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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2answers
20 views

Range of a function - trigonometric

Question: Find the range of the function: $$\sin^4 x + \cos^4 x$$ I really have no idea how to initiate this question. Please help me find a solution!
3
votes
3answers
41 views

find the side of an equilateral triangle given only the distance of an arbitrary point to its vertices

Triangle ABC is an equilateral triangle and P is an arbitrary point inside it. The the distance from P to A is 4 and the distance from P to B is 6 and the distance from P to C is 5. How to find the ...
0
votes
4answers
42 views

Physics problem, stuck in algebra.

I end up with the equations; $$u=u_1' \cos(a)+u_2' \cos(b)$$ $$u_1' \sin(a)=u_2' \sin(b)$$ $$u^2=u_1'^2+u_2'^2$$ I have to show that $$a+b=\frac{\pi}{2}$$ $x'$ isn't the derivative of $x$, it's a ...
-1
votes
2answers
74 views

Scratching my head whith a problem of infinity.

The first equation in a) gives a sum of 1 and the second equation starts with a sum equal to $\pi$ or $180$ in radian and degree mode.By removing the sign from $\sqrt x$ in b) the value of y is still ...
2
votes
2answers
35 views

Evaluating an inv. tan function

The problem: Evaluate the inv. function by sketching a unit circ., finding the angle, and eval. the correct pair on the circle. Function: $\tan^{-1}(-1)$ I found a solution for this, but my ...
1
vote
2answers
35 views

Simplifying a trigonom. expression

The problem: Simplify the expression. Specify the range of $x$ for which the simplification holds: $\cos(\tan^{-1}x)$. So we know that, $\tan^{-1}x$ is the angle $\theta$ for which $\tan\theta ...
1
vote
3answers
29 views

Rotating an object around any origin

I want to extend my program that generates PDF and I need like to rotate an object (for example -30deg clockwise): 1: original 2: rotated object (origin is bottom left) The first problem is, that ...
3
votes
2answers
116 views

How to prove this $\frac{\sin{(nx)}}{\sin{x}}\ge\frac{\sqrt{3}}{3}(2n-1)^{\frac{3}{4}}$

let $n<\dfrac{\pi}{2\arccos{\dfrac{c}{2}}},c\in (0,2),c=2\cos{x}$, show that $$\dfrac{\sin{(nx)}}{\sin{x}}\ge\dfrac{\sqrt{3}}{3}(2n-1)^{\frac{3}{4}}$$ where $0<x<\dfrac{\pi}{2}$ My idea: ...
0
votes
1answer
12 views

Find catesian coordinate of T-point $P(-\frac{65\pi}{2}) $

Find the Cartesian coordinates of T-point $P(-\frac{65\pi}{2}) $. It is easy when there is no negative sign. I don' t know how to do with negative sign.
1
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2answers
56 views

Why must closest approach occur when relative velocity is perpendicular to motion?

The first part i) I can solve correctly, but I need some advice and intuition on how to solve the second part ii). Here is the mark-scheme for the question: But for part ii) I do not understand ...
1
vote
0answers
51 views

Would this thinking about the dot product hold?

Background today I completed the chapter on the dot product of vectors. But in trying to figure out exactly what the dot product is. I came to the conclusion that it can be interpreted as the length ...
1
vote
1answer
33 views

Trigonometric inequality question [on hold]

Let $0 < A < \frac {\pi}{2}$ and $0 < B < \frac {\pi}{2}$. (a) prove that $\sec^2 A + \csc^2 A \cdot \csc^2 B \cdot \sec^2 B \geq 9.$ (b) determine values of $\sec A$ and $\sec B$ when ...
2
votes
5answers
51 views

If angular velocity $\omega=\sqrt{\frac{3g\sin\theta}{2a}}$ can I find angular acceleration $\alpha$ by differentiating $\omega$?

It was my understanding that angular acceleration is the derivative of angular velocity. The reason I ask is Thanks.
2
votes
5answers
99 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 ...
-2
votes
1answer
27 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
56 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 ...
-1
votes
2answers
41 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
32 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 ...
7
votes
3answers
103 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: ...
9
votes
1answer
64 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 ...
1
vote
2answers
50 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 ...
2
votes
4answers
93 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
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2answers
21 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
55 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 ...
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
2answers
78 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?
1
vote
5answers
50 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
88 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?
2
votes
1answer
35 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 ...
1
vote
2answers
56 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!
1
vote
2answers
43 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 ...
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!
3
votes
2answers
34 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 ...
3
votes
6answers
138 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 ...
0
votes
1answer
26 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
2answers
40 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 ...
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
-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
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
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 ...
2
votes
1answer
53 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 ...
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
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
3
votes
3answers
312 views

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

In a triangle $ABC$ $$(b + c)\cos A + (c + a)\cos B + (a + b)\cos C=?$$
0
votes
2answers
68 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 ...
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 ...
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
1answer
37 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