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
20 views

Find an algebraic expression for sin(arccos(x))

I have a question that asks me to find an algebraic expression for sin(arccos(x)). From the lone example in the book I seen they're doing some multistep thing with the identities, but I'm just not ...
0
votes
5answers
35 views

How to prove $\displaystyle\lim_{x \to 0} \dfrac{\sin^{-1} x}{x} = 1$?

How to prove this? Is there any geometrical proof? I have proved , btw, $\displaystyle\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ by Sandwich Theorem and little geometry.
-1
votes
1answer
34 views

trigonometry - identities and formula, proving

Prove: $$\sin(x) - \sin(x) \cos^2(x) = \sin^3(x).$$ I swear it is easy, but I don't know what I'm forgetting to look at?
0
votes
1answer
32 views

Solving triangle

If side $a$ is known and the angles are given as functions of two variables (let's call them $x$ and $y$), what is the easiest way to find $y$ as a function of $x$. To make things easier, let one of ...
4
votes
1answer
62 views

How prove that $\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$ [duplicate]

How prove that sum $$\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$$
0
votes
2answers
58 views

show that $(1 +\sin{A}) + \cos^2{A} = 2(1 + \sin{A})$? [on hold]

It is an identity. The question is asking me to show that the left hand side is equal to the right hand side. $$(1 +\sin{A}) + \cos^2{A} = 2(1 + \sin{A})$$
1
vote
2answers
34 views

Show that: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 +1 } )$

could someone Please give me some hint of how to do this question thanks
1
vote
1answer
60 views

Express $C_n = \cosh(0) + \cosh(1) + \cosh (2) + \dots + \cosh(n)$

Could someone give me some hint of how to do this question please. I've been stuck for more than $3$ hours on this question.
1
vote
1answer
24 views

Trig and Geometry problem

I have this problem to solve. There is a triangle ABC containing a line segment bisecting Angle C with length s. The side opposite angle A is length a, across angle B is length b and the measure of ...
-2
votes
3answers
58 views

Complex Number to a power

I asked this question yesterday, but the answers did not actually answer what I wanted to know since I asked the question in the wrong way. I have $e^{i\frac{2014\pi}{12}}$. I know Euler's formula, ...
0
votes
1answer
25 views

Find an equation for a sinusoid with minimum and maximum

Here's my problem: Find an equation for a sinusoid that has a minimum at (30°,-1) and an adjacent maximum at (75°,7). Please help! I've tried everything I can think of, but I'm really drawing a ...
0
votes
2answers
48 views

finding angle value inside this triangle

I need a method to calculate the angle X in the image below, I know its value (30 degree) but how ?!! thank you.
2
votes
0answers
18 views

Can trigonometric functions for double precision be implemented in terms of those for single precision?

In some program environments like GLSL there is support for single and double precision numbers for arithmetic and square roots computation, but only single precision trigonometric functions are ...
4
votes
0answers
43 views

How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin z}=\root{12}\of{\sin 2y} $?

Let $\sin(x+y) = 2\sin\left(\dfrac{x-y}{2}\right)$ and $\sin(y+z) = 2\sin\left(\dfrac{y-z}{2}\right)$. How prove $\root 4\of{\frac{1}{2}\sin x\cos z}+\root 4\of{\frac{1}{2}\cos x\sin ...
-1
votes
0answers
16 views

Coordinate Geometry Help (circles + trigonometry)

Question : Find all points $(x, y)$ {if there are too many then number of points is enough} which lie on or inside the circle $x^2 + y ^2 = 9$ and satisfying the equation $\tan^4 (x) + \cot^4 (x) + 2 ...
0
votes
0answers
24 views

Dirichlet integral using real Analysis

The teacher made this approach to solve the Dirichlet integral , $$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin ...
2
votes
4answers
46 views

Simplifying the expression $2\cos^{2}6x-1$

I am trying to simplify the expression $2\cos^{2}6x-1$. The book got the answer of $\cos 12 x$ by doing $2\cos^{2}6x-1 = \cos2\left(6x\right) = \cos12x$ It said the double angle is $12x$. I don't ...
0
votes
1answer
30 views

Textbook error or my own error, Law of sines question.

A triangle with side 8 inches and corresponding angle 13 degrees, side x inches and corresponding angle 120 degrees. To answer this I set up a proportion: $sin(13)/8=sin(120)/x$ ...
-1
votes
3answers
42 views

Write the product of two trig equations equal to one?

Solve: Write the product of two trig equations that is equal to one. This one confuses me because I can think of trig equations that equal one, but I can't think of trig equations that I could ...
1
vote
1answer
58 views

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? [duplicate]

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? So if it converges then $\lim_{b \to\infty}\int_{0}^{b}\sin(x^2)\;\mathrm dx$ exists and our integral converges to this ...
3
votes
1answer
55 views

How can you find the integral of $\frac{cos(2t)}{2t^2}$ between 1 and infinity?

How can you find the integral of $\frac{\cos(2t)}{2t^2}$ between 1 and infinity? $$ I = \int\limits_1^\infty \frac{\cos(2t)}{2t^2} dt $$ My problem is that I just simply do not know how to handle ...
0
votes
1answer
63 views

Don't understand proof of why $\cos x$ is a contraction mapping on $[0, 1]$

I've read a couple proofs of why $\cos x$ is a contraction mapping on $[0,1]$ but none of them are clear enough for me to understand. What if we have something like $\lvert \cos x - \cos y \rvert = w ...
2
votes
6answers
69 views

Solve for $x:1 + \tan^2(x) = 8\sin^2(x)$

I have a tricky problem , I tried several methods and I can't seem to get a definite answer. $1 + \tan^2(x) = 8\sin^2(x), x \in [\frac{\pi}{6} , \frac{\pi}{2}]$ I got to ...
1
vote
2answers
31 views

Solutions for the following equation.

Let us have an $n$ positive integer. How many solutions do we have for the following equation in the interval $(0,\frac{\pi}{2})$? $$\underbrace{\cos(\cos(\ldots(\cos x)\ldots))}_{n\text{ times ...
0
votes
3answers
38 views

Trig and derivatives: If condition holds for derivative, does it hold for the original equation?

Let's say I have some trigonometric identity such as $\sin(x) + 1 = -\cos(y)$. As we can see, the derivative of this identity gives $\cos(x) = \sin(y)$, which implies that $x + y = \pi/2$. Does that ...
5
votes
2answers
1k views

What is cosine to the power of zero?

I was doing a question relates to substitution rule under integration. The question is as follow: Evaluate $\int{1\over{(1+x^2)^n}}dx, n\in \mathbb{Z}^+$ We have seen that ...
0
votes
5answers
47 views

Expressing $\cos^5(x)$ using trigonometric addition formulas

If $\cos(3x) = 4\cos^3(x)-3\cos(x)$, and $\cos^3(x) = \frac{1}{4}(\cos(3x) + 3\cos(x))$, how can we express $\cos^5(x)$ in the same way?
0
votes
1answer
26 views

Integration: Find length of curve using NINT

Here are the questions - For question 4, part (b) gives a unit circle. But I'm unable to proceed with parts (a) and (c), since the curve is double valued for -0.5 Also, for question 6, integration ...
1
vote
2answers
56 views

Method to integrate $\cos^4(x)$

Here my attempts for integrating $\cos^4(x)$ in few methods. 1st method. $(\cos^2x)^2=(\frac{1}{2})^2(1+\cos2x)^2$ $=\frac{1}{4}(1+2\cos2x+\cos^22x)=\frac{1}{4}(1+2\cos2x)+\frac{1}{4}(\cos^22x)$ ...
2
votes
1answer
32 views

If $ \cos(\theta) = - \frac{2}{3} $ and $ 450^{\circ} < \theta < 540^{\circ} $, find…

If $ \cos(\theta) = - \frac{2}{3} $ and $ 450^{\circ} < \theta < 540^{\circ} $, find: The exact value of $ \cos \! \left( \frac{1}{2} \theta \right) $. The exact value of $ \tan(2 \theta) $. ...
2
votes
0answers
28 views

Trigonometrical functions and complex numbers

(This question will at first appear too broad. However, the overall philosophy will be explained below in a way that asks specific questions, which I hope will be conducive to this being a reasonable ...
2
votes
3answers
40 views

Solve: $\tan(2\theta-36^\circ) = \sqrt{8}$

$\tan(2\theta-36^\circ) = \sqrt{8}$ in degrees. I tried making $\sqrt{3}$ into $60$ degrees, and then the answer was $47$ degrees but I don't think that is right.
-3
votes
4answers
49 views

Solve: $\sin(4x)\sin x - \cos(2x)\cos x = 6$ [on hold]

Thank you for attempting to answer my question! Solve: $$\sin(2x)\sin x - \cos(4x)\cos x = 6, \textrm{ where } x\in(-\pi,\pi)$$
-1
votes
3answers
34 views

The graph of $y=6\cos\theta+10\sin\theta$ would be a sinusoid if it were plotted… [on hold]

The graph of $y=6\cos\theta+10\sin\theta$ would be a sinusoid if it were plotted. What would be the first positive value of $\theta$ at which there is a high point? Calc says it is 1.29 but I'm not ...
0
votes
1answer
35 views

Evaluation of an integral.

$$I_1=\int \frac{a^2\sin^2(x)+b^2\cos^2(x)}{a^4\sin^2(x)+b^4\cos^2(x)}dx$$ I tried writing it as $$\int \frac{a^2+\cos^2(x)(b^2-a^2)}{a^4+\cos^2(x)(b^4-a^4)}dx$$ But I don't know how to proceed. ...
1
vote
0answers
42 views

Is this simplification 'allowed'?

I've just been doing a problem that involved this equation: $$ \frac{1}{\sin\left(\frac{\theta}{2}\right)}\left( \sin\left(b\theta-\frac{\theta}{2}\right)-\sin\left(a\theta-\frac{\theta}{2}\right) ...
0
votes
0answers
31 views

Evaluating $ \int \cot^{2} \! \left( \frac{\pi}{\lfloor \log(x) / \log(3) \rfloor} \right) ~ \mathrm{d}{x} $.

How would I evaluate $$ \int \cot^{2} \! \left( \frac{\pi}{\lfloor \log(x) / \log(3) \rfloor} \right) ~ \mathrm{d}{x}? $$
0
votes
2answers
20 views

Express the polar equation in parametric form

$r= \sin(\theta)+2\cos(\theta)$ We did not go over very much of this topic. I do not understand the example in my book.
-3
votes
1answer
48 views

Please help me simplify! [on hold]

tanx-cotx/sinxcosx Please help me simplify
2
votes
4answers
64 views

Integral of $\tan(x)$ from $0$ to $2\pi$

I had a disputation with my friend. He said that $$ \int_0^{2\pi} \tan(x) \ dx $$ is undefined. While I admit that the Integral from $0$ to $\frac{\pi}{2}$ goes to infinity, I don't know if the ...
2
votes
2answers
76 views

Prove $\sin(45^\circ) + \sin(15^\circ) = \sin(75^\circ)$

I rewrote the statement as $$ \sin(30^\circ + 15^\circ) + \sin(15^\circ) = \cos(15^\circ) $$ Then I got $$ (\sqrt{3}-2) \sin(15^\circ) = \cos(15^\circ) $$
2
votes
1answer
17 views

Use de Moivre's theorem to obtain an expression for $\sin^6x$ as a sum of terms in the form $\cos ax$

I'm not exactly sure if I'm on the right lines but I've started with a binomial expansion: $(\cos x+i\sin x)^6=\cos 6x +i \sin 6x= \cos^6 x + i(6\cos^5x \sin x)-15\cos^4x \sin^2x-i(20\cos^3x \sin^3 ...
2
votes
5answers
54 views

Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable. Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$ Consider $g(x):= ...
0
votes
0answers
15 views

Cartesian extremities of a 3d segment

I have a segment in 3d space and I want to calculate its extremities. I know the cartesian coordinates (x,y,z) of the segment's middle point, the segment's length L and the segment's orientation using ...
3
votes
2answers
55 views

Rational parametrization of circle in Wikipedia

In http://en.wikipedia.org/wiki/Circle but also in the corresponding article in the German Wikipedia I find this formulation ( sorry, I exchange x and y as I am accustomed to it in this way ) : "An ...
4
votes
3answers
264 views

Rewriting $\sin(2\cos^{-1}{(x/3)})$ as a fraction?

I'm looking to simplify $$\sin(2\cos^{-1}{(x/3)})$$ I know it simplifies to $\frac{2x}{3}\sqrt{1-\frac{x^2}{9}}$, but I am unsure of the required steps. Thanks for any help
2
votes
4answers
45 views

Find the type of triangle from equation.

In triangle $ABC$, the angle($BAC$) is a root of the equation $$\sqrt{3}\cos x + \sin x = \frac{1}{2}.$$ Then the triangle $ABC$ is a) obtuse angled b) right angled c) acute angled but not ...
0
votes
2answers
11 views

What is the relationship between the trigonometric secant and the geometric secant of a circle?

What is the difference between the geometric secant(the line that cuts two points of a curve) of a curve, and the trigonometric secant(=1/cosinex) ? If they are the same, can you explain how they are ...
4
votes
3answers
45 views

Inverse of an ordered pair?

Let $f: A \to B$ be a bijective function where $A = [0, 2\pi)$ and $B$ is the unit circle. Find the inverse of $f(\theta) = (\cos\theta, \sin\theta)$. I don't understand what it means to take the ...
1
vote
1answer
53 views

Integral using trigonometric substitution

I'd like to ask for feedback on my calculation for this integral: $$\int{\frac{dx}{2-\cos{x}}}$$ Using half-angle substitution: $$t = \tan{\frac{x}{2}}$$ $$\cos{x} = \frac{1-t^2}{1+t^2}$$ $$dx = ...