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
31 views

how to find the value of the trigonometric function in the question

$$\text{if }\frac{\sin\theta}{\sin\phi}=\frac12 \text{ , }\frac{\cos\theta}{\cos\phi}=\frac32 \text{ ; if both the angles are the acute angle, then find } \tan\theta \text{ and } \tan\phi.$$ this ...
2
votes
8answers
83 views

Prove the trigonometric identity $\cos(x) + \sin(x)\tan(\frac{x}{2}) = 1$

While solving an equation i came up with the identity $\cos(x) + \sin(x)\tan(\frac{x}{2}) = 1$. Prove whether this is really true or not. I can add that $$\tan\left(\frac{x}{2}\right) = ...
1
vote
1answer
37 views

Dividing a trigonometric expression

Given: $$\sin {x} ⋅ \cos {3x} = \sin {x} ⋅ 2\sin {3x} ⋅ \cos {3x}$$ Can I divide by $\sin {x} ⋅ \cos {3x}$ ? If I check $\sin {x} ⋅ \cos {3x} = 0$ I get 2 more answers that are correct to the ...
0
votes
1answer
42 views

I apply the sum-to-product identity for $\sin$, but my result differs from the textbook's

I don't understand the last transformation here: $$\sin x - \cos 3x = 0\iff \sin x -\sin\left(\frac\pi2 - 3x\right) =0\iff 2\sin\left(\frac\pi4-x\right)\cos\left(2x-\frac\pi4\right)=0$$ When I apply ...
-1
votes
2answers
42 views

Simplify $\tan3x/\tan x$. Answer given is $(2\sin 2x +1)/(2\sin 2x-1)$ [closed]

The question is to simplify $\displaystyle \frac{\tan{3x}}{\tan x}$. The answer given in my book is $\displaystyle \frac{2\sin 2x+1}{2\sin 2x- 1}$ but I am not getting this answer by solving it. Can ...
1
vote
1answer
33 views

Inverse sum representation of sine

The other day I was playing with functions of the form $$ f(x) = \frac{1}{\frac{1}{a_0(x-b_0)} + \frac{1}{a_1(x-b_1)} + \cdots + \frac{1}{a_n(x-b_n)}} $$ and I found particularly that $$ ...
1
vote
4answers
63 views

Trignometric Identities and Equations

For the following problem(s) I cannot get any answer(s). I would appreciate your help very much. $$\tan { \theta -\sec { \theta } =\sqrt { 3 } } $$ TI get 30 degrees as the reference angle. What ...
0
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2answers
42 views

Infinite series of trigonometric ratios

The question is to compute: $$(1+\cos A)+2(1+\cos A)^2 + 3(1+\cos A)^3+\ldots = \sum_{k=1}^{\infty}k(1+\cos A)^k.$$ I tried by setting $1+\cos A=y$, then the serie becomes $$y+2y^2+3y^3+\ldots = ...
2
votes
1answer
43 views

Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$.

Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$. Can I use Taylor's polynomial?
2
votes
1answer
47 views

Gaussian function in the limit of trigonometric functions

I've noticed that $$ (\sin\theta \cos\phi)^{2n} + (\sin\theta \cos\phi)^{2n-1} $$ increasingly resembles a Gaussian function of $(\theta, \phi)$ as $n$ goes to infinity. In particular, when I take ...
0
votes
0answers
15 views

Trigonometric identities for Bessel Functions?

I'm wondering if there exists extensions of trigonometric identities to special functions like Bessel? For example, is there an alternative way to express the following? $J_0((a+b)x) = ?$ Thanks
3
votes
1answer
46 views

Should cosecant be defined as $\csc \theta = \frac{1}{\sin \theta}$, specifying the constraint: $\sin \theta \neq 0$?

I'm studying trigonometry on my own, and I keep noticing that the trigonometric functions are never defined with constraints to deal with divide-by-zero issues. As an example, I've seen cosecant ...
2
votes
2answers
71 views

Prove $\frac{\sin (2016x)}{2016x}<\frac{\sin x}{x}$

How can we prove that $$\frac{\sin (2016x)}{2016x}<\frac{\sin x}{x}$$ with $x$ close to/near $0$ I don't know where to start from, but I think that we need to examine distinct cases of $x>0$ ...
1
vote
3answers
40 views

If $f’(x) = \sin x + (\sin4x)(\cos x)$, then $f’(2x^2 + \pi/2) $is?

If $$f'(x) = \sin x + \sin4x \cdot \cos x,$$ then $$f'(2x^2 + \pi/2)$$ is? Given answer: $$4x\cos(2x^2) – 4x\sin(8x^2) \sin(2x^2)$$ I tried and I'm getting the answer as $\cos(2x^2) - ...
2
votes
2answers
70 views

Solving $\cos^2{\theta}-\sin{\theta} = 1$

Can someone please help me solve this? $$\cos^2{\theta}-\sin{\theta} = 1, \quad\theta\in[0^\circ, 360^\circ]$$
0
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2answers
95 views

How to go about proving that $\cos(\frac{\pi}{2}-x) = \sin(x)$?

I have very little experience writing proofs so I don't know how to begin. I recognize that the statement is always true, but I can't go about proving it without using circular reasoning. How could ...
1
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2answers
37 views

Working out the length of the 3rd side of an isosceles triangle- Pythagoras' theorem

I have been revising some maths equations and see that you can work out the third side of an isosceles triangle using the formula $\sqrt2 x$ $x$ being one of the equal sides. Could someone explain ...
0
votes
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
56 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
22 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
534 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
votes
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
45 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
votes
2answers
23 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 ...
-1
votes
2answers
24 views

Straight Lines and Trigonometry [closed]

If the line segments joining the points A$(a,b)$ and B$(c,d)$ subtends an angle $x$ at the origin, then $\cos x= ?$
-2
votes
1answer
32 views

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

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
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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 ...
-6
votes
0answers
52 views

math problem question [closed]

If $$\frac{\tan(A-B)}{\tan A}+\frac{\sin^2C}{\sin^2A}=1$$ then prove that $$\tan A\cdot\tan B=\tan C$$
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
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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 ...
-2
votes
0answers
54 views

What does Mathematica show in contrast to WolframAlpha online? [closed]

If I enter $$tan((4*1-1)*pi/(4*5))$$ in the search field of WolframAlpha online I get under "Alternate forms" an expresion in radicals $$-1+\sqrt{5}-\sqrt{5-2 \sqrt{5}}$$ which is great. If I enter ...
1
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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
93 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 ...
1
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0answers
30 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
65 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
212 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
vote
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
45 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?
1
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0answers
15 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
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4answers
32 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
403 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 ...