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

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-2
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1answer
42 views

Evaluating a trigonometric integral

Show that $$ \int \frac{y^2\, dx - x^2\, dy}{x^2 + y^2} = -\frac{4a}{3} $$ where Life is a semi circle at $x = a\cos t$ and $y = a\sin t$ from $t = 0$ to $t = \pi$. I tried it but this is where I ...
1
vote
1answer
43 views

minimum value of a trigonometric equation

Q: What is the minimum value of $$\cos(2x)+3\sin(2x)$$ ? (a question from my book) I know minimum value of $\cos$ and $\sin$ is $-1$. So I thought that minimum value of this equation is $-1 + 3*-1 = ...
0
votes
3answers
36 views

Trigonometric elimination involving 4 variables [closed]

Eliminate $\theta$ from the two equations $$\cos \alpha=\cos \beta.\cos \phi=\cos \gamma.\cos \theta$$$$\sin \alpha=2\sin \phi/2.\sin \theta/2$$ The answer is in the form of $\tan$ of half angles...I ...
2
votes
2answers
56 views

Trigonometric Identities help please

$$ \frac {\sec \theta}{ \csc \theta - \cot \theta } - \frac { \sec \theta }{ \csc \theta + \cot \theta } = 2\csc \theta $$ I really have no idea how to verify. I try then but can't make sense ...
4
votes
3answers
57 views

Trying to prove a trigonometric identity

I've been trying to solve it for quite some time but I still don't get it why it is true. The original equation is: \begin{equation*} 1-\frac{\sin{^2}\theta}{1-\cos\theta}=-\cos\theta. ...
0
votes
2answers
55 views

Calculus problem on limits giving weird answers!

I got this problem as homework and I just don't seem to solve this problem: $$\lim\limits_{x \to \pi/2}\frac{\cot x}{2x - \pi} $$ The answer to the problem as said by the book is $\frac{1}{2}$. I ...
-3
votes
1answer
50 views

Please help with this question about angle of elevation and depression [closed]

A surveyor needs to determine the height of a building. She measures the angle of elevation of the top of the building from the two points, 38cm apart. The surveyor's eye level is 180cm above the ...
1
vote
1answer
23 views

$f(x)= sin(x)^{3}+cos(x)^{3}$ prove ${f}''(x)= \frac{3}{2}(cos(x)+sin(x))(3sin(2x)-2)$

$f(x)= \sin(x)^{3}+\cos(x)^{3}$ prove that ${f}''(x)= \frac{3}{2}(cos(x)+sin(x))\, (3sin(2x)-2)$ I tried to solve it but I can't complete it.
0
votes
2answers
22 views

Finding the points of intersection on a circle

Before addressing my issues, below is the question from a past examination paper along with a diagram I dre in order to facilitate readers. 3(a) A circle has center $C(5, 8)$ and radius ...
-4
votes
3answers
54 views

Establishing the Trigonometric Idenitity [closed]

How do you establish the trigonometric identity? $$\frac{(2\cos^2\theta-1)^2}{\cos^4\theta - \sin^4\theta} = 1-2\sin^2\theta$$
3
votes
1answer
22 views

How to compute $\cos(x)$ within $n$ digit accuracy when $x = \sqrt{y}$ with $y \in \mathbb{N}$

How does one compute $\cos(x)$ within desired $n$ digit accuracy when $x = \sqrt{y}$ with $y \in \mathbb{N}$ and $x$ is not rational? The reason I am asking this question is that calculators ...
0
votes
4answers
56 views

How to establish the identity

How do I establish the identity in this problem? Struggling with this one at the moment. $$\frac{\sin\theta\cos\theta}{\cos^2\theta-\sin^2\theta}=\frac{\tan\theta}{1-\tan^2\theta}$$
0
votes
1answer
17 views

% of traversal for a point between two other points along a line

I'd like to solve for some "phase" or percentage, involving an arbitrary location (xC, yC) between two points. I'm not familiar with how to phrase this question, so please excuse my ignorance. Not ...
16
votes
11answers
2k views

Proving a trigonometric identity: $\frac{\cos x}{1-\sin x} -\tan x = \sec x$

I am trying to prove a trig identity that is confusing me. The identity is $$\frac{\cos(x)}{(1-\sin(x))}-\tan(x)=\sec(x)$$ Here is my attempt. I did ...
4
votes
0answers
36 views

query about the cosine of an irrational multiple of an angle?

de Moivre's identity $$ (\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta $$ only applies as written when $n \in \mathbb{Z}$. if the exponent is a fraction $\frac{m}{n}$ then there will ...
0
votes
1answer
17 views

Find all $x$ and $y$ $(0 \le x, y \lt 2\pi)$ so that the following equation is true.

Find all $x$ and $y$ $(0 \le x, y \lt 2\pi)$ so that the following equation is true. (Enter your answers as a comma-separated list.) $$(\sin^2x + 1) + i\tan y = 2 \sin x + i$$ $x =$ $y =$ My work: ...
3
votes
1answer
40 views

Determine the amplitude and phase shift of $f(x) = \sqrt{3} \cos2x-\sin2x$

Question: Determine the amplitude and phase shift of $f(x) = \sqrt{3} \cos2x-\sin2x$ Attempted solution: The amplitude can be calculated by: $$A = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{4} = 2$$ ...
0
votes
2answers
107 views

How is $ \cos (\alpha / \beta) $ expressed in terms of $\cos \alpha $ and $ \cos \beta $?

If it is possible to express $ \cos n \alpha $ in terms of $ \cos \alpha $ as a power series for integer $n$ ... I like to see an expression for the quotient angle that obviously tallies when $ ...
6
votes
6answers
498 views

What is the purpose of the compound angle identity in trigonometry?

This may be a silly question, but one that I am confused about nonetheless. With regards to the compound trig identities such as $\cos(A+B)=\cos A\cos B - \sin A\sin B$ etc., I'd like to know why ...
0
votes
4answers
39 views

Finding a Trigonometric Form of Complex Number

I need to find the trigonometric form of the complex number: $1-i\sqrt{3}$ I found that $r = 2$ which means the trigonometric form is $2 ( \cos \alpha - i\sin \alpha)$ and I need to find the ...
1
vote
1answer
22 views

Parametric equation of clock hands

I am trying to draw a clock with both hour and minute hands in a computer program. The movement of the clock hands would mirror a traditional wall clock (hours from $12, 1, 2, 3,..., 11$ and back to ...
2
votes
1answer
125 views

Find $x$ and $y$

If $\frac{\tan 8°}{1-3\tan^{2}8°}+\frac{3\tan 24°}{1-3\tan^{2}24°}+\frac{9\tan 72°}{1-3\tan^{2}72°}+\frac{27\tan 216°}{1-3\tan^{2}216°}=x\tan 108°+y\tan 8°$, find x and y. I am unable to simplify the ...
1
vote
0answers
20 views

Finding the value of a trigonometric equation

Find the numerical value of $$\tan(3\pi/11)+4\sin(2\pi/11)$$ without actually calculating the values. How to start? Please help.
1
vote
4answers
50 views

Minimum value of trigonometric equation

Find the minimum value of the expression $$y=\frac{16-8\sin^{2} 2x +8\cos^{4} x}{\sin^{2} 2x}$$ When I convert the expression completely into 2x, cross multiply and make the discriminant of the ...
0
votes
2answers
64 views

Using l'Hopital's rule to find the limit .

I need a hint to evaluate the following limit: $$\lim_{x \to 0} \frac{x^3\sin\left(\frac{1}{x^2}\right)}{\cos x}$$
0
votes
0answers
41 views

Fourier Expressions

In the Fourier series, what are all the ways we can express: $\displaystyle\sin\left(\frac{n\cdot\pi}2\right)$ $\displaystyle\cos(n\cdot\pi)$ I know we can express as $(-1)^{(n+1)}$, and as ...
5
votes
2answers
45 views

A trigonometric product

I have to prove: $$\prod_{i=1}^6 \left(2\cos\left(\frac{2^{i}\pi}{13}\right)-1\right)=1$$ I really have no idea about starting with this one. With the help of Wolfram Alpha, I noticed that: ...
2
votes
2answers
43 views

Finding angles of hyperbolic triangles

I am trying to learn about how to find the angles of hyperbolic triangles. Now below is a problem: It has all the steps but I am not understanding the concept (the ones that are underlined in green ...
-1
votes
1answer
39 views

Equivalent Trigonometry, help me please [closed]

I have been try simplify this, could you help me? $$\dfrac{\csc x - \cos x}{\sec x - \sin x}$$ Thank you so much
0
votes
1answer
36 views

How to determine the height of a pole using a mirror and a subject's height? [closed]

You, $5$ feet tall, come upon $2$ friends arguing about the height of a telephone pole $20$ feet away. You want to help them determine the height but only have a $10$ ft tape measure, and a mirror. ...
1
vote
1answer
13 views

Expectation of trigonometric functions involving random variables.

This is more a formulation question. I need help making a sales pitch (lol). I am working on an practical engineering problem where I encounter functions of the form: $\cos(\phi + d_\phi)$, $ ...
2
votes
2answers
53 views

If $ \tan(20^{\circ}) = p $, find $ \frac{\tan(160^{\circ}) - \tan(110^{\circ})}{1 + \tan(160^{\circ}) \tan(110^{\circ})} $.

I applied the $ \tan(A - B) $-formula to make it $\tan(50^{\circ}) $, then I split it to $ \tan(30^{\circ} + 20^{\circ}) $. My answer came out to be $ \dfrac{\sqrt[3]{p + 1}}{\sqrt[3]{- p}} $, but ...
31
votes
3answers
673 views

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

I'm interested in integrals of the form $$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for ...
1
vote
0answers
22 views

Realistic Bounce (Using Trig?)

background: I am making a graphics program where the major purpose of it is to have a ball (traveling on an arbitrary slope) to bounce realistically off of a line (which is also at a arbitrary slope). ...
3
votes
1answer
32 views

arccos and arcsin integral contradiction:

I am shown: $$f(x) = \arcsin x \implies f'(x) = \frac{1}{\sqrt{1-x^2}}$$ $$f(x) = \arccos x \implies f'(x) = -\frac{1}{\sqrt{1-x^2}}$$ These two derivatives can be very readily derived by a bit of ...
3
votes
1answer
72 views

Finding $\prod_{k=1}^{n-1}\cos\frac{2k\pi}n$

Finding $$\mu=\prod_{k=1}^{n-1}\cos\frac{2k\pi}n$$ I thought $$z^n=1=e^{i2\pi}\implies z=\cos\frac{2k\pi}n+i\sin\frac{2k\pi}n\quad k\in\{1,2,...,n-1\}$$ Now we have: ...
0
votes
1answer
63 views

Find largest possible value of $x+y$

If $4\sin x. \cos y + 2\sin x+2\cos y+1=0$, find the largest possible value of the sum $(x+y)$. How do I manipulate my expression? I am not getting $(x+y)$ form. Thanks.
4
votes
4answers
92 views

Evaluate $\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$ using a double integral

I was given the following problem: Evaluate the following integrate using a double integral: $\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$. The professor told us off the bat the answer was ...
1
vote
0answers
29 views

Converting solutions to separation constant to Cosh and Sinh

The Laplace's equation inside a rectangle is: $$u_{\text{xx}}+u_{\text{yy}}\text{=0}$$ The IC's are: $${u(0,y)=g(y)}$$ $${u(L,y)=0}$$ $${u(x,0)=0}$$ $${u(x,H)=0}$$ Via method of separation we have ...
1
vote
1answer
36 views

Limit of functions involving trigonometry as n approaches infinity

By graphing these functions, I know that P(n) approaches pi as n tends towards infinity. However, is there a mathematical way for proving this? I am doing a maths exploration on Archimedes' ...
0
votes
1answer
32 views

Is there an algorithm to determine if an arc through 3 points is concave up or concave down?

Armed with only the three points in 2-dimensional space, $X = \{x_1, x_2, x_3\}$, is there a simple inequality or algorithm that can return whether or not an arc $A$ through these three points is ...
3
votes
3answers
32 views

Equivalence of Solutions to Wave Equation

The differential equation $$\ddot x = -\omega^2 x$$ apparently has solutions of $$x = Ae^{i\omega t} + Be^{-i\omega t} \tag{1}$$ AND $$x = A\sin(\omega t) + B\cos(\omega t) \tag{2}$$ AND $$x = ...
0
votes
1answer
22 views

Smooth function, lateral limit and trigonometry

Let $\theta: (t_0-\varepsilon, t_0)\to\mathbb{R},\ \theta\in C^{\infty}((t_0-\varepsilon,t_0))$. Knowing that the following two limits exist: $\lim\limits_{t\nearrow t_0} \cos^2 \theta(t)$ and ...
6
votes
6answers
77 views

System of equations involving sin and cos

I'm trying to solve the following system: $$ \sin(x) + \cos(y) = 0.6\\ \cos(x) - \sin(y) = 0.2\\ $$ Solving for y in terms of x: $$ y=\arccos(0.6-\sin(x))=\arcsin(\cos(x) -0.2) $$ Therefore: $$ ...
3
votes
5answers
154 views

Trigonometric Functions Limit

In my assignment I have to solve the following question. I know the answer, but I keep getting it wrong, and I don't know how to solve it. $$\lim_{x \to 0} \frac{1-\cos x}{x\sin x}$$ I have tried ...
1
vote
3answers
19 views

Find the values of $x$ satisfying $\sin^{-1}(|\sin x|)-\cos^{-1}(\cos x)\ge0$ in $[0, 2\pi]$

Find the values of $x$ satisfying $\sin^{-1}(|\sin x|)-\cos^{-1}(\cos x)\ge0$ in $[0, 2\pi]$. I think it would be better explained by drawing the graphs. Kindly help me in this question.
-1
votes
2answers
43 views

The number of pairs $(x,y)$ of real numbers satisfying $|\tan(\pi y)|+ \sin^2(\pi x)=0$ and $x^2 + y^2\le2$ [closed]

Here I have a question: Find the number of pairs $(x,y)$ of real numbers satisfying the following: $$|\tan(\pi y)|+ \sin^2(\pi x)=0\quad\textrm{and}\quad x^2 + y^2\le 2$$ The answer is ...
0
votes
1answer
19 views

Converting cos(x-30) to two terms

I have a worked example in front of me for a particle kinetics and kinematics question. In the working it has a an equation: $$0.6=\frac{V_0\sin(60)+V^`cos(x-30)}{10cos(30)}$$ And on the next line ...
0
votes
1answer
22 views

Find sides of triangle

I already know two sides $a_2$, $b_2$ and the angle $C$. I don't know angles $A$, $B$ and sides $a_1$, $b_1$, $c_1$, $c_2$. How can I find $a_1$, $b_1$, $c_1$? Or How can I find $c_2$? Here, ...
-1
votes
0answers
41 views

Write the word or phrase that best completes the statement or answers the question. [duplicate]

Write the word or phrase that best completes the statement or answers the question. $$\cos^2\theta-\sin^2\theta=1+\sin\theta$$ i have tried manipulating the equation by substituting $x^2$ and $y^2$ ...