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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0answers
6 views

Difficult problem involving a percentage of the period of a sinusoid

Im having difficulty intuitively understanding how to solve this problem: $x(t) = A*cos(\Omega*t + \phi)$ $A > 0$ $\phi$ range is $(−\pi,\pi]$. $x(t) ≥ 2.4$ for $18$% of each period takes ...
-1
votes
2answers
46 views

If $A+B+C=π$, prove that

If $A+B+C=π$, prove that $$\cos^2A+\cos^2B-\cos^2C=-2\cos A\cdot\cos B\cdot\cos C.$$ ATTEMPT: Given $$A+B+C=π,$$ $$A+B=π-C$$ Taking "cos" on both sides $$\cos(A+B)=-\cos C.$$ Now, ...
2
votes
1answer
27 views

How would one evaluate $\sin(72\pi/11)$?

How would one evaluate $\sin(\frac {72\pi} {11})$?. The prime number in the bottom is getting me stuck. I couldn't see how to use it using the sum of two angles trig identity.
0
votes
2answers
23 views

How to simplify inverse trigonometric function

How to simplify the following equation: $$\sin(2\arccos(x))$$ I am thinking about: $$\arccos(x) = t$$ Then we have: $$\sin(2t) = 2\sin(t)\cos(t)$$ But then how to proceed?
2
votes
0answers
34 views

Expanding trigonometric functions with binomial expansion

I was challenged to take $\cos^{\pi}(\pi)$ and expand it using binomial expansion and $\cos(x)=\frac{e^{xi}+e^{-xi}}2$, which I tried: $$\cos^{\pi}(\pi)=\left(\frac{e^{\pi i}+e^{-\pi ...
0
votes
3answers
38 views

If $\frac{m+1}{m-1}=\frac{cos(\alpha-\beta)}{sin(\alpha+\beta)}$„ then

If $$\frac{m+1}{m-1}=\frac{\cos(\alpha-\beta)}{\sin(\alpha+\beta)}$$, prove that : $$m=\tan(π/4 +\alpha).\tan(π/4 +\beta)$$. My attempts/ Here .. ...
1
vote
3answers
24 views

Prove that $m\tan (\theta-30°)=n\tan (\theta+120°)$

If $m\tan (\theta-30°)=n\tan (\theta+120°)$ then prove that : $$\cos 2\theta=\frac{m+n}{2(m-n)}$$ My attempt\ Here, $$m\tan (\theta-30°)=n\tan (\theta+120)$$ $$\frac{\tan (\theta-30°)}{\tan ...
2
votes
4answers
51 views

Proving $\tan A=\frac{1-\cos B}{\sin B} \;\implies\; \tan 2A=\tan B$

If $\tan A=\dfrac{1-\cos B}{\sin B}$, prove that $\tan 2A=\tan B$. My effort: Here $$\tan A=\frac{1-\cos B}{\sin B}$$ Now $$\begin{align}\text{L.H.S.} &=\tan 2A \\[4pt] &=\frac{2\tan ...
1
vote
0answers
40 views

Find $\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$

What is the value of the following sum? $$\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$$ where $\gcd$ is the greatest common divisor.
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0answers
18 views

Proving that If $A+B+C=π$ then, [duplicate]

If $A+B+C=π$ then prove that : $\sin(B+2C)+\sin(C+2A)+\sin(A+2B)=4\sin\frac{B-C}{2}.\sin\frac{C-A}{2}.\sin\frac{A-B}{2}$. My attempts: Here $A+B+C=π$ Now, $$\begin{align} ...
1
vote
1answer
27 views

Evaluating the integral of a sine function

I am having some trouble with part (b) and part (c) of this: (b) I know that I have to differentiate it and I get $\cos (\frac{\pi}{x})$ and by using the definite integral I get $\cos (\pi n)-\cos ...
1
vote
1answer
31 views

Showing that $\alpha$ satisfies the equation $\sin 2x=x$

This is an A level question. For better understanding, I will attach a screenshot of the question and the mark scheme. Question: Here's what I have done: $$A(OBA) = \frac 12r^2α$$ [basic ...
0
votes
2answers
30 views

What is this procedure called for angle radians?

So, my lecturer says that $-\cos(\frac{\pi}{8}) = \cos(\frac{9\pi}{8})$. What did he do to get that? Please recommend a source where I can brush up on my knowledge of angles.
0
votes
3answers
40 views

How do I compute the angles of a pyramid from the angle between its sides?

I have been given the following problem to solve: In a right pyramid whose base is an equilateral triangle, the angle between 2 side-faces is 70 degrees. Compute the base angle of a side-face. I ...
0
votes
2answers
52 views

How does $\frac{\sin\theta}{\cos\theta}$ become $\frac{y}{x}$

I ended up in the wrong math class (trigonometry) for my level but am trying to survive by catching up on some more basic principles. I'm wondering if the same principle (and if so, what is it) is ...
0
votes
0answers
32 views

let $\alpha \in \Bbb{R} $ and $\cos(\alpha \pi) = \frac{1}{3}$, prove $\alpha $ is irrational [duplicate]

Let $\alpha \in \Bbb{R} $ and $\cos(\alpha \pi) = \dfrac{1}{3}$, prove $\alpha$ is irrational. (Proof by contradiction) If we consider $\cos \left(\dfrac{m\pi}{n} \right)=\cos \left(\dfrac{ m\pi ...
0
votes
2answers
51 views

Evaluate the definite integral $\int_{0}^{a}\frac{dx}{(a^2+x^2)^{3/2}}$

I'm trying to solve this integral with trigonometric substitution but am having a ton of trouble: $$\int\limits_{0}^{a}{\frac{dx}{(a^2+x^2)^{\frac{3}{2}}}}$$ I tried $x=a\tan{\theta}$ and thus ...
3
votes
5answers
95 views

Why does $\DeclareMathOperator{arccot}{arccot}\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$ diverge?

Why does $$\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$$ diverge? In my textbook it says that from the positive side it's zero, and from the negative side it's $\pi$. However, when entering ...
0
votes
1answer
24 views

Trigonometry markup

Imagine we have the following problem; $$\cos(x) = \cos(a) \Rightarrow x=a+k\times 2\pi\\ or \\x=-a+k\times 2\pi$$ And we have the following answers.. : $$a=\frac{\pi}{3} \\or \\a=-\frac{\pi}{3}$$ ...
0
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1answer
21 views

How do you calculate the change in thickness of a cylinder, if you shave off a flat section?

I have a piece of steel, cylindrical (hollow), 200mm outside diameter with 160mm inside diameter (...
0
votes
2answers
35 views

Solving $\sec(3\alpha+30^\circ)=\csc(7\alpha-40^\circ)$ [on hold]

Can you solve for $α$ in degrees/radians and tell me exactly how to do so? $$\sec(3\alpha+30^\circ)=\csc(7\alpha-40^\circ)$$
0
votes
0answers
32 views

How can I change a summation of cosines to a product of cosines for higher degree functions?

I was wondering if I can get an alternate form of a sum over cosines $$\sum_{n=1}^m\cos{f(n)}$$ and I found that I can. We must however make a modification to the upper limit with $m\rightarrow2^m$. ...
2
votes
3answers
28 views

New coordinates after clockwise rotation of triangle?

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X-Y$ plane about the vertex $P$ by angle ...
1
vote
1answer
39 views

Trigonometric solution for $\int_{0}^{2 \pi} \sin^n (x) \cos^m (x) dx $?

At home I came across the exercise and had to compute: $\int_{0}^{2\pi} \sin^n (x) \cos^m (x) dx $ with $m$, $n \in \mathbb{N} $ My current set of tools for solving problems of that kind is rather ...
0
votes
1answer
30 views

Conditional Proof in Trigonometry

If $\sin\theta + \sin\alpha=m$ and $\cos\theta + \cos\alpha=n$, prove that: $$\frac{\sec(\theta+\alpha)}{2}=\frac{\sqrt{m^2+n^2}}{2}$$ My attempt\ given: $$\sin\theta+\sin\alpha=m$$ $$2 ...
0
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1answer
20 views

Trigonometry Proving

If $\sin\theta + \sin\alpha=x$ and $\cos\theta + \cos\alpha=y$, prove that ; $$\frac{\tan(\theta - \alpha)}{2} = \pm\sqrt{\frac{4-x^2-y^2}{x^2+y^2}}$$ Attempts: Here $\sin\theta + \sin\alpha=x$ ...
4
votes
2answers
47 views

How do I show that $n=2$ is the only integer satisfy :$\cos^n\theta+ \sin^n\theta=1$ for all $\theta$ real or complex?

It is well known that :$\cos²\theta+ \sin²\theta=1$ for all $\theta$ real or complex ,I would like to ask about the general equality :$\cos^n\theta+ \sin^n\theta=1$ if there is others values of the ...
2
votes
4answers
90 views

Show if $A^TA = I$ and $\det A = 1$ , then $A$ is a rotational matrix

Show if $A^TA = I$ and $\det A = 1$ where $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $, then $A =\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & ...
2
votes
0answers
34 views

Find elevator height given rope length?

This question is deceptively difficult. I feel like it's probably some classic example somewhere, but I'm not sure how to describe it in enough detail to get valid results in searching online. ...
2
votes
4answers
44 views

Prove that $16\cos^5A-20\cos^3A+5\cos A=\cos5A$

Prove that $$16\cos^5A-20\cos^3A+5\cos A=\cos5A$$ My solution begins here; $$ \begin{align} \text{RHS} & =\cos5A \\ & =\cos(A+4A) \\ & =\cos A\cos4A-\sin A\sin4A \\ & =\cos ...
7
votes
5answers
447 views

Solve the following trigonometric integral [on hold]

Calculate: $$\int _{0}^{\pi }\cos(x)\log(\sin^2 (x)+1)dx$$
0
votes
0answers
13 views

Prove Bernoulli Function is Constant on Streamline

I have an incompressible, inviscid fluid, under the influence of gravity, with a velocity potential: $$ \mathbf{u} = (-\cos(x)\sin(y), \sin(x)\cos(y), 0) $$ Using Euler's equations, $$ \mathbf{u} ...
0
votes
3answers
38 views

Why dividing by trigonometric functions gives wrong answer when solving trigonometric equations?

Hello I have a problem with solving Trigonometric equations. Why this is not true for $0\le\theta\le360$ $$2\sin\theta\cos\theta=\sin\theta$$ $$2\cos\theta=1$$ Set of solutions $\theta=60,360$ and ...
2
votes
2answers
27 views

Find Length of line which has rotating object.

I have 3 Images. A, B and C. if I place it on graph, its look something like this. Now main image is A and I place B and C on that image's (A) center point. For easy understanding, let's consider ...
0
votes
1answer
41 views

Finding if there is a maximum or minimum on a curve?

My apologies for being very brief with this question, the reason for this is because I don't know where to start. The question is as follows: A curve has the equation $\lambda \cosh(x) + \sinh(x)$, ...
2
votes
1answer
64 views

Proof that there are no solutions this equation. (3 variables, Square root and Sine) [on hold]

Hypothesis: There do not exist three different positive integers $a,b,c$ such that $$ -\sqrt{ab}\cdot \sin(p \cdot (a-b))+\sqrt{ac}\cdot \sin(p \cdot (a-c)) -\sqrt{bc}\cdot \sin(p \cdot (b-c)) =0 $$ ...
0
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1answer
49 views
+100

Trigonometric position function and intersection

I have the following position function for a point: $x(t) := C_x - (S_x-C_x) \cdot \cos(t\cdot\theta) + (S_y-C_y) \cdot \sin(t\cdot\theta) + t \cdot v_x$ $y(t) := C_y - (S_x-C_x) \cdot ...
6
votes
3answers
78 views

What is the integral value of $\frac{\tan 20^\circ+\tan40^\circ+\tan80^\circ-\tan60^\circ}{\sin40^\circ}$?

I have tried possibly all approaches. I first expressed $80$ as $60+20$ and $40$ as $60-20$ and then used trig identities.I later used conditional identities expressing $\tan ...
1
vote
1answer
47 views

How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$

Suppose $g$ is a function that has its derivatives everywhere and $G(x)=\int_0^x g(t)dt$. How to compute $\lim\limits_{x \rightarrow 0} \frac{1}{x^2}\int_0^{G(x)} \arctan(s+2s^2) ds$? To start ...
0
votes
0answers
25 views

Find the rotation angles of a 2-D rotation matrix between two vectors

I am trying to solve the following to find $\theta$. I was given two vectors $\begin{bmatrix}-4.95 \\ -.7\end{bmatrix}$ and $\begin{bmatrix}3 \\ 4 \end{bmatrix}$ and asked to compute the rotation ...
0
votes
0answers
23 views

Find the third angle

Three planes are orthogonal to each other. I have found the rotation about the 2 axis (x and y). Is there a way to find the third angle around z provided the angles around x(70 degrees) and y(-1 ...
2
votes
2answers
91 views

Proving $x>\sin(x)$ without calculus for $x>0$

The starting problem was to prove $$\sin 26^{\circ}\sin 58^{\circ}\sin 74^{\circ}\sin 82^{\circ}\sin 86^{\circ}\sin 88^{\circ} \sin 89^{\circ}>\frac{45\sqrt{2}}{64\pi}\\\cos 1^{\circ}\cos ...
3
votes
2answers
63 views

How do I prove that sin is not defined implicitly by an algebraic equation?

How do I prove that sin is not defined implicitly by an algebraic equation? In essence, there does not exist rational functions $f_0,\ldots,f_{n-1}$ that satisfies ...
3
votes
2answers
50 views

How to explain why the angle between two vectors in $\mathbb{R}^n$ is defined the way it is.

It is given in couple of the textbooks I have seen that they just define the angle between two vectors $\vec{x}, \vec{y} \in \mathbb{R}^n$ to be $\theta$ such that $$ \cos \theta = \frac{\vec{x} ...
0
votes
5answers
63 views

If $\tan^2(\theta)+2\sec^2(\theta)=5$. Find the value of $\sin^2(\theta)$

I have a trig problem which i can't really understand where to start. It says If $$\tan^2(θ)+2\sec^2(θ)=5.$$ Find the value of $$\sin^2(θ).$$ I think it has something with to do with Pythagorean ...
7
votes
8answers
499 views

Convergence of $\sin{\pi\sqrt{n}}$

Revising for an exam: Let $a_n = \sin{(\pi\sqrt{n})}.$ Show that: (i) $a_{n+1} - a_{n} \rightarrow 0$ (ii) The sequence $(a_n)$ is bounded. (iii) $(a_n)$ does not converge. My ...
3
votes
4answers
42 views

Identity of $8\sin^2(t)\cos^2(t)$

I know this probably has a simple answer, but I am having trouble understanding the steps to find the identity for this problem. This is the answer I was provided: $$8\sin^2(x)\cos^2(x) = ...
-1
votes
0answers
25 views

parametr with trigonometric equations

can you help me with this equation: $$ \frac{4p+3}{6} - \frac{\sin{8x}}{2} - (p+\frac{2}{3})\sin(4x-\frac{\pi}{4})=0 $$ for which values of $p$ equation has has 3 distinct solutions in range ...
6
votes
1answer
155 views
+50

Formula for cos(k*x)

I need to prove that: \begin{align} c_k =&\; \cos(k\!\cdot\!x)\\ c_k :=&\; c_{k-1} +d_{k-1}\\ d_k :=&\; 2d_0\!\cdot\!c_k +d_{k−1}\\ d_0 :=&\; −2\!\cdot\!\sin^2{(x/2)}\\ \end{align} ...
7
votes
4answers
697 views

Two different trigonometric identities giving two different solutions

Using two different sum-difference trigonometric identities gives two different results in a task where the choice of identity seemed unimportant. The task goes as following: Given $\cos 2x ...