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
2answers
36 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
164 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 - ...
0
votes
1answer
11 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
14 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
47 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 ...
1
vote
1answer
38 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
21 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
35 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
27 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 ...
1
vote
1answer
28 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 ...
0
votes
1answer
50 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
265 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 $\sqrt{x}$ ...
4
votes
2answers
75 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
34 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
0answers
27 views

Did I do these trig problems correctly?

1) Find the area of a sector formed by central angle theta = 2.4 in a circle of radius r = 3cm. I got 10.8cm^2 2) a) sin 217degrees = -0.6018 b) tan -114 degrees = -2.2460 c) csc 7pi/4 = ...
0
votes
1answer
46 views

How do I simplify this trig problem?

Simplify the expression. Use exact values and show all steps. ...
7
votes
3answers
92 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 ...
0
votes
0answers
25 views

How does one define appropriate branch cuts for arcsin(z) in the complex plane?

In learning about complex numbers, I have come across the following: http://upload.wikimedia.org/wikipedia/commons/b/be/Complex_arcsin.jpg Clearly, arcsin is multi-valued because it is a function of ...
3
votes
0answers
51 views

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

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
vote
2answers
41 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
28 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
28 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 ...
0
votes
3answers
47 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
66 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
96 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
votes
1answer
30 views

simplify $ 1- (sin^2 a/1+cosa)+(1+cosa/sin a)-(sina/1-cosa)$ [closed]

simplify $ 1- (sin^2 a/1+cosa)+(1+cosa/sin a)-(sina/1-cosa)$
1
vote
2answers
48 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
34 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
39 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
36 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
19 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 $2\cos{\theta}$ ...
-2
votes
3answers
90 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
34 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 ...
0
votes
4answers
54 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
55 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
28 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
19 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
13 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
48 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
0answers
27 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
96 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 ...
0
votes
3answers
21 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
38 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 ...
0
votes
1answer
32 views

Finding Angles counterclockwise

I have a robot arm like this: and I have to write a program that will move the arm to the point (2,0). I am having trouble expressing angles $\phi_2$ and $\phi_3$ in terms of $\phi_1$. ($\phi_1$ is ...
0
votes
0answers
42 views

$Sin(a) Sin(b) == A (Sin(a) + K Sin(b))$

When $b$ is close to $\pi/2$, can we find an approximation like $$M \sin(a) \sin(b) = A (\sin(a) + K \sin(b))$$ or $$M \sin(a) \sin(F*b) = A (\sin(a) + K \sin(b))$$ or $$M \sin(F*a) \sin(b) = A ...
1
vote
2answers
51 views

Solve $\cos(5y) + \cos(3y) +\cos(y) = 0.5$ for real $y$.

Well $\cos(3y)=\cos(y+2y)=\cos(y)\cos(2y)-\sin(y)\sin(2y)$. That's all I got. I've tried putting it in the equation but it doesn't seem to work out. How to solve this?
2
votes
3answers
161 views

Unclear step in the proof of half-angle formula for tangent

I wonder how could $$2\cos^2\left(\frac a2\right)$$ be transformed into $$1+\cos(a)$$ This is from a step in my textbook's proof of the tangent half-angle formula: $$tan\left(\frac a2\right) = .. ...
1
vote
2answers
35 views

Unclear step in half-angle formula derivation (trigonometric identities)

In deriving the half-angle formulas, my textbook first says: "Let's take the following identities:" $$\cos^2\left(\frac a2\right)+\sin^2\left(\frac a2\right)=1;$$ $$\cos^2\left(\frac ...
0
votes
1answer
53 views

Find all real x such that $\cos x$, $\cos2x$, $\cos 4x$, $\cos 8x$, etc. ($\cos 2^n$ for all non negative $n$) are all negative

I think I got that $|\cos (2^nx)|$ must be less than $|\sin (2^nx)|$ for all non negative $n$.
1
vote
0answers
23 views

Practical determinations of trigonometric identities

I am looking for articles, or any reference, that detail practical determinations of trigonometric identities, with particular emphasis on trigonometric functions raised to the power of 3 or higher. ...