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
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5answers
45 views

Showing the $n$-th derivative of $\cos x$ by induction

I was asked to show that the $n$-th derivative of $\cos x$ is $\cos(\frac{n\pi}{2} + x)$. My progress : By induction, I proved it was true for $n=1$. Then I assumed it was true for $n = k$ so now I ...
0
votes
0answers
13 views

How do you find the bearing of closest approach?

The first part i) I can solve correctly, but I need some advice and intuition on how to solve the second part ii). Could someone please talk me through this step by step in simple English as I ...
6
votes
2answers
81 views

Evaluate trig functions without a calculator

My precalculus test asked me to determine which was greater: $\tan (53)$ or $\sec (38)$. I looked at it like this, but it seems so close that it's difficult to imagine that they would ask this: ...
58
votes
10answers
6k views

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim\limits_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my ...
-2
votes
1answer
22 views

How to find slope on line that known only point and angle

How to find slope on line that known only point and angle Image will describe more clearly I'm wont to find the orange line slope to find point on it ( b , c , d ) suppose that A and angle are ...
0
votes
4answers
52 views

Why $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$?

I am working out a Fourier Series problem and I saw that the suggested solution used $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$ to simply the expressions while finding the Fourier Coefficients ...
4
votes
4answers
182 views

Alternative ways to solve this trigonometric inequality?

The inequality is: $$ \frac{\sin{\theta}+1}{\cos{\theta}}\leq 1 \text{ with } \cos{\theta}\neq0 \land 0\leq \theta\lt 2\pi$$ I've tried splitting it up into cases of $\theta$ that make ...
1
vote
2answers
47 views

Induction Proof of trig inequality

This is for a course, I don't want the answer just a prod in the right direction! I've got a problem that states let n be an integer such that $$n\gt0$$ $$\text{Prove: }\sum_{k=0}^n |\cos k| \ge ...
-1
votes
2answers
37 views

How do I change $\cos(\frac{\theta}{2})$ into $\cos(\theta)$ in an equation?

Just give me an example. eg. $\cos(\frac{\theta}{2})=\frac{1}{2}.$ I want to make $\cos(\frac{\theta}{2})$ become $\cos(\theta)$. Thanks.
1
vote
2answers
29 views

simplification of the area of a hyperbolic circle (BONOLA, S 53)

I'm trying to understand the S-53 of "Non-Euclidean Geometry" (BONOLA, R.) in which the formula for the area of a circle of radius r: $$2\pi k^2(\cosh\frac rk -1)$$ is somehow reduced by only applying ...
1
vote
2answers
72 views

Proof of trigonometric identity $\sin(A+B)=\sin A\cos B + \cos A\sin B$

All the proofs I've seen are geometrical, assuming that $A+B$ is less than $90$ degrees. How can you prove this identity for $A+B$ greater than $90$ degrees, or more generally, any arbitrary value?
9
votes
1answer
61 views

How were the sine, cosine and tangent tables originally calculated?

As I understand it... ahem... the (cosine, sine) vector was calculated for (30 degrees, PI/6), (45 degrees, PI/4) and (60 degrees, PI/3) angles etcetera, however, I would like know the original ...
4
votes
2answers
230 views

How to find the maximum diagonal length inside a dodecahedron

I am trying to find the maximum length of a diagonal inside a dodecahedron with a side length of $2.319914107\times10^{89}$ meters. I am not sure if any other information than that is needed, if it is ...
2
votes
4answers
82 views

Find $\sec \theta + \tan \theta$.

If $\tan \theta=x-\frac{1}{x}$, find $\sec \theta + \tan \theta$. This was the question ask in my unit test. My Efforts: $\tan^2 \theta=(x-\frac{1}{x})^2$ $\tan^2 \theta= (\frac {x^2-1}{x})^2$ ...
1
vote
2answers
20 views

Finding Principale period of $\cos$ function

Find principle period of $3\cos (2x-3)$. Today I have learned about principle period of various trigonometric function. I know that principle period of cos is $2 \pi$. Please someone can help me ...
3
votes
2answers
53 views

can't seem to understand $\sin{\theta} = y$ on a unit circle

So I've been working very hard on my trigonometry on khan academy. However I'm constantly getting stumped by one type of question in particular. There is some fundamental flaw in my understanding. ...
3
votes
1answer
55 views

Nonsensical result in the midst of calculating an integral via substitution.

I was just calculating an integral via a trigonometric substitution and ended up with $\color{red}{ \text{something pretty nonsensical} }$ but $\color{blue}{ \text{reversing the substitution} }$ ...
5
votes
1answer
47 views

To find a trigonometric limit without Wallis' integrals

What is the limit $$ \lambda =\lim\limits_{n \to \infty}{n\int_0^{\frac{\pi}{2}}(\sin x)^{2n} dx}$$ I would like to find it without Wallis' integral formula: I mean without evaluating the closed ...
3
votes
6answers
133 views

Evaluate$ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$

Find the value of the integral $ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) $ I tried putting $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but am unable to proceed further. I think the following integral can be ...
1
vote
2answers
38 views

Naive proof that $\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$ [duplicate]

As part of a larger proof, I must show that: $$\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$$ I have thought about this but can't figure out how to get my hands on the value since I don't know any ...
1
vote
5answers
47 views

Solve for $\theta$: $a = b\tan\theta - \frac{c}{\cos\theta}$

This question was initially posted on SO (Link). I'm not sure the answer given there was correct. I cannot get the results from those expressions to match my CAD model. The title pretty much sums ...
2
votes
3answers
86 views

Trigonometric equation $2\sin x+\cos x+1=0$

I have to calculate $\dfrac{d}{dx}\dfrac{1+\cos x}{2+\sin x}=0$. I have already simplified to: $2\sin x+\cos x+1=0$, but I have no idea how to go further.. Could someone give a hint?
3
votes
4answers
135 views

What is $\frac{d(\arctan(x))}{dx}$?

Let $v= \arctan{x}$. Now I want to find $\frac{dv}{dx}$. My method is this: Rearranging yields $\tan(v) = x$ and so $dx = \sec^2(v)dv$. How do I simplify from here? Of course I could do something like ...
1
vote
1answer
31 views

At the instant of release of an object from rest. Is the only force that can act its weight? [on hold]

This is the third question from a mechanics exam past paper: I can do parts i) and ii) but for iii) in finding the angular acceleration, i used $C=I\alpha$, where $C$ is the applied couple or ...
4
votes
5answers
55 views

If $\sin( 2 \theta) = \cos( 3)$ and $\theta \leq 90°$, find $\theta$

Find $\theta\leq90°$ if $$\sin( 2 \theta) = \cos( 3)$$ I know that $\sin 2\theta = 2\sin\theta\cos \theta$, or alternatively, $\theta = \dfrac{\sin^{-1}(\cos 3)}{2}$. Can somebody help me?
1
vote
6answers
264 views

Range of a trigonometric function

Question: Prove that: $$0 \leq \frac{1 + \cos\theta}{2 + \sin\theta}\leq \frac{4}{3}$$ I have absolutely no idea how to proceed in this question. Please help me!
33
votes
1answer
709 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
1
vote
2answers
55 views

Trigonometry and triangle proof

Question: Prove that in an acute angle triangle ABC: $$\tan A\tan B +\tan A \tan C + \tan B \tan C \geq 9$$ I have no idea where to even begin this question. Please help me!
2
votes
2answers
253 views

Goat tethered in a circular field

I need help on this question. I was thinking of counting the area of the whole circle and then subtracting it with the area that is not eaten by the goat. But I don't know how to find this particular ...
1
vote
2answers
37 views

Expressing $ 12\sin( \omega t - 10) $ in cosine form

$$ 12\sin( \omega t - 10) $$ I understand how it's solved when using the graphical method, however I'm having trouble understanding something about the trigonometric identities method. The solution ...
0
votes
2answers
38 views

Why does resolving forces in one direction give a completely different answer to resolving the opposite way?

I can solve parts i), ii) and am able to show that $R=0$ for part iii). In this question $g$ is the acceleration of free fall taken to be $9.8$ Using Newtons 2nd law [$F=ma$] for the last part I ...
3
votes
2answers
32 views

Solving for an angle

I was never good in trigonometry. I have a rectangle with dimensions $L_1$ and $W_1$. I want to rotate it so that it fits inside another rectangle with dimensions $L_2$ and $W_2$. I need to find the ...
0
votes
3answers
44 views

Implicit differentiation of trig functions

I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation: $$\cos\cos(x^3y^2) - x \cot y = -2y$$ I figured that the calculation requires the chain ...
0
votes
1answer
25 views

Use $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial?

I am reading Barbeau's Polynomials and I found the following problem: Use the identity $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial. But I really have no idea on how ...
0
votes
1answer
24 views

In a triangle ABC,a:b:c is 4:5:6.The ratio of the radius of the circumcircle to that of incircle is

In a triangle ABC,a:b:c is 4:5:6.The ratio of the radius of the circumcircle to that of incircle is
1
vote
2answers
26 views

Rearranging equation $t = \frac{T}{2\pi} (\psi - \epsilon \sin \psi)$ in terms of $\psi$

I was playing around with the maths for orbits and trying to make a parametric equation that, well.. worked. I found a worksheet with parametrics with another variable ($\psi$), but I wanted to be ...
-5
votes
1answer
30 views

In a triangle ABC,a:b:c is4:5:6.The ratio of the radius of the circumcircle to that of incircle is [on hold]

In a triangle ABC,a:b:c is 4:5:6.The ratio of the radius of the circumcircle to that of incircle is
22
votes
2answers
236 views

Disproving an “almost true” trigonometric identity

The plausible looking "identity" $$\sin(\frac{\pi}{51})+\cos(\frac{\pi}{74})=\frac{3}{2\sqrt 2}$$ is not true, but it is close indeed: $$LHS=1.0606\color{blue}{598...}$$ ...
1
vote
2answers
35 views

When can I and when can I not use complex replacement?

If I want to calculate: $$(2 cos(t))^3$$ Can I not replace cos(t) with $Re(e^{it})$ and calculate $(2e^{it})^3$ to be $8e^{3it}$ and thus the real part of this becomes 8cos(3t)? But that answer is ...
1
vote
1answer
52 views

Is $f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$ strictly positive?

I would like to have an advice for this exercise. Let $x\in[0,\pi]$ For which values of $x$ this function $$f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$$ is strictly positive ? I tried to ...
12
votes
3answers
305 views

$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$

How can I find the following product using elementary trigonometry? $$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ.$$ I have tried using a substitution, but nothing ...
0
votes
0answers
16 views

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle , then P^-1+P^-2+P^-3 is equal to [duplicate]

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle , then P^-1+P^-2+P^-3 is equal to note s=a+b+c/2 area of triangle rs/2
0
votes
2answers
67 views

In any triangle ABC, the expression (a + b + c) (a + b - c) (b + c - a) (c + a - b)$ is equal to

In any triangle ABC, give an equivalence to the expression $$(a + b + c) (a + b - c) (b + c - a) (c + a - b)$$ Can somebody help me? Note that ...
3
votes
3answers
307 views

In a triangle ABC, (b + c) cos A + (c + a) cos B + (a + b) cos C is equal to

In a triangle $ABC$ $$(b + c)\cos A + (c + a)\cos B + (a + b)\cos C=?$$
7
votes
2answers
104 views

$\frac{\pi}{4}=k\arctan \frac{1}{m}+l\arctan \frac{1}{n}$ has only four solutions?

Is the following true? "$$\frac{\pi}{4}=\arctan \frac{1}{2}+\arctan \frac{1}{3}$$$$\frac{\pi}{4}=2\arctan \frac{1}{2}-\arctan \frac{1}{7}$$$$\frac{\pi}{4}=2\arctan \frac{1}{3}+\arctan ...
1
vote
0answers
34 views

Using two chords and an angle to find center and radius of a circle

Hello, I am trying to solve the problem below. Is it possible to solve for the Center and Radius of the circle given the information provided, or is there something missing? I know how it's simple ...
0
votes
2answers
19 views

Vectors, How to measure total force and direction.

I am currently looking for some math help that I am quite struggling with. The problem is: (Vectors) A fisherman use his pole and line to pull a fish out of the water. The line exerts a force on ...
1
vote
1answer
36 views

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle, then P^-1+P^-2+P^-3 is equal to

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle , then P^-1+P^-2+P^-3 is equal to
1
vote
2answers
156 views

Why is $\sin \theta$ just $\theta$ for a small $\theta$? [duplicate]

When $\theta$ is very small, why is sin $\theta$ taken to be JUST $\theta$?
2
votes
3answers
65 views

Skewed Trigonometric Function

What would be an expression for a periodic function (period $2\pi$) that essentially behaves just like a negative sine function, but it has the following quirk: It's $0$s lie on the usual places ...