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

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2
votes
1answer
42 views

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
98 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
68 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
38 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
50 views

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

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
54 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
17 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 ...
1
vote
2answers
93 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
46 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
37 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
147 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
46 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
44 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,...
18
votes
3answers
437 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
35 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 ...
3
votes
2answers
234 views
+50

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) = \sqrt 0 = 0$ $ \tan(22.5) = \sqrt 2 -1$ $ 3 \cdot \tan(30 ^\circ) =\...
-3
votes
0answers
48 views

Are a cirlcle and a triangle limited to seven or do they have their own pattern? [closed]

there are 7 triangles in which the perimeter and the measurement of angles are: $5+5+1=11$ $5+5+2=12$ $5+5+3=13$ $5+5+4=14$ $5+5+5=15$ $5+5+6=16$ $5+5+7=17$ The sequence of seven angles of ...
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?
0
votes
4answers
38 views

Proving this trigonmetric identity

I have tried to prove this identity but I keep getting unhinged by the cubed expression. $$ \sec(x)(\sin^3(x) + \sin(x)\cos^2(x)) = \tan(x) $$ I have tried multiplying out the sec, but doesn't ...
-6
votes
2answers
41 views

Find $\theta$ if $2 \sin^2 \theta - \cos^2\theta = 2$. [closed]

Find $\theta$ if $2 \sin^2 \theta - \cos^2\theta = 2$.
0
votes
2answers
18 views

I can't find an appropriate piecewise function for this graph [closed]

On one of my piecewise questions I've split a graph into an exponential function, a cosine function and a parabolic function. I've done fine for exponential and parabola but I'm totally stuck on ...
0
votes
1answer
32 views

How to calculate the sides of a triangle when only the area and 2 angles are given?

I'm reading the book 'Trig without tears' by Stan Brown and there is a table mentioning the method for finding the sides of a triangle (not only right triangles) when only the area and 2 angles are ...
0
votes
3answers
36 views

Coordinates of two parallel lines knowing their distance

I am trying to draw parallel lines knowing the distance between them. The lines are finite, I know the (x, y) coordinates of their origin and ending points, so I need to calculate, somehow, the (x, y) ...
-4
votes
1answer
41 views

What formulas can I use that can help me solve this? [closed]

A ferris wheel is elevated 1 m above the ground. When the car reaches its highest point on the ferris wheel, its altitude from the ground level is 31 m. How far away from the center, horizontally, ...
0
votes
4answers
74 views

Can someone explain to me why and how $a\cos x+b\sin x$ is the cosine or sine of an angle multiplied by a scalar?

I would like to ask a pair of questions regarding this function. First question: How is this expression derived? Let's consider that $f(x)=a\cos x+b\sin x$. Now, I tried hard to derive the ...
0
votes
1answer
34 views

Trigonometry Mind Blank

Hey guys I'm a bit rusty with my math. I have four coordinates. I have found all the lengths of of distance between the four coordinates however now I want to calculate each of those 8 individual ...
2
votes
2answers
56 views

How to find $\tan x $ from $(a+1)\cos x + (a-1)\sin x=2a+1$?

How do I find $\tan x$ from this equation? $$(a+1)\cos x + (a-1)\sin x=2a+1$$ Thanks for any help!!
4
votes
1answer
56 views

Solving $\frac{1}{2} < \cos \theta < \frac{\sqrt{3}}{2}$

Find the values of theta which satisfy the given condition on a unit circle $$\frac{1}{2} < \cos \theta < \frac{\sqrt{3}}{2}$$ I'm able to plot the points and answer according to me ...
0
votes
1answer
27 views

What's the relation between earth coordinates and angles?

I've been looking for an answer for a specific question, a part of my question maybe related to this: Calculate the angle of a vector in compass (360) direction However, my question is more specific, ...
0
votes
2answers
50 views

Finding the sum of $\cos45°$ + $i\cos135°$ + … + $i^{n}\cos(45+90n)°$ + … + $i^{40}\cos3645°$

My question is as follows: If $i^{2}$ = -1, find the value of $$\cos45° + i\cos135° + \ ...\ + i^{n}\cos(45+90n)° + \ ...\ + i^{40}\cos3645°$$ without the aid of a calculator. In terms of my attempts ...
-2
votes
0answers
11 views

How to draw diagrams on math.stackexchange [migrated]

I am asking this question here because I don't have enough reputation to ask it on meta.math.stackexchange. How do I draw diagrams to answer questions on trigonometry and geometry. I mean something ...
2
votes
3answers
128 views

Evaluate $\cos 36^\circ - \cos 72^\circ$ without the aid of a calculator [duplicate]

I have a quick question about a difficult trigonometric functions problem that I have been assigned. The problem is as follows: Evaluate $$\cos36° - \cos72°$$ without the aid of a calculator. In terms ...
1
vote
1answer
47 views

How do I prove this derivation of a definite integral?

Q,How to prove that $\int_{0 }^{\Pi /2}\sin ^{m}x \cos ^{n}x dx =\left [{(m-1)(m-3)(m-5)...2 or 1}\right ]\left [ \left ( n-1)\left ( n-3 \right )..2 or 1 \right ) \right ]\div \left [ \left ( m+n)(...
1
vote
1answer
84 views

The strange “turning point” of $\arctan(x)/\arctan(\sqrt{x})$

After looking at an interesting graph: $$y=\frac{\arctan(x)}{\arctan(\sqrt x)}$$ There seemed to be a turning point around $(3{,}88;1{,}198)$ (https://www.desmos.com/calculator/58wloddve3) <- A ...
2
votes
5answers
63 views

On the proof $\tan 70°-\tan 20° -2 \tan 40°=4\tan 10°$

I am currently studying in class 10 and I am unable to do this problem. $$\tan 70 ° -\tan 20° -2 \tan 40° =4\tan 10°$$ Can anybody please help me. Thanks!
1
vote
2answers
68 views

Can $\dfrac{\sqrt{3}}{\sin20^{\circ}}$ - $\dfrac{1}{\cos20^{\circ}}$ have two values?

I would like to confirm a solution. The question goes as: Show that $\dfrac{\sqrt{3}}{\sin20^{\circ}}$ - $\dfrac{1}{\cos20^{\circ}}=4$ Firstly I combined the two terms to form something like: $$\...
2
votes
4answers
79 views

Help with the integral $\int x\sqrt{\frac{1-x^2}{1+x^2}}dx$

I would like to know what is $$\int x\sqrt{\frac{1-x^2}{1+x^2}}dx.$$ I put $x=\tan(y)$ to get integral of $\displaystyle \int \frac{\sin(y)}{\cos^3(y)}.\sqrt{\cos(2y)}dy$ I don't know whether $\sin(x)...