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
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1answer
35 views

Geometry formulas, how to show identities.

Given $d$ is integer: How do I show: $$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$ How do I rewrite and show, for $k$ is an integer: $$ ...
0
votes
1answer
28 views

Given only angles and area of triangle, find side length.

The area of a triangle is $60$ square inches. Find the length of the side included between $A = 25°$ and $C = 110°$. (Round your answer to one decimal place.)
-4
votes
3answers
61 views

Prove that $\sin^2 \theta + \sin^2 \beta= \sin(\theta + \beta)$ when $\theta+\beta = 90^\circ$ [on hold]

If $\theta, \beta$ are two acute angles prove that : $$\sin^2 \theta + \sin^2 \beta= \sin(\theta + \beta) $$ when $\theta, \beta$ are complementary angles, i.e. $\theta + \beta = 90°$. My try... ...
0
votes
1answer
25 views

Peripendicular Line at distance d from point in a given direction

I have a line given by $Ax + By + C= 0$, and a point $x_0,y_0$. From that point $x_0,y_0$ in the direction of the line up to distance $d$, I want to find the equation of the line that is perpendicular ...
5
votes
2answers
51 views

Relationship between $\sin(a+b)$ and derivative product rule?

I noticed this interesting correlation between the sine angle addition formula and the derivative product rule. The sine addition formula is $$\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$$ The ...
-1
votes
2answers
54 views

How do I find the solution(s) to $\cos^2x=2-2\sin x$?

In the interval of $[0,2\pi)$, how do I find the solutions to $$\cos^2x=2-2\sin x$$ I know that $\cos^2x=1-\sin^2x$, but don't see how to advantageously use this.
0
votes
3answers
68 views

Definite integral with the squared cosine under the square root

I can't solve this $$\int_{0}^{5}{\sqrt{1+\left(\dfrac{\pi}{2}\cos(10 \pi x)\right)^2}dx}$$ My approach: If $10\pi x =u \to 10\pi dx=du$, so ...
0
votes
4answers
29 views

Points $A$, $B$, and $C$ are on the circumference of a circle with radius 2

Points $A$, $B$, and $C$ are on the circumference of a circle with radius $2$ such that $\angle BAC = 45^\circ$ and $\angle ACB = 60^\circ$. Find the area of $\triangle ABC$. I've drawn a circle ...
-3
votes
0answers
20 views

Trigonometric problem with two angles [on hold]

$$(a+b) \tan( \theta -\phi) = (a-b) \tan( \theta +\phi)$$ and $$a \cos \phi + b \cos 2\theta = c$$ Prove $a^2 -b^2 +c^2 = 2ac \cos 2\phi$
13
votes
4answers
150 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
1
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3answers
46 views

Find the number of solutions of the trigonometric equation in $(0,\pi)$

Find the number of solutions of the equation $$\sec x+\csc x=\sqrt {15}$$ in $(0,\pi)$. The question is easy. But when you solve, you get would get $4$ as the answer. I am sure the method gives $4$ as ...
0
votes
0answers
18 views

How to calculate the radius of a circle which must have a number of nodes at its ends [on hold]

Hi I am trying to create a text wheel very similar to this. here's my image The large ring is made up of circular nodes each 80 units in diameter. How can I calculate the radius of the large circle ...
0
votes
3answers
42 views

differential equation with substituion

Solve for y: $y'\tan(x+y)=1-\tan(x+y)$ so far I have made the substituion $u=x+y$, which yields $\frac{du}{dx}=1+\frac{dy}{dx}$. However, I am not sure what to do from here.
-1
votes
4answers
75 views

If $\cos x = \frac{3}{7}$, then $\sin\frac{x}{2} = ?$ [on hold]

If $\cos x=\frac{3}{7}$, then find $\sin\frac{x}{2}$. I tried everything, but it seems I'm stuck forever in this problem.
2
votes
2answers
42 views

How do I properly read a clinometer?

If the weight hangs down at roughly 42 degrees, would the angle be 90 degrees - 42 degrees = 48 degrees?
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votes
2answers
32 views

Find the rang of $\sin (a) + \sin (b)$ [on hold]

If : $a+b=\frac{\pi }{2}$, Find the range of $$\sin (a) + \sin (b)$$
4
votes
2answers
83 views

Show that in any triangle, we have $\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$

Show that in any triangle, we have $$\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$$ where $R$ is the circumradius of the triangle. Here is my work: ...
0
votes
0answers
17 views

Figure out the component of a value in X and Y coordinates using trigonometry.

Alright. It's been long that I studied trigonometry and did Laws of Motion and Free Body Diagrams, and I was decent good at them, but somehow I am having trouble in understanding the following. Note ...
0
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0answers
43 views

Is it possible to define an inverse of the main three trig. functions without domain restrictions?

Ok, I know that the main three main trigonometric functions, that is the tangent, sine, and cosine, are periodic and thus not one-to-one, but onto. And, since an inverse requires a function to be onto ...
1
vote
3answers
222 views

A confusion in a calculation with complex numbers

Consider the followings: $$ 1+e^{ix}+e^{2ix}+e^{3ix}= \dfrac{1-e^{4ix}}{1-e^{ix}} $$ Then, we take absolute square to the both sides $$ |1+e^{ix}+e^{2ix}+e^{3ix}|^{2}= \dfrac{1-\cos4x}{1-\cos x} $$ ...
1
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2answers
33 views

Prove that $\tan \left ( \sum_{k=1}^{n} \theta_k \right ) \geq \sum_{k=1}^{n} \tan (\theta_k)$

I'm trying to prove by induction that $$\tan \left ( \sum_{k=1}^{n} \theta_k \right ) \geq \sum_{k=1}^{n} \tan (\theta_k)$$ provided that $$\sum_{k=1}^{n} \theta_k < \frac{\pi}{2}$$ So in ...
7
votes
0answers
41 views

Why do people prefer cosine to sine when speaking of harmonic oscillation?

In almost all of the physics textbooks I have ever read, the author will write the oscillating function as $$x(t)=\cos\left(\omega t+\phi\right)$$ My question is that, is there any practical or ...
2
votes
1answer
38 views

Help with Definite integral question

Anyone please help with this question: (a) Show that: \begin{align} \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx \end{align} (b) Hence show that: \begin{align} \int_{0}^{\frac{\pi}{4}} ...
6
votes
3answers
110 views

Is it true that $\sin x > \frac x{\sqrt {x^2+1}} , \forall x \in (0, \frac {\pi}2)$?

Is it true that $$\sin x > \dfrac x{\sqrt {x^2+1}} , \forall x \in \left(0, \dfrac {\pi}2\right)$$ (I tried differentiating , but it's not coming , please help)
0
votes
0answers
15 views

Find marginal distribution (Integral Solution)

I have derived bivariate exponential distribution in term of polar coordinate system. Now I need to derive marginal distribution of $f(\theta)$ from joint $f(r,\theta)$ for this we have to eliminate ...
1
vote
1answer
29 views

How high above sea level do your eyes have to be to see a point that is 4.1 miles away “as the crow flies”?

There's a fireworks show going on tonight at a little town that's 4.1 miles away from my house, and I want to watch it from a hill near my house. So I thought I'd set up a simple geometry problem to ...
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votes
2answers
70 views

Express the number $4$ and $5$ and $6$ and $7$ and $8$ [on hold]

Express the number $4$ and $5$ and $6$ and $7$ and $8$ with trigonometric identities or series or equations. example: Express the number $1$, $$\cos^2 x + \sin^2 x=1$$ Express the number $2$, ...
0
votes
1answer
69 views

Why does the following limit give two answers?

I want to calculate $$ \lim_{t \to 0} \frac{t^2}{\sin^2(t)}$$ and I proceed as follows $$\stackrel{H}{=} \lim_{t \to 0} \frac{2t}{2\sin(t)\cos(t)} \implies \lim_{t \to 0} \frac{2t}{\sin(2t)}$$ ...
-1
votes
2answers
63 views

Six variables. System of equations.

$$ \begin{align} x & =\frac{R+\frac{G+B}{-2}}{R+G+B} \\[10pt] y & =\frac{\frac{(G-B) \sqrt{3}}{2}}{R+G+B} \\[10pt] z & =R+G+B \end{align} $$ How do I get the formula for ...
2
votes
3answers
131 views

How do i evaluate this integral $ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $?

Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$ Note :By mathematica,the result is : $\frac{Gamma\left(\frac1 ...
1
vote
1answer
29 views

Find the density

Suppose that radius $R$ of one sphere is a continuous random variable with density $$f_R(r)=6r(1-r) I_{[0,1]}(r)$$ Find $f_V(v)$ and $f_S(s)$ the densities of volume and surface area I did ...
0
votes
2answers
18 views

Problems identifying harmonic motion

Not sure why I am having so much trouble with this. I have a function f(t) = -cos(t) + 3sin(t-pi/6). I am trying to find the amplitude, period, and phase angle. But, I am under the impression that ...
2
votes
3answers
65 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
8
votes
2answers
121 views

Does $\tan (x)$ equal $\frac{-1}{x-\frac{\pi}{2}}+\frac{-1}{x+\frac{\pi}{2}}+\frac{-1}{x-\frac{3\pi}{2}}+\frac{-1}{x+\frac{3\pi}{2}}+…$?

I set my Year 12 students a question involving the sums of rational functions $\frac{1}{x-n}$. The graph of a sum of these functions looks an awful lot like a tan graph. This led me to ask: Does ...
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votes
2answers
52 views

general solution to trigonometric equation, help!!! [on hold]

if $$\sin\left(\frac {π}{4} \cot\theta\right)=\cos\left(\fracπ4\tan\theta\right)$$ then find general solution of $\theta$
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votes
2answers
82 views

How to evaluate $\int \frac{\mathrm dx}{1+\sin x−\cos x} $?

Is there someone show me how I evaluate this integral:$$\int\frac{\mathrm{d}x}{1+\sin x−\cos x} $$ I used $t=\tan\frac{x}{2}$ but i didn't succeed . Thank you for any help .
1
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1answer
39 views
+50

How to find a real function from a complex function.

I have the complex function $z\left(n\right) = i^{n} = \cos\left(\theta\left(n\right)\right) + i \sin\left(\theta\left(n\right)\right), \theta\left(n\right) = \frac{n \pi}{2},$ and I know that, on an ...
1
vote
6answers
57 views

Does the equation $2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$ have real solution?

Do the equation $$2\cos^2 (x/2) \sin^2 (x/2) = x^2+\frac{1}{x^2}$$ have any real solutions? Please help. This is an IITJEE question. Here $x$ is an acute angle. I cannot even start to attempt ...
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votes
0answers
33 views

Questions of multiple angles [on hold]

$2\sin A/\cos3 A+2\sin3A/\cos9A+2\sin9A/\cos27A= \tan27A-tanA $
1
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2answers
28 views

How are arc components of a spherical system derived?

I am studying a flight dynamics book (see Flight Dynamics by Stengel) and am rusty on spherical coordinates. Commonly, aerospace coordinates use a North/East/Down right-hand system. So $z=-h$, ...
3
votes
4answers
147 views

Is integration of $x\operatorname{cosec}(x)$ defined?

Is integration of $x\operatorname{cosec}(x)$ possible? If yes, then what is its closed form; if not, then why is it non-integrable ?
1
vote
1answer
24 views

Find the measurement of line BD

So I was trying to find the measurement of $BD$ I drew green lines to make myself some angles, the measurement $3$ is from the point A to C, If only I can line $AE$ or $CE$ then I will just use the ...
2
votes
1answer
109 views

Trigonometric ratio of multiple and sub multiple angles

Given that $a$ lies in 1st quadrant and $$ \sin a +\cos a +\operatorname{cosec} a+\sec a+\tan a+\cot a=7$$ then we have to prove that $\sin(2a)$ is a root of $$x^2-44x-36.$$ I have tried to break all ...
0
votes
4answers
57 views

trigonometry expression simplification with inverse cosine

While working on a problem, I ended up with this expression for y: $$ y=x\sin\left(\arccos\left(\frac{\sqrt{x^2-y^2}}x\right)\right) $$ Is there any way to express $y$ in terms of $x$ only, with no ...
0
votes
2answers
38 views

What angle does the board need to be cut at?

If someone has a 2'' wide board and a 1 1/2'' wide board, and they want to cut the narrower board at an angle so the cut is 2'' long and the boards will fit together, what angle do they need to cut ...
-2
votes
2answers
41 views

If a 16' ladder is placed correctly on a level surface, how high up will the ladder reach?

So i have just began learning about sin cos and tan, and i came across this problem and for some reason I'm having trouble figuring it out. *** When using a straight ladder, it is recommended that ...
1
vote
2answers
29 views

How to scale a 2D vector and keep direction

I want to take any vector in R2 and scale its length to 1 while keeping the original direction (ratio of x component to y). As an example of my goal, let's say I have the vector (1,1), it would become ...
0
votes
1answer
33 views

Do I need to use different trig functions in different quadrants?

I don't have any formal education in Trigonometry or Calculus, but I'm studying a book on Pre-calc before school begins this fall. I've completed College level Algebra too, so math isn't something ...
1
vote
2answers
31 views

Equation with sine and cosine - coefficients

I have some trouble with the conceptual understanding of the way we solve this kind of equations. Let's say we have: $$(3-3b^2)\sin(bx)+3a\cos(2x)=6\cos(2x)$$ The method employed on classes was ...
3
votes
2answers
52 views

Explanation of derivation made at wikipedia.

in this wikipedia article A deriviation to convert true and eccentric anomaly. I am however quite stunned by a single line - trying to reproduce but after half a dozen sheets of paper I can't find how ...