would like to plot, but I can't find the much needed documentation for them because I don't know how to correctly name them in english . This formulas assume that ...
1
vote
1answer
157 views

How do we define the branch cuts for $\sin^{-1}z = \frac{1}{i} \log(\sqrt{1-z^2} + iz)$ as $(-\infty,-1)$ and $(1,\infty)$?

As $\sin^{-1}z$ is a function of complex $\log$, it is multivalued. The branch cuts to make $\log$ single-valued are defined conventionally as $-\pi < Arg(z) \leq \pi$. Why wouldn't this carry over ...
2
votes
4answers
281 views

How to deduce the following trig relation?

How can I deduce: $$\sqrt{|x|}\sin(\frac{1}{x}) \le \sqrt{|x|}$$?? I know of the relation. $$\sin(u) \le u$$ $$u = \frac{1}{x}$$ $$\sin(1/x) \le \frac{1}{x}$$ But nothing related to $\sqrt{x}$ ...
5
votes
2answers
109 views

Evaluation of $\int \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\ $

How do we find $$\int \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\ $$ NB: It is not mandatory that $ax^2+bx+c$ has only a single root
1
vote
1answer
45 views

In $\triangle ABC$, if $\sin^2{A}+\cos^2{C}=\cos^2{B}$,then $C=?$

In $\triangle ABC$, if $\sin^2{A}+\cos^2{C}=\cos^2{B}$,then $C=?$ We have $\sin^2{A}+\cos^2{C}=\cos^2{B} \implies 2\sin^2{A}+2\cos^2{C}=2\cos^2{B} \implies 1-\cos{2A}+\cos{2C}-1=\cos{2B}-1 \...
0
votes
1answer
55 views

How do I simplify this trig problem?

Simplify the expression. Use exact values and show all steps. $$8\left[\left(\cos\dfrac{3\pi}4\right)\left(\sin-\dfrac\pi4\right)\right]+\dfrac1{\sin\frac{5\pi}6}-\left(\tan\pi+\cot\dfrac{7\pi}{4}\...
7
votes
3answers
137 views

Calculate trigonometric integral $ \int_{-\infty}^{\infty}{\sin(x^2)}\,dx$

Recently, I came across the following integral: $$ \int_{-\infty}^{\infty}{\sin(x^2)}\,dx=\int_{-\infty}^{\infty}{\cos(x^2)}\,dx=\sqrt{\frac{\pi}{2}} $$ What are the different ways to calculate such ...
5
votes
1answer
108 views

Random Wolfram|Alpha identity related to $\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{1}{k^2}}$

I was watching a Numberphile video (on how $\tan^{-1}{1} + \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{2}$) and I thought about whether the series $$\sum_{k = 1}^{\infty}{\tan^{-1}}{\frac{...
1
vote
2answers
301 views

Misunderstanding about the definition of a limit (Spivak Calculus)

In Spivak's text, I quote: "In general, if $\epsilon > 0$ to ensure that $|x^2\sin(\frac{1}{x})| < \epsilon$ we need only require that $|x| < \epsilon$ and $x \ne 0$" This can easily be ...
0
votes
1answer
45 views

How can I determine the range of the graph $\arccos(1/x^2)$

EDIT: All that is required for me to understand how to graph the function, is how to determine its range As the title implies, I am unsure of how to graph $\arccos(1/x^2)$. So far, I have found ...
0
votes
3answers
44 views

If $2−\cos^2θ=3\sinθ\cosθ$ then $\sin \theta \neq\cos\theta$ then $\tan \theta$ is…

If $2−\cos^2θ=3\sinθ\cosθ$ then $\sin \theta\neq \cos\theta$ then $\tan \theta$ is … What is $\tanθ$? My work is: $2−\cos^2θ=3\sinθ\cosθ$ $2\sinθ\cosθ=\sin2θ$ or dividing both sides by $\cosθ$ or $\...
1
vote
3answers
80 views

why tan θ > sin θ for range 0 to 90

(i) Prove the identity $$\tan^2 \theta - \sin^2 \theta \equiv \tan^2 \theta\sin^2\theta$$ (ii) Use this result to explain why $\tan θ > \sin θ$ for $0^\circ < \theta < 90^\circ$ I only need ...
1
vote
1answer
93 views

Why $\sin(\pi)$ sometimes equal to $0$?

Simplify the statements. Which variables are free and which are bound? If the statement has no free variable, find out if the statement is true or false. Justify your answer. This was the ...
5
votes
2answers
114 views

Evaluation of $\int \frac{x\sin(\sin x)}{x+5} \ dx$

How do we find $$\int \frac{x\sin(\sin x)}{x+5} \ dx\ ,$$ is there any way to take that $\sin x$ out from parent $\sin(\cdot)$ ?
1
vote
2answers
82 views

Integral of $\int \frac{\cos \left(x\right)}{\sin ^2\left(x\right)+\sin \left(x\right)}dx$

What is the integral of $\int \frac{\cos \left(x\right)}{\sin ^2\left(x\right)+\sin \left(x\right)}dx$ ? I understand one can substitute $u=\tan \left(\frac{x}{2}\right)$ and one can get (1) $\int \...
0
votes
2answers
44 views

What is important to know in regards to trig functions?

I believe I forgot everything I learned in pre calculus 3 years ago, and I need to fine tune my studies. I just took a look at the book I will be using this spring and it has a few questions stating $\...
0
votes
1answer
48 views

Analytic Trigonometry

Find the exact values. $A)$ $\tan 60^\circ + \tan 225^\circ$ $B)$ $\tan 285^\circ$ (use $285^\circ = 60^\circ + 225^\circ$) I'm just confused on how to do these kind of problems when they are in ...
0
votes
2answers
39 views

Which identity has been used here?

I have this written down in my notes, but I cannot remember how it came about: $$\sin(3t)\cos(10t) = 0.5(\sin (13t) + \sin (-7t))$$
0
votes
1answer
35 views

Area of region in polar coordinates

I have to verify a point: I'm supposed to find the area of the region given in polar coordinates $$\sec{\theta}\le r\le 2\cos{\theta}$$ I plotted the curves of $\sec{\theta}$ and $2\cos{\theta}$ ...
-1
votes
3answers
140 views

How do you get $\alpha$ from $\tan{\alpha}$?

How do you get $\alpha$ from $\tan{\alpha}$? Hello, I want to know how to obtain $\alpha$ from $\tan{\alpha}$. I mean, what is a formula (if there is one)? I know that it is schemes where it ...
0
votes
1answer
46 views

Unclear step in a textbook trigonometric identity proof

This is a step in the proof of a trigonometric identity: $$\frac {1+cos\left(\frac {\pi}{2}-a\right)}{1-cos\left(\frac {\pi}{2}-a\right)}=\frac {2\cos^2\left(\frac{\pi}{4}-\frac a2\right)}{2\sin^2\...
0
votes
4answers
173 views

Find the exact value of $\sin\left(\arcsin(0.5)+\arctan(-4)\right)$

Find the exact value of $\sin\left(\arcsin(0.5)+\arctan(-4)\right)$ My calculator gives a decimal for $\arctan(-4)$ so I don't know what answer is expected.
0
votes
3answers
68 views

Convert $1 + e^{2i}$ to $e^i \cos(1)$

Reading a long solution I saw this a step that converts $1 + e^{2i}$ to $e^i \cos(1)$. How is this done? How do I generalize this?
1
vote
1answer
596 views

Bearings and distances

Two ships leave port at the same time. One travels at $5$ km/h on a bearing of $46$ degrees. The other travels at $9$ km/h on a bearing of $127$ degrees. How far apart are the ships after $2$ hours?
0
votes
2answers
22 views

Determining the value of the trigonometry expression

If $\sin(x) + \cos(x) = 1/2$, what is the value of $\sin^3(x) + \cos^3(x)$ ? I started by cubing my first equation but I was found some difficulty in finding value for $\sin(x)\cos(x)$
0
votes
1answer
19 views

Trigonometric equations with cosec

If $\frac{3\pi}{2}<t<2\pi$ and $\\cost=\frac{3}{\sqrt{10}}\\$ , find the value of $\\cosec t+cos2t$
1
vote
1answer
211 views

Calculate $\sin(x)$, $\cos(x)$, and $\tan(x)$ without calculator

I know: $$\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\tan(x) = \frac{\text{opposite}}{\text{adjacent}}$$ but how do you ...
0
votes
1answer
54 views

is there an existing formula in finding the area of a rhombus wherein only the side is given?

is there an existing formula in finding the area of a rhombus wherein only the side is given? No measure of angles, no lengths of diagonals , height, etc. is given.
3
votes
4answers
158 views

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$

Calculation of $\displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$ $\bf{My\; Try}::$ Let $\displaystyle I = \displaystyle \int\frac{1}{\tan \frac{x}{2}+1}dx$, Now let $\displaystyle \tan \frac{x}{2}=t\...
0
votes
3answers
27 views

The relation between hyperbolic sine and hyperbolic cotangent

I was wondering if someone can verify (or not) the correctness of the following function? $$\frac{1}{\sinh^2X}=\coth^2X-1$$ I saw it in a paper but I am weak in math, so I am unsure if it is correct ...
0
votes
3answers
53 views

Integrating $\cos(x)^3dx$

My attempt at integrating $\cos(x)^3dx$: $$\begin{align}\;\int \cos^3x\mathrm{d}x &= \int \cos^2x \cos x \mathrm{d}x \\&= \int(1 - \sin^2 x) \cos x \mathrm{d}x \\&= \int \cos x dx - \int ...