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

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-1
votes
0answers
23 views

Trig funct graphs check (amplitude, period)

Hi for the following questions I was wondering if I was correct in my answers and if I am incorrect, please correct me. Thank You My solutions, please correct me if I am wrong. 2cosx, amplitude ...
-2
votes
0answers
14 views

Start and endpoint of line, creating arrow heads [closed]

I have a start point(5.6,4) and an endpoint (6.1,3.15) I want to make an arrow head at the start point that is an equilateral triangle(60 degrees) with a length of .1. How can I accomplish this? ...
3
votes
2answers
50 views

Simplifying trig expression $\frac{1}{1-\cos \theta}$

I need help with the following trig problem, I'm getting the first part, but can't seem to complete it. $$\frac{\cos \theta}{1-\cos^2 \theta}- \frac{1}{1-\cos \theta}$$ The first part is going to ...
1
vote
3answers
84 views

Find $f'(x)$in terms of $f(x)=|\cos(x)|\sqrt{1-\cos(x)}$

I am trying to solve the following exercise : Let $f$ be the function defined by : $$\forall x\in]0,\pi[\;\;\;\;\; f(x)=|\cos(x)|\sqrt{1-\cos(x)}$$ calculate $f '(x)$ in terms of $f(x),$ for all $x\...
0
votes
1answer
40 views

Doubt regarding signs in trigonometry equations

I have been trying to solve some equations, and for the same I found an online answer. Here's the link - http://citeseerx.ist.psu.edu/viewdoc/downloaddoi=10.1.1.456.6096&rep=rep1&type=pdf#page=...
0
votes
1answer
16 views

Plotting triangles based on a single point with distance and angle.

I'm tasked with creating an arrowhead within a pdf program. I have a single point with at $x=5.6$, $y=4$ this would be point A of my triangle I want to make the sides equal at $90$ degrees angles ...
1
vote
3answers
63 views

$\sin \alpha = \frac{3}{5} $ and $\cos \beta = -\frac{12}{13}$ . Find the values that $\cos(\alpha+\beta )$ can get.

$\sin \alpha = \frac{3}{5} $ and $\cos \beta = -\frac{12}{13}$ . Find the values that $\cos(\alpha+\beta )$ can get. Here $0<\alpha < \frac{\pi}{2}$ and $\frac{\pi}{2}<\beta<\pi$. Yes I ...
4
votes
1answer
48 views

Maximum and minimum of $f(x)=\cos(\sin(x))-\sin(\cos(x))$

Given the function: $$f(x)=\cos(\sin(x))-\sin(\cos(x))$$ it has absolute maxima at $x=(2k+1)\pi$ with $k=0,1,..N$ and relative maxima at $x=2k\pi$. It is not clear where are the minima. Putting the ...
0
votes
0answers
11 views

Length of elliptical segment given starting and ending points and slope

I would like to represent the flight path of a turning aircraft with an ellipse. Initially, the baseline turn is 180 deg, with a constant radius. The speed of the aircraft is constant. During the ...
1
vote
1answer
33 views

Angles of lines tangential to a circle

I am looking to find the angles of line features relative to the tangent of a circle. Please see this example for general idea. Angles to line features (purple) I am looking for are (poorly drawn) ...
0
votes
1answer
24 views

How to solve $7.51\tan{\theta} - 2.656(\sec{\theta})^2=0$

I'm trying to solve $7.51\tan{\theta} - 2.656(\sec{\theta})^2=0$ and the way that it's been done in my notes is by somehow changing the equation to $7.51\tan{\theta} - 2.656(\tan{\theta})^2 - 2.656=0$ ...
0
votes
0answers
25 views

What maths would most likely have used for this game's horizontal bullet spread? Firing at 90° y causes the marks to line up perfectly.

While playing Doom, a game with a lot of mathematical techniques for various things, if I aim my x-as-well-as-y-spreading shotgun up at a 90° on the y view angle (x and y angles are used to look ...
1
vote
3answers
54 views

Find all the angles $v$ between $-\pi$ and $\pi$

Find all the angles $v$ between $-\pi$ and $\pi$ such that $$-\sin(v)+ \sqrt3 \cos(v) = \sqrt2$$ The answer has to be in the form of: $\pi/2$ (it must include $\pi$) I have tried squaring but I get ...
4
votes
6answers
86 views

I was trying to find out the intervals where $\sin ^{-1}x > \cos ^{-1}x$

I was trying to find out the intervals where $\sin ^{-1}x > \cos ^{-1}x$ The easiest way was to just look at the graph and I found out that the region is $x \in ({1\over \sqrt{2}} , 1]$ But I ...
2
votes
2answers
43 views

How to prove that a sum of $\cosh(kx)$ is equal to a formula? [duplicate]

I need to prove that $$\sum_{k=0}^{n}\cosh(kx) = \frac{\sinh((n+1/2)x) + \sinh(x/2)}{2\sinh(x/2)}$$ Can you help me out? How do I even start?
0
votes
1answer
28 views

Finding $f(x)$ in $\cos^2(x)f(x)=x^2-2\int_1^x \sin(t)\cos(t)f(t) \, \mathrm{d}t$

I need to find a valid $f(x)$ such that: $$\cos^2(x)f(x)=x^2-2\int_1^x \sin(t)\cos(t)f(t) \, \mathrm{d}t$$ I can apply the FToC and I get: $$(2\cos(x)-\sin(x)f(x))+(\cos^2 x f'(x))=2x\sin(x)\cos(x)...
6
votes
7answers
163 views

If $\sin x + \sin y = 1$ and $\cos x + \cos y = 0$, solve for $x$ and $y$

$\sin x + \sin y = 1$ $\cos x + \cos y = 0$ Any valid pair of $(x, y)$ is fine, as the restrictions on the board in the image below are obscured. I got the question from chapter 26 of a comic ...
0
votes
0answers
39 views

Math precalculus/trig

Circle $O$ below has radius 1. Eight segment lengths are labeled with lowercase letters. Six of these equal a trigonometric function of $\theta$. Your answer to this problem should be a six letter ...
2
votes
1answer
49 views

What is the problem in my computation of $\sin 18^{\circ}$?

I needed to compute $\sin 18^{\circ}$. Now, these two relations hold for every $x$: $\cos 5x=16\cos^5x-20\cos^3x+5\cos x$ $\sin5x=16\sin^5x-20\sin^3x+5\sin x$, which can be easily proved using the ...
1
vote
5answers
80 views

Prove the following trigonometric result

If $\theta_1,\theta_2(0\leq\theta_1,\theta_2<2\pi)$ are two solutions of $\sin(\theta+\phi)=\frac{1}{2}\sin(2\phi)$, prove that $$\frac{\sin(\theta_1)+ \sin(\theta_2) }{ \cos(\theta_1)+ \cos(\...
2
votes
2answers
67 views
+50

Changing the period of sine versus arc length

Let's consider $ y = \sin x $. Let $ s \in \mathbb{Q} $ and $ s > 1 $. One may calculate the arc length of sine between $ 0 $ and $ 2\pi s$ using the formula: $$ L = \int_0^{2\pi s} \sqrt{1 + \...
-4
votes
3answers
101 views

Find the value of $6P_{10} - 15P_8 + 10P_6+7$ for $P_n=\sin^n x+\cos^n x$

If $P_n=\sin^n x+\cos^n x$ where $n$ is a whole number and $x$ is a real number. Find the value of $6P_{10} - 15P_8 + 10P_6+7$ I tried this: $$P_6 \Longrightarrow \sin^6 x + \cos^6 x = (\sin^2 x + \...
1
vote
4answers
71 views

find minimum value of $2^{\sin^2(\theta)}+2^{\cos^2(\theta)}$

find minimum value of $2^{\sin^2(\theta)}+2^{\cos^2(\theta)}$ I have found the minimum value using derivative method : Let $f(\theta)=2^{\sin^2(\theta)}+2^{\cos^2(\theta)}$. Then calculate $f'(\...
1
vote
1answer
39 views

Find $\tan 2x$, given $\tan(x+y)=3$ and $\tan(x-y)=2$

I am having a hard time to solve this trigonometric system of equations. The equations is as follows: We are given $$\tan(x+y)=3$$ $$\tan(x-y) = 2$$ and we need to find $$\tan2x$$ I have ...
0
votes
1answer
14 views

Bearings question confusion

At 12.00pm , a ship was spotted at a point P , 30 km due north of an island , L . The ship was sailing on a bearing of 120 degree at 32km/h . How far was the ship from the island at 12.30pm ? My ...
0
votes
1answer
51 views

Why don't we take $\sin x$ as negative square root of $1-\cos^2x$? [closed]

I am confused of using $\sin x$ as as negative square root of $1-\cos^2x$. Can anyone help please?
1
vote
1answer
18 views

Sides of a triangle are in Arithmetic Progression, then find $\tan (\alpha+ \frac{\beta}{2})$

The sides of a triangle are in Arithmetic Progression. If the smallest angle of the triangle is $\alpha$ and largest angle of the triangle exceeds the smallest angle by $\beta$, then find the value of ...
3
votes
3answers
55 views

How to find $ \tan \left(\frac{x}{2}\right) $ knowing that $\cos \left(x\right)+\sin \left(x\right)=\frac{7}{5} $

Good evening to everyone. I don't know how to find $ \tan \left(\frac{x}{2}\right) $ knowing that $$\cos \left(x\right)+\sin \left(x\right)=\frac{7}{5} $$ and x$\in (0,\frac{\pi}{3})$ Here's what I've ...
0
votes
0answers
18 views

Hypocycloid with an outer ellipse

I have tried to change the traditional hypocycloid a bit. What I've basically done is that a circle now rolls inside an ellipse. I am trying to find the equation for the same. I am mostly done, ...
0
votes
0answers
15 views

Cycloids with ellipse

I have been researching about the epitrochoids and hypotrocoids lately. I was wondering if it would be possible for us to have an ellipse rolling around a circle? If so, then how could one derive its ...
2
votes
3answers
131 views
+200

A trigonometric problem when calculating distance to the boundary of a convex hull

Suppose we have a sphere and a point outside of the sphere. We denote the point outside as $v$ and the origin of the sphere as $x$. The convex hull of the sphere and $v$ should be like an ice cream ...
1
vote
1answer
26 views

How to solve this implicit differentiation problem concerning arcsin?

My overarching question is about differentiating when you have these inverse trig functions, but listed below is the specific question I am trying to solve. If you help me with the problem, it'll help ...
0
votes
2answers
49 views

Check my work: Evaluating $\tan\frac{7\pi}{8}$ using a half-angle formula

I am doing a trig problem involving half-angle identities, and I am not sure if my solution is correct. Can someone please check my work? The question: Find the exact value of $\tan\frac{7\pi}{8}...
2
votes
1answer
40 views

Seeking advice from the more experienced on which trig identities are crucial to memorize and which can be derived quickly

This is a bit of a two part question. I also have read some of the related questions, but I think mine is different as whether they can be derived quickly, rather than whether they can be derived, is ...
1
vote
2answers
52 views

How does $\int_{\pi/3}^{\pi/2} \frac{1-\cos^2x}{\sqrt{\sin^2(x/2)}}dx$ simplify to $\int_{\pi/3}^{\pi/2} 4\sin(x/2)\cos^2(x/2)dx$?

$$\int_{\large{\frac{\pi}{3}}}^{\large{\frac{\pi}{2}}} \frac{1-\cos^2x}{\sqrt{\sin^2\left(\frac x2\right)}}dx$$ How does the above simplify to the below? $$\int_{\large{\frac{\pi}{3}}}^{\large{\frac{...
0
votes
5answers
79 views

Prove $\tan(\frac{\pi}{2} -\theta ) = \cot \theta$

$$\tan\left(\frac{\pi}{2} -\theta \right) = \cot \theta$$ I can prove this by changing into $\cos$ and $\sin$ .But I want to know that , is it possible to prove it using relation given below . If ...
0
votes
0answers
9 views

Rotation of axes formulas work differently depending on the type of object they are applied to?

I'm currently working through the rotation of axes section in my trig textbook, and I'm having trouble understanding why the rotation of axes formulas rotate the axes in different directions depending ...
2
votes
1answer
17 views

What determines the signs of the trigonometric functions in the quadrants of the xy-plane? [duplicate]

In the xy-plane, how can we determine the signs of the trigonometric functions in each quadrant? For example, sine is positive in Quadrant I, cosine is negative in Quadrant II, etc. How can we ...
0
votes
0answers
15 views

Angle between two segments

I'm scanning a target and when I scan it I know which direction it is heading, 0 deg is north 90 E, etc. I know my heading and and which way I need to turn to "face" the target (so -150 means turn to ...
1
vote
3answers
66 views

Solve $ \int{x\sin^2(x)}\ dx $

I need to solve this integral: $$ \int{x\sin^2(x)}\ dx $$ I SOLVED it by writting: $$ \sin^2(x) = \frac{1-\cos(2x)}{2} $$ and used integration by parts for $x\cos(2x)$, and the result is: $$ \frac{1}...
7
votes
4answers
149 views

inequality $\sqrt{\cos x}>\cos(\sin x)$ for $x\in(0,\frac{\pi}{4})$

How can I prove the inequality $\sqrt{\cos x}>\cos(\sin x)$ for $x\in(0,\frac{\pi}{4})$ ? The derivative of $f(x):=\sqrt{\cos x}-\cos(\sin x)$ is very unpleasant, so the standard method is ...
0
votes
2answers
48 views

What is the geometrical definition of the $\sec\theta$

This is the geometrical definition of the $\sec\theta$: My problem with this definition is when the angle $\theta$ is in the forth quadrant. The $\sec \theta$ is positive but the geometrical ...
0
votes
1answer
24 views

How to calculate the shortest rotation from current to the target angle? [closed]

In the following situation: My current angle is 40*, my target angle is 130*. How should I calculate the rotation that should be done to reach the target angle from the current one? I've done the ...
0
votes
2answers
45 views

How are the trigonometric ratios geometrically defined for non-acute angles?

The usual way trigonometric ratios are geometrically defined is always relative to an acute angle. So this way inside a right triangle, the trigonometric ratios are defined by the ratios of hypotenuse,...
22
votes
3answers
465 views

The entry-level PhD integral: $\int_0^\infty\frac{\sin 3x\sin 4x\sin5x\cos6x}{x\sin^2 x\cosh x}\ dx$

I hope you find this integral interesting. Evaluate $$\int_0^\infty\frac{\sin\left(\,3x\,\right)\sin\left(\,4x\,\right) \sin\left(\,5x\,\right)\cos\left(\,6x\,\right)}{x\,\sin^{2}\left(\,x\,\...
0
votes
0answers
26 views

Applying distortion to Bézier surface

I am trying to simulate the image warp effect, that is used in Adobe Photoshop. The rectangular image is warped according to a cubic Bézier surface (in 2D, all Z components are 0). Having any Bézier ...
1
vote
3answers
74 views

How to integrate $\int \dfrac{1}{\sin^4 x \cos^4 x} dx$

The integral in question is: $$\int \dfrac{1}{\sin^4 x \cos^4 x} dx$$ I tried using $1 = \sin^2 x + \cos^2 x$, but it takes me nowhere. Another try was converting it into $\sec$ and $\csc$, but ...
0
votes
0answers
37 views

What determines the signs of the trigonometric functions in the quadrants of the $xy$-plane? [closed]

In the $xy$-plane, how can we determine the signs of the trigonometric functions in each quadrant? For example, sine is positive in Quadrant I, cosine is negative in Quadrant II, etc. How can we ...
4
votes
2answers
326 views

The relationship between tan(x) and square roots

Please note: I am working in DEGREES I think the easiest way to illustrate my point is by showing some examples: $ \tan(0^\circ) = \sqrt 0 = 0$ $ \tan(22.5^\circ) = \sqrt 2 -1$ $ 3 \cdot \tan(30 ^\...
0
votes
5answers
61 views

Find the minimum and maximum value of $\frac{1}{\sin{x} -3\cos{x} +5}$ [closed]

Find the maximum value and minimum value of $$\frac{1}{\sin{x} -3\cos{x} +5}.$$ Any tips?