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
48 views

find the coefficient

If $n$ is an odd natural number, and $\sin(n\theta) = \Sigma_{r=0}^{n} b_r \sin^r\theta$, then find $b_r$ in terms of $n$. I have tried this using trigonometric expansion but unable to find solution ...
3
votes
2answers
58 views

If $f(x) = \cos x\cos2x\cos4x\cos8x\cos16x$, then $f’(\pi/4)= ?$

If $f(x) = \cos x\cos2x\cos4x\cos8x\cos16x$, then $f’(\pi/4)= ?$ Ans: $\sqrt{2}$
0
votes
1answer
23 views

converting cos to sin and tan in specific quadrants

I'm having issues understanding as to how to go about doing this. I cant seem to figure out how to find the values of sin and tan in terms of the given cos value in the 3rd quadrant. Thanks with any ...
6
votes
4answers
538 views

How can I simplify this complex number to get a real number?

$$\large \frac {e^{i \frac{\pi a}{2}}[1-e^{i\pi a}]} {[1-e^{i2\pi a}]}$$ I am trying to arrive at $$\frac {1}{2\cos\left(\frac{\pi a}{2}\right)}$$ I've tried dividing top and bottom by one of the ...
0
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1answer
28 views

Get Distance Between Point and Side of Ellipse

So I have an ellipse where I know the two foci, the length and the width and all the relevant information. I then have a point somewhere in the ellipse. This point is known and an arbitrary angle ...
3
votes
1answer
46 views

Prove that $IL,JK$ and angle bisector of angle $BCD$ are concurrent

Given a convex quadrilateral $ABCD$. In $\Delta ABC$, $I$ is the incentre and $J$ is the excentre opposite to vertex $A$. Similarly, $K$ is the incentre and $L$ is the excentre opposite to vertex $A$ ...
0
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2answers
28 views

Given three sides of an isosceles trapezoid, find the smaller base side

I've been surprised at how challenging this problem is. Given an isosceles trapezoid, with the larger base b, the four angles, and the two equal sides c know, find the length of the shorter base a. Is ...
-2
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1answer
32 views

Find hypotenuse interceting points and length to those points from origin [closed]

I need the logic to know the lengths OA and OB from the attached diagram which shows triangle inscribing rectangle. I know only the sides of rectangle and outer rectangle (mentioned in the diagram). ...
1
vote
1answer
25 views

How do we draw the trigonometric function when the y variable is a fraction?

How do we draw the trigonometric function when the y variable is a fraction? For example: $\frac1y=4\cos (\pi x)$ As I am able to draw normal graphs when the $y$ variable is just $y$, but am ...
1
vote
1answer
42 views

Solving an algebraic equation with tangents and sinuses.

I got myself into a rather complicated algebraic equation with multiple instances of tangents and sinuses. While my calculator is able to solve it numerically, I would like to solve it algebraically. ...
3
votes
1answer
41 views

Prove that the ratio of the areas of the triangles $A'B'C'$ and $ABC$ is $2\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

If the bisectors of the angles of a triangle $ABC$ meet the opposite sides in $A',B',C'$,prove that the ratio of the areas of the triangles $A'B'C'$ and $ABC$ is $2\sin \frac{A}{2}\sin \frac{B}{2}\sin ...
1
vote
3answers
49 views

How is that a rotation by an angle θ about the origin can be represented by this transformation matrix?

$$ \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} $$ How was this matrix derived? I know how to use it, but where did it come from? Can someone prove why ...
1
vote
1answer
31 views

How to prove that: $\sin(t)\div\cos(\frac t2) = 2\sin(\frac t2)$

$$\frac{\sin(t)}{\cos(\frac t2)} =2\sin(\frac t2)$$ I'm not really sure how to tackle this. I've tried expressing $\cos(\frac t2)$ as $\sin(\frac \pi2 - \frac t2)$ and $\sin(t)$ as $\sin(2\pi + t)$ ...
1
vote
1answer
34 views

Methods for solving definite trig. integrals?

I am studying Fourier series and there is a lot of integration going on, specifically with trigonometric functions involved. When solving for the Fourier coefficients, often times, the definite ...
1
vote
1answer
50 views

Are these two functions equivalent?

I'm working my way through some 'Graphs of trigonometric functions' on khanacademy.org and came across something that I found to be a little confusing, and I wanted to know if my intuition is correct ...
2
votes
1answer
94 views

Simplify $\sin(2x -x)$

can $\sin(2x - x)$ be simplified to $\sin(x)$, or do I have to use a compound angle formula (with $\cos$ and $\sin$) to do subtraction here? The context. I had this $$ \frac{\sin(2x-x)}{\sin (x) \cos ...
3
votes
1answer
44 views

Solving $a \sin 2x = \sin (x + \gamma)$

I am trying to solve the following equation: $$a \sin 2x = \sin (x + \gamma)$$ or, equivalently: $$2 a = \frac{\cos \gamma}{\cos x} + \frac{\sin \gamma}{\sin x}$$ where $a$ and $\gamma$ are ...
2
votes
0answers
32 views

Characterization of cosine of rational multiples of $\pi$

Given an algebraic number $x$ such that $-1 \leq x \leq 1$ is there a characterization to figure out whether $\cos^{-1}(x)$ is a rational multiple of $\pi$ or not? One characterization would be that ...
4
votes
2answers
66 views

Solve $\cos \frac{4x}{3}=\cos x+1$

Solve the equation \begin{equation} \cos \frac{4x}{3}=\cos x+1\tag 1\end{equation} I had tried by taking $\cos\dfrac x3=t$ and from this we have ...
4
votes
6answers
213 views

$\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$ [duplicate]

How can I show that $\arctan (x) + \arctan(1/x) =\frac{\pi}{2}$? I tried to let $x = \tan(u)$. Then $$ \arctan(\tan(u)) + \arctan(\tan(\frac{\pi}{2} - x)) = \frac{\pi}{2}$$ but it does not ...
2
votes
4answers
45 views

Inequality between altitude and sides in triangle

Let $a,b,c$ be the side lengths and $h_a,h_b,h_c$ the altitudes each connect a vertex to the opposite side and are perpendicular to that side. Then we need to prove ...
1
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2answers
24 views

Angle of elevation and distance to an object

A person walking a straight slope sees (from ground level) an Object across the valley at an angle of $45^{\circ}$, after another 50 meters walking the angle is $60^{\circ}$, How far away is the ...
1
vote
1answer
46 views

What is the meaning of $x=0$ in this trigonometric expression?

Given: $$ \tan(2x) = \tan(2x+20°) $$ The solution should be: $2x = 2x + 20° + 180°k$ But then $2x$ is canceled. My question is: what is the meaning of the expression when there's no $x$ in it?
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0answers
16 views

Derivation/equation for solid angle factor correction

Derivation/equation for solid angle factor correction Summary: I want to determine a correction for the Solid Angle Factor (SAF) due to partially overlapping 'outer' spheres (of different sizes), as ...
1
vote
4answers
34 views

If $a_1+a_2\sin x+a_3\cos x+a_4\sin 2x+a_5\cos 2x=0$ is an identity in $x$,then prove that $(a_1,a_2,a_3,a_4,a_5)=(0,0,0,0,0)$

If $a_1+a_2\sin x+a_3\cos x+a_4\sin 2x+a_5\cos 2x=0$ is an identity in $x$,then prove that $(a_1,a_2,a_3,a_4,a_5)=(0,0,0,0,0)$ I tried:$a_1+a_2\sin x+a_3\cos x+a_4\sin 2x+a_5\cos 2x=0$ $a_1+a_2\sin ...
2
votes
5answers
72 views

Prove $(1+ \tan{A}\tan{2A})\sin{2A} = \tan{2A}$.

Prove the following statement $$ (1+ \tan{A}\tan{2A})\sin{2A} = \tan{2A}. $$ On the left hand side I have put the value of $\tan{2A}$ and have then taken the LCM. I got $\sin{2A}\cos{2A}$. How do I ...
1
vote
1answer
20 views

How to simplify this expression and try to make $m=0$

How to simplify this expression $\cosh(2n s) \cos( 2 m s) = 2 \cosh (w/2) \cos(m s) \sinh(n s)$ I tried to find $m=?$ I try this solution $\frac{\cos( 2 m s)}{\cos(m s)}=2 \cosh ...
3
votes
2answers
78 views

Evaluation of $\int_{0}^1 \frac{1}{x} \log^3{(1-x)}dx =-\frac{\pi^4}{15}$ and $\int_{-\pi}^{\pi} \log(2\cos{\frac{x}{2}}) dx =0$

In the following encyclopedia, http://m.encyclopedia-of-equation.webnode.jp/including-integral/ I found the relations below \begin{eqnarray} \int_{0}^1 \frac{1}{x} \log^3{(1-x)}dx ...
0
votes
1answer
25 views

Using Tan to find the area of a triangle

I have come across a question that I can't seem to figure out. If tanA = 3/4, find the area of the given triangle without using a calculator The given triangle is an scalene triangle with a ...
4
votes
3answers
404 views

Find the longest side of the triangle.

The sides $a,b,c$ of a $\triangle ABC$ are in $GP$ whose common ratio is $\frac{2}{3}$ and the circumradius of the triangle is $6\sqrt{\frac{7}{209}}$.Find the longest side of the triangle. I used ...
1
vote
0answers
85 views

Expressing $\sin{73^\circ}$ in terms of $\sin 5^\circ$, $\sin 49^\circ$, and $\sin 87^\circ$ [closed]

Let $a=\sin{5^\circ},b=\sin{49^\circ},c=\sin{87^\circ}$. Show that $$\sin{73^\circ}=\dfrac{a^2-b^2+ac}{4a(a^2-b^2+ac)-(a-b+c)}$$ Can you explain why this holds? Thanks.
0
votes
1answer
41 views

Finding angular acceleration

Given: $\mu_B=0.52$ $\theta=30^{\circ}$ Weight- $25$ lb $\omega=0$ $l=6$ ft $1/\kappa=3\sqrt 2$ radius of curvature. Find $\alpha$ My Equations of motion are the following: $\xleftarrow{+}\sum ...
-3
votes
0answers
39 views

Trigonometric Question [closed]

If $\cos(x-y)=t$ then which of the following must be $-t$ A. $\cos(y-x)$ B. $\cos(180^\circ-x+y)$ Or C. $\cos(90^\circ+x-y)$
4
votes
2answers
71 views

$a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle

If in a triangle $ABC$,$c$ is the longest side and $a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle. $a^2+b^2=2Rc\Rightarrow ...
7
votes
2answers
44 views

Evaluate $\sum_{r=0}^n \binom{n}{r}\sin rx \cos (n-r)x$

Evaluate $$ \sum_{r=0}^n \left[\binom{n}{r}\cdot\sin rx \cdot \cos (n-r)x\right] $$ I tried to use binomial identities, but since there are trigonometric terms, I don't have the idea ...
1
vote
1answer
23 views

Extending flange on bent bar so horizontal tangent moves to correct place

I'm bending a bar as in the image below, but i need to extend the left flange with x to get the top of the bend (green tangent at the bend) to move up to the other green line. v, u, a, b, d and r is ...
2
votes
3answers
85 views

Finding $\int_0^{\pi/8} x\sin 2x\,dx$

I have a trigonometric equation, when integrated and evaluated should be a specific value. I cannot get that value. The question: $$\int_0^{\pi/8} x\sin 2x\,dx$$ The answer should be ...
2
votes
2answers
42 views

$\lim_{n\to \infty}\frac{S_1+S_2+S_3+…+S_n}{n}=\frac{1}{2}\cot\frac{\theta}{2}$

Let $S_n=\sin \theta+\sin 2\theta+\sin 3\theta+.......+\sin n\theta$.Prove that $\lim_{n\to \infty}\frac{S_1+S_2+S_3+.....+S_n}{n}=\frac{1}{2}\cot\frac{\theta}{2}$ $\lim_{n\to ...
3
votes
3answers
42 views

$\cos \alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+…+\cos(\alpha+(n-1)\beta)=0 $

If each side of a regular polygon of $n$ sides subtend an angle $\alpha$ at the center of the polygon and each exterior angle of the polygon is $\beta$,then prove that $\cos ...
3
votes
3answers
72 views

Given $x\in \left(0; \frac\pi2\right)$. Prove that $\sin x>\frac{2x}{\pi}$

Given $x\in \left(0; \frac\pi2\right)$. Prove that $$\sin x>\frac{2x}{\pi}$$ This is my try: Let $y=\sin x-\frac{2x}\pi\implies y ' = \cos x - \frac2\pi\implies y ''=-\sin x <0; \forall ...
1
vote
2answers
35 views

Convert $(-1.0, 1.0)$ to degrees

I'm trying to convert an analog stick from a game controller into degrees. It gives me a range from $-1$ to $+1$ on the $x$ and $y$ axes. I can get values for $x$ and $y$. If dead right is $0$ ...
0
votes
1answer
36 views

Addition Theorem Simplify

Could you help me use addition theorem formulas to express the following trig. function in terms of $t$ when $\tan (a/2)=t$ ($t$ cannot $=1$) $1$) $\cos a$ so far I have $\tan^2 (a/2)=\dfrac{1-\cos ...
2
votes
1answer
50 views

A circle's sine wave is an ellipse's…

We all know what a sine wave is, and how it relates to a circle. What is the vertical and horizontal distance when I take a point and drag it along the perimeter of the ellipse? It definitely has to ...
7
votes
5answers
130 views

Find $\int_0^\frac{\pi}{2} \frac {\theta \cos \theta } { \sin \theta + \sin ^ 3 \theta }\:d\theta$

My Calc 2 teacher wasn't able to solve this: $$\int_0^\frac{\pi}{2} \frac {\theta \cos \theta } { \sin \theta + \sin ^ 3 \theta }\:d\theta$$ Can someone help me solve this?
8
votes
1answer
241 views

The case of Captain America's shield: a variation of Alhazen's Billard problem

I'm sure a lot of you are acquainted with Alhazen's Billiard problem, which involves finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in ...
1
vote
1answer
50 views

Is the count for graphing trigonometric functions always 1/4 of a period? [closed]

I saw this video in YouTube since I'm studying for a quiz and found out that the count used for graphing trigonometric functions is 1/4 of a period. Is it always like that?? Sine, cosine, tangent, ...
1
vote
1answer
53 views

Trig limit in Spivak's Calculus

$$\lim_{x\rightarrow 1} (x-1)^3 \sin\frac{1}{(1-x)^3} = 0$$ To prove that this is true, the chapter on limits has things like $\lim_{x\rightarrow a}(f\cdot g)(x) = \lim_{x\rightarrow a}f(x)\cdot ...
0
votes
1answer
29 views

Mind refresher on a few simple algebra-geometry problems

I feel silly for asking this, but I've completely forgotten some steps on how to do a few of these simple algebra/geometry problems. 1) Simplify $\sqrt{18x}-4\sqrt{x^3}$. I rearranged this to ...
1
vote
2answers
33 views

How to express $\phi$ in terms of $R\text{, }x\text{ and }\theta$

Let $S$ be a circle with radius $R$ and center at $O$. Let $P$ be any arbitrary point inside circle such that its distance from $O$ is $x$ and the ray $\overrightarrow{OP}$ cuts the circle $S$ at ...
-1
votes
5answers
58 views

Trigonometric equation. [closed]

Please help in Solving the Trigonometric Equation: $$\cos^2x - \sin^2x = \cos3x$$