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

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5
votes
2answers
51 views

Solve $\sin x = x - 2 \pi/3$

What is $x$ if $\sin x = x - 2 \pi/3$? The answer is $x \approx 2.61$ but how do I work that out (without Taylor series - this is homework for 10th grade)? Thanks.
1
vote
1answer
46 views

Integral with simple fractions

I have a problem with this integral $$\int_\ \frac{\cos x }{\sin x \sqrt{1+\cos^2x}} \, dx$$ Using substitution $u = \sin x $ we get $$\int_\ \frac{1 }{\ u \sqrt{2-u^2}} \, du$$ I think the ...
0
votes
4answers
73 views

Calculate the Limit as x approaches 0

I am asked to calculate the following limit $$ \lim_{x\to0}\frac{\ln(1+\sin x)}{\sin(2x)} $$ First, I tried expressing $1+\sin x=t$, then express $x$ from that equation but my equation seemed to just ...
1
vote
1answer
44 views

Evaluate $\int_{\pi}^{3 \pi/2}\frac{1}{1-\rho \sin{2 \theta }} d\theta$

I would like to evaluate $$\int_{\pi}^{3 \pi/2}\frac{1}{1-\rho \sin{2 \theta }} d\theta$$ For $-1<\rho <1$. Unfortunately nothing I have tried has got me very far so I would appreciate ...
0
votes
1answer
45 views

Sampling the Sine Function

Consider the sampled sine function, $f(n)=\sin(\omega n)$, where $n$ is an integer. If $\omega_2 = 3\pi/2$, does there exist an $0 \leq \omega_1 \leq \pi$ such that $\sin(\omega_1 ...
0
votes
2answers
43 views

De Moivres Theorem question and complex numbers

Question is: Find the cube root of $27 (\cos 30° + i \sin 30°)$ that, when represented graphically, lies in the second quadrant. I did this: ...
1
vote
1answer
37 views

Solving $\int\sec^3(x) \, dx$ example, trouble getting from step to step

I'm doing an example problem that is solving the integral of $\sec^3(x) \, dx$. The first step shown in the problem takes it from that to: $$\sec(x)\tan(x) - \int \sec(x)\tan^2(x) \, dx$$ The book ...
0
votes
5answers
62 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 ...
-1
votes
2answers
30 views

Given the real number $t = - \frac 5 4 \pi$, give the values of the sine, cosine and tangent.

I've found that these problems are quite hard for me, such as this: given the real number $t = - \frac 5 4 \pi$, give the values of the sine, cosine and tangent. Is there any good explanation on HOW ...
0
votes
1answer
46 views

As x approaches infinity, why does $ \lim_{x \to \infty}\arctan \left(\frac{x-2}{2}\right) = \frac{\pi}{2} $ [closed]

Just wondering why the following is true! $$\lim_{x\to \infty } \arctan\left(\frac{x-2}{2}\right) = \frac{\pi}{2} $$ Thanks!
3
votes
3answers
68 views

Proving whether the series $\frac{\cos(n)}{n}$ is absolutely convergent

I have the infinite sum $$\sum_{n=1}^\infty \frac{\cos(n)}{n}$$ and I am able to show that it is conditionally convergent by using the Dirichlet Test (and the Lagrange Trig Identity to show the ...
2
votes
3answers
24 views

How Angle AOP' is equal to (90° - θ) in the second figure?

I'm learning Trigonometry right now with myself and at current I'm understanding how to find the trigonometric ratio of the angle (90°- θ) in those of θ. I'm little bit confused right now in second ...
3
votes
0answers
28 views

Is it always possible to find the roots of $P(z)=az^4+bz^3+cz^2+bz+a$, where $a,b,c \in \mathbb{R}^*$, by first dividing both sides by $z^2$?

A classic way to solve quartics in the form $P(z)=az^4+bz^3+cz^2+bz+a$, if we know that the roots lie on the unit circle, is to divide both sides by $z^2$ and then use the fact that if $$z=\cos \theta ...
0
votes
1answer
28 views

Find $\sinh^{-1}x$

The hyperbolic sine function, $\sinh(x)$ , is defined by the equation: $$ \sinh(x) = \frac {e^x-e^{-x}} {2}$$ Find a formula for its inverse, $$ \sinh^{-1}(x) $$
0
votes
1answer
27 views

Sum of transformations of continuous uniform random variable

Let $X$ be uniformly distributed on $(a,b)$. I want to find the cdf of $$ \sin^2(X) + \cos^2(X) $$ My feeling is that since $\sin^2(X) + \cos^2(X) = 1$, the cdf will be: $$F(1 \le x)= \begin{cases} ...
0
votes
2answers
36 views

Find all solutions to the equation $3 \sec^2(x) - 4 = 0$

I know this is possible using the quadratic formula, but I want to find a cleaner way (If possible) to solve this problem. Any help is appreciated, Thanks.
-1
votes
1answer
29 views

A trigonometry based triangle problem

In the triangle ABC below, side a is 10 units, and side b is 12 units. cos(angleACB) = 1/5. Find the value of cos(angleCBA). I'm pretty sure that I should use the law of sines, or the law of cosines, ...
0
votes
3answers
32 views

Non Permissible values of $\cot(x)$

Why is it that the non-permissible values of cotangent $x$ is only where $\sin(x) = 0$ and not also where $\cos(x)=0$
0
votes
2answers
20 views

A Limits Problem [closed]

Using the fact that $$\lim_{h \to 0}\dfrac{\sin(h)}{h} = 1$$ and $$\lim_{h\to 0}\dfrac{\cos(h)-1}{h}=0\text{,}$$ Compute the following limits: $\lim_{h\to 0} \dfrac{\sin(x+h)-\sin(x)}{h}$ ...
2
votes
4answers
30 views

Find the minimum value of following trigonometric expression.

Find the minimum value of $\sin^{2} \theta +\cos^{2} \theta+\sec^{2} \theta+\csc^{2} \theta+\tan^{2} \theta+\cot^{2} \theta$ $a.)\ 1 \ \ \ \ \ \ \ \ \ \ \ \ b.)\ 3 \\ c.)\ 5 \ \ \ \ \ \ \ \ ...
1
vote
1answer
81 views

Why do derivatives of functions exist?

Consider following function: $f(x)=x^2 \sin{\frac{1}{x} }$ if $x\neq 0$ and $f(0)=0$. Why does the derivative of $f(x)$ exist? Find the deriviative and determine whether or not it is continous. ...
2
votes
4answers
50 views

Prove that $2\sin^{-1}\sqrt x - \sin^{-1}(2x-1) = \frac{\pi}{2}$.

Prove that $2\sin^{-1}\sqrt x - \sin^{-1}(2x-1) = \dfrac\pi2$. Do you integrate or differentiate to prove this equality? If so, why?
1
vote
1answer
21 views

Why arsin function has range $[-\pi/2,\pi/2]$ [duplicate]

While studying in P.75 of inverse trigonometric functions it tells we have to restrict our domain before finding the inverse.But I can't get why we choose $[-\pi/2,\pi/2]$?Why can't we choose ...
0
votes
0answers
16 views

Counting zeros of trigonometric functions of functions

There is not any context for this problem, it is a general question: In General: If you are given a trigonometric function of a function $\sin(f(x))$ with an arbitrary function f(x), is there any ...
0
votes
2answers
46 views

Tangent line of Lissajous curve?

I'm trying to find at how many points the tangent line of $(\cos(3t),\sin(2t))$ goes through the point $(3,0)$. My attempt: This is the same thing as saying for how many values of $t$ do we have ...
0
votes
1answer
94 views

Calculate the Integral $\int _0^{\frac{\pi }{2}}\:\frac{\sin^{7/2}x}{\sin^{7/2}x+\cos^{7/2}x}dx$ [duplicate]

I have to calculate this Integral - $$\int _0^{\frac{\pi }{2}}\:\frac{\sin^{7/2}x}{\sin^{7/2}x+\cos^{7/2}x}dx$$ I have no idea how to start, Hint someone ? Thanks.
0
votes
4answers
79 views

What's the period of $\sin \left(x^\frac{3}{2}\right)$?

How can one tell the period of $\sin \left(x^\frac{3}{2}\right)$? Is it $(2\pi)^\frac{2}{3}$?
0
votes
1answer
33 views

Calculate curve using 3 points

I'm trying to find a way of creating a curve using 3 points in x,y space so I can use it to find other points. Essentially I have a unit which moves up and down on the y axis and I want to be able to ...
0
votes
2answers
55 views

The sine/cosine/tangent of 0, 180 and 360 degree angles

If the ratios sine, cosine and tangent are applicable to right angled triangles, then how can they be applicable for the angles of 0, 180 and 360 degrees? They don't seem to have a right angle, since ...
0
votes
0answers
11 views

Is finding vertices is possible from angle between the medians and circumcenter in triangle (2D) and in tetrahedron (3D)?

With the aid of triangle (2D) and tetrahedron(3D) shapes, am developing a security protocol on which angle between the medians and circumcenter are considered as the security parameters. How to prove ...
4
votes
3answers
89 views

Solve $\sin(3x)=\cos(2x)$

Question: Solve $\sin(3x)=\cos(2x)$ for $0≤x≤2\pi$. My knowledge on the subject; I know the general identities, compound angle formulas and double angle formulas so I can only apply those. With ...
1
vote
1answer
38 views

$\epsilon - \delta$ proof of limit of $\arctan \left(\frac {x+z}{y}\right)$ as $(x, y, z) \to (1, 2, -3)$

Find the limit and prove the limit for $$\lim_{(x,y,z)\rightarrow(1,2,-3)}\arctan \left(\frac{x+z}{y}\right).$$ This is a homework problem. While I am comfortable with general epsilon-delta proof ...
2
votes
3answers
47 views

Solution for $\sum_{i=0}^{n} \sin(\frac{i\pi}{2n})$? [duplicate]

While I was trying to find the formula of something by my own means I came across this sum which I need to solve, however I don't know if there is a solution for it, maybe it doesn't mean anything and ...
2
votes
5answers
63 views

Find the least value of $4\csc^{2} x+9\sin^{2} x$

Find the least value of $4\csc^{2} x+9\sin^{2} x$ $a.)\ 14 \ \ \ \ \ \ \ \ \ b.)\ 10 \\ c.)\ 11 \ \ \ \ \ \ \ \ \ \color{green}{d.)\ 12} $ $4\csc^{2} x+9\sin^{2} x \\ = \dfrac{4}{\sin^{2} x} ...
2
votes
6answers
64 views

Solving integral using trig substitution $\tan(x/2)=t$

I have problems with solving the following integral: $$ \int{{\sin x - \cos x}\over {\sin x + \cos x}} \, dx$$ Could anybody please help me to find the solution and show me the method how it can be ...
0
votes
1answer
28 views

For what values of $x$ is it possible to compute $\cos(x \phi)$ with vectors?

When we have two vectors $a = (a_{1}, a_{2}, \dots, a_{n})$ and $b = (b_{1}, b_{2}, \dots, b_{n})$ of the same length, we can compute the cosine of the angle between them by means of the following ...
0
votes
2answers
45 views

A farmer's water trough problem. How to find dh/dt?

Here is a question about the water trough, it goes like this: A farmer has a water trough of length 8m which has a semi-circular cross-section diameter 1m. Water is pumped to trough at a constant rate ...
7
votes
2answers
56 views

Why is $\cos^2(2\pi/5) + \cos^2(4\pi/5)=3/4$?

Suppose $\theta=2\pi/5$. Apparently it is true that $1+ \cos^2 \theta + \cos^2(2\theta) + \cos^2(3 \theta) + \cos^2 (4\theta) = 5/2$, or equivalently, $\cos^2(2\pi/5) + \cos^2(4\pi/5)=3/4$. What is ...
1
vote
1answer
29 views

Proof that $8\cos^4(\phi)=\cos(4\phi)+4\cos(2\phi)+3$

Hi fellow mathematicians, I want to proof that $8\cos^4(\phi)=\cos(4\phi)+4\cos(2\phi)+3$ but somehow I cant figure out how to do that. Do you know an elegant proof? Help appreciated!
7
votes
3answers
109 views

How to compute $\cos(\arctan(2)) = 1/\sqrt{5}$

I'm doing matrices and I rotated a line about an angle. The gradient of my line I'm rotating to the $x$-axis is $2$, from $y=2x$. So obviously the angle that the line $y=2x$ makes with the $x$-axis is ...
-1
votes
0answers
52 views

Lost proof of trigonometric formula

The following formula seems to be true for odd positive integers $n$ but i forgot the way I proved it $$\sum_{k=1}^n\tan(\alpha+\frac{k2\pi}{n})=n\tan(n\alpha)$$ Maybe someone can deliver the ...
0
votes
5answers
32 views

Complex Numbers Euler's Identity

I am trying to follow a solution to a problem and it says that $\left(\frac{1+i\tan\theta}{1-i\tan\theta}\right)^a = \left(\frac{\cos\theta+i\sin\theta}{\cos\theta-i\sin\theta}\right)^a$ However, I ...
2
votes
3answers
22 views

Finding the exact value of $b$ when given the argument in $z=(b+i)^2$

In the answer to the above question, there are two methods. My method was that I expanded $(b+i)^2$ out and I do $\arctan \frac{2b}{b^2+1}$ and then solve quadratic equation. However, my friend did ...
0
votes
2answers
17 views

Find the height of the stump of a tree, given the angle which the broken-off part makes with the ground

A portion of a $30$m long tree is broken by tornado and the top struck up the ground making an angle $30^{\circ}$ with ground level. The height of the point where the tree is broken is equal to: ...
1
vote
3answers
21 views

How can I expand the range of $\arcsin$ and $\arccos$ in a piecewise manner?

Suppose I have a set of points $U = [0,2\pi]$ and I consider the set of points $\sin U = y$, which maps to $[-1,1]$. I now want to find an inverse for $\sin$ that allows me to recover all the points ...
2
votes
2answers
57 views

Integral of Lorentzian type with trigonometric function

Consider the following Riemann integral $$ \int_0^\infty \mathrm{d}x \frac{\alpha^2}{(x-x_0)^2+\alpha^2} \frac{\sin\left[{\left(x - x_1\right) t }\right]}{x-x_1} $$ with the displacements $x_0,x_1 \in ...
0
votes
1answer
136 views

How does Mathematica calculate sin(Pi/5)? [closed]

Consider: Sin[Pi/5] Which returns: $$\sqrt{\frac58-\frac{\sqrt5}{8}}$$ Does anyone know how to figure out what trigonometric identities are used by Mathematica ...
2
votes
4answers
33 views

limit involving trig functions

I'm not sure how to solve this limit. $$ \lim_{x\to 0} \frac{\tan 6x}{\sin 2x} $$ After some rearranging I get this. $$ \lim_{x\to 0} \frac{\sin 6x}{\cos 6x} \cdot \lim_{x\to0} \frac{1}{\sin 2x} $$ ...
2
votes
0answers
25 views

Pattern of collision of bouncy balls in a sphere?

Suppose that you have two infinitely bouncy golf balls that exist inside a perfect sphere in weightless suspension, and both golf balls start bouncing at a random angle and are 10 or 100 times ...
0
votes
2answers
20 views

$A=90^\circ$ creating a problem in the $\tan(A\pm B)$ formulas

As the title suggests I am having problem using $\tan(90^\circ)$ in identities of $\tan(A+B),\tan(A-B)$. We can easily see that we get $\frac{\infty}{\infty}$. Suppose I don't know value of ...