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
3answers
75 views

Is there no such identity as $\csc^2+\sec^2=1$?

$$\csc^2+\sec^2=1?$$ I thought I could just use reciprocal from the other formula $\sin^2+\cos^2=1$, can you explain what's wrong?
2
votes
2answers
47 views

What is the symbol to denote that two triangles are similar?

Does there exist a unique symbol to denote that two triangles are similar to each other without resorting to using the phrase "is similar"?
5
votes
3answers
54 views

Trig Equations Using Identities

How would you solve: $2\csc^2x=3\cot^2x-1$ I said: Turn the cosecant to $1+\cot^2~x$. Distribute to get $3=\cot^2~x$. Turn it into tan. To get $\tan x=\frac{1}{\pm \sqrt3}$. Is ...
1
vote
2answers
55 views

What is the difference between these two limits, one with $\lim\limits_{x\to0^{+}}$, the other with $\lim\limits_{x\to 0}$?

I don't need an exact answer, I just need to know how these two limits would affect the answer and if there is a huge difference on how they are worked out, if they have a different step-by-step ...
0
votes
1answer
37 views

Evaluating a Trigonometric Expression involving Periodicity

Evaluate: $$\dfrac{\csc(90+\theta)+\cot(450+\theta)}{\csc(450-\theta)-\tan(180+\theta)}+\dfrac{\tan(180+\theta)+\sec(180-\theta)}{\tan(360-\theta)-\sec(-\theta)}$$ I simplified this into ...
1
vote
4answers
92 views

Simplifying $ 2\cos^2(x)\sin^2(x) + \cos^4(x) + \sin^4(x)$

who can simplify the following term in a simplest way? $$ 2\cos^2(x)\sin^2(x) + \cos^4(x) + \sin^4(x)$$ (The answer is 1). Thanks for any suggestions.
2
votes
3answers
47 views

Is this the correct period?

What is the period for the following: $$ y = 10 \sin\Bigl(\frac{2\pi}{365}(x-50)\Bigr) $$ Is the period $$ \frac{2\pi}{\frac{2\pi}{365}} $$ which would be $365$?
4
votes
6answers
75 views

Solving $6 \cos x - 5 \sin x = 8$

My attempt: Using the formula for linear combinations of sine and cosine: $$A \cos x+B \sin x=C \sin (x+\phi)$$ $$ \sqrt{51} \left(\frac{6}{\sqrt{51}} \cos x - \frac{5}{\sqrt{51}}\sin x\right) = 8 ...
6
votes
3answers
149 views

Evaluating $\lim_{n \to \infty}\frac{n}{2}\sqrt{2-2\cos\left(\frac{360^\circ}{n}\right)}$

I was thinking about different ways of finding $\pi$ and stumbled upon what I'm sure is a very old method: dividing a circle of radius $r$ up into $n$ isosceles triangles each with radial side length ...
2
votes
4answers
44 views

How to find if a point is outside a circle circumference area?

I'd like to know if it's possible to calculate if a point is inside or outside the circle circumference area based on it's $x$ and $y$ values ? Example, $(x, y)= (0.85, -0.9)$ and the radius is $1$
0
votes
1answer
15 views

Find the sum of the radii of inscribed and circumscribed circles for an n-sided in terms of cot

The sum of the radii of inscribed and circumscribed circles for an n-sided regular polygon of side 'a', is (a) $ a.cot(\frac{\pi}{n}) $ (b) $ \frac{a}{2}cot(\frac{\pi}{2n}) $ (c) $ ...
3
votes
2answers
89 views

Integrate area of the shadow?

Today I found an interesting article here. It computes (approximately) area of the shadow. I was wondering what is exact value of the area. My first thought was to use integrals but it doesn't seem ...
2
votes
4answers
101 views

Find the principal solutions of the trigonometric equation $\cos x-\sin x+\sin 2x+3\cos2x+1=0$

I am unable to simplify the expression. If I simplify the double angles, it leaves me with a nasty expression, $\cos x-\sin x+2\sin x\cos x+6\cos^2 x-2=0$. What do I do next. Some hints, please. ...
-1
votes
2answers
57 views

$\Re(e^{-j\theta}A)-\Re(e^{-j\theta}B)\gt0 \Leftrightarrow cos(\theta-\angle(A-B))\gt0$

I have $\cos(\theta cdot (A-B)$. Is it valid to say that $\cos(\theta) \cdot (A-B)= \cos(\theta -\angle{(A-B)})$ and if yes which rule is used το come up with such result? edit: We know that ...
1
vote
3answers
75 views

General Solution of $\sin(mx)+\sin(nx)=0$

Problem: Find the general solution of $$\sin(mx)+\sin(nx)=0$$ My attempt: $$$$ $$\sin(mx)=-\sin(nx)$$ $$=\cos\left(\dfrac{\pi}{2}-mx\right)=\cos\left(\dfrac{\pi}{2}+nx\right)$$ Using ...
3
votes
2answers
47 views

How To Tackle Trigonometric Proofs involving $4$th and $6$th powers?

How do I prove that $\cos^4A - \sin^4A+1=2\cos^2A$ $\cos^6A + \sin^6A =1-3\sin^2A\cdot\cos^2A$ I was going through a very old and very rich book of Plane Trigonometry to build a nice foundation for ...
0
votes
5answers
80 views

Euler's Identity in Degrees

Since we have a simple conversion method for converting from radians to degrees, $\frac{180}{\pi}$ or vice versa, could we apply this to Euler's Identity, $e^{i\pi}=-1$ and traditionally in radians, ...
1
vote
3answers
75 views

simplify and evaluate $\frac{\tan80^\circ-\tan20^\circ}{1+\tan80^\circ\tan20^\circ}$ [closed]

How do you simplify and evaluate $\dfrac{\tan80^\circ-\tan20^\circ}{1+\tan80^\circ\tan20^\circ}$? What is the problem asking?
0
votes
0answers
48 views

Need to solve for an angle, do I need to use numerical methods??

I need to run a simulation on MATLAB where I need to solve two equations for two unknowns. The equations look something like this, $$x_{comp} = G\cdot \cos^2 b \cdot \cos(a+b).$$ $$z_{comp} = ...
-7
votes
3answers
58 views

Find exact values of $\tan(105^\circ)$ and $\tan(11\pi/12)$ without calculator [closed]

How do you find the exact values of the following without using a calculator? $$\tan(105^\circ) \qquad \tan(11\pi/12)$$
2
votes
3answers
73 views

When is $\tan(a+b)$ undefined? [closed]

For what values of $a$ and $b$ is $\tan(a+b)$ undefined? What is the relationship between $a$ and $b$ when it is undefined? What about for $\tan(a-b)$?
1
vote
1answer
38 views

Solve triangle point given base, point height and difference of sides

I have the intuition that one should be able to calculate the position of the circle in the image below (or the equivalent, solve a and b). We have the following information: h and d is known as well ...
0
votes
1answer
41 views

About the convexity of $\sin x$ for $\pi\leq x\leq 2\pi$ [closed]

To prove the convexity of $\sin x$ over $[\pi,2\pi]$ through the second derivative is easy, but I would be interested in a (possibly) simple proof of convexity that avoids derivatives. Can you provide ...
3
votes
4answers
111 views

Find the matrix $\mathbf{A}$ if $A\binom{7}{-1} = \binom{6}{2}.$

Find the $2\times2$ matrix $A$ where $A^2=A$ and $$A\begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.$$ I tried plugging in: $A= ...
2
votes
2answers
51 views

Longest pipe that fits around a corner. [duplicate]

While studying, I came upon the problem "Two corridors of widths $a$ and $b$ intersect at right angle. What is the length of the longest pipe that can be carried across the two corridors, touching the ...
-5
votes
3answers
66 views

Prove that $\sin x =2t/(1+t^2) $ and $\cos x =(1-t^2)/(1+t^2), t=\tan(x/2)$ [closed]

Prove that $\sin x =\dfrac{2t}{1+t^2}$ and $\cos x =\dfrac{1-t^2}{1+t^2}$, where $t=\tan\left(\frac{x}{2}\right)$.
0
votes
6answers
71 views

Find the value of theta so that: $\sin(\theta + 30^\circ ) = \cos 50^\circ$

Can you please explain how to solve this question please, I already have the answer but I do not know the process in achieving it. Find the value(s) of $\theta$ such that: $\sin(\theta + 30^\circ ...
0
votes
4answers
48 views

Prove the trigonometric identities $\cot(x)- \cot(2x) =\csc(2x)$

Prove $\cot(x) - \cot(2x) =\csc(2x)$. I start to solve from LHS, and change all the terms into $\sin$ and $\cos$, but I could not prove it into $\csc(2x)$.
1
vote
3answers
80 views

Solve the trigonometric equation: $\sin {3x} = 4 \sin^2 x$

Solve the equation $\sin{3x} = 4 \sin^2 x$. I tried to change the $\sin{3x}$ to $3\sin x\cos x$ then solve it, but I could not find the correct answer.
3
votes
1answer
48 views

Rotation of complex numbers in a complex plane. Check my work?

Say that $c_1 = -i$ and $c_2 = 3$. For this problem, let $z_0$ be an arbitrary complex number. We can rotate $z_0$ around $c_1$ by $\pi/4$ counterclockwise to get $z_1$. Next, we canrotate $z_1$ ...
1
vote
1answer
75 views

Prove that $(n-1) \sum_1^n \cot(\theta_i) \leq \sum_1^n \tan(\theta_i) $

n is a positive integer and $\theta_i$ is such that $ 0^\circ \leq \theta_i \leq 90^\circ $ for all positive integers $i \leq n$ and $\sum_1^n \cos^2(\theta_i) = 1$. Prove that $(n-1) \sum_1^n ...
0
votes
2answers
19 views

calculating rotation direction between two angles

Consider the following scenario: Say I have a robot positioned at (0,0) and his current angle is 70. I need an algorithm that given two angles - the current angle and the target angle, will give ...
4
votes
0answers
59 views

If $x_1, x_2,…,x_{10}$ are such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} \cos(x_i)$ [duplicate]

Take $x_1, x_2,...,x_{10}$ such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$ with $x_1, x_2,...,x_{10}$ on $\left[0,\frac{\pi}{2}\right]$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} ...
5
votes
4answers
70 views

Find $\frac{d^2y}{dx^2}$ as a function of $x$ if $\sin y+\cos y=x$

Find $\frac{d^2y}{dx^2}$ as a function of $x$ if $\sin y+\cos y=x$ Ok bit confused as my textbook gives the answer to this problem as: $$\frac{d^2y}{dx^2}=\pm\frac{x}{\sqrt{(2-x^2)^3}}$$ So I ...
1
vote
1answer
59 views

Find $a$ when $\sin a= \cfrac{3}{5}$ where $0<a<\cfrac{\pi}{2}$ without a calculator

I have been trying to find $a$ when $\sin a= \cfrac{3}{5}$ where $0<a<\cfrac{\pi}{2}$ by using exact values but I can't seem to find a particular method to evaluate it. My original question is ...
2
votes
0answers
47 views

Prove that a complicated parametric equation is a portion of a circle

In a nuthshell, I would like to prove that some complicated parametric equation is that of a portion of a circle. Let $ P = \left[ \begin{array}{ccc}X \\Y \\Z \end{array} \right] $ be a 3D point ...
4
votes
5answers
106 views

How to solve $\sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6^\circ$

Question: $ \sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6° $ I have partially solved this:- $$ \sin78^\circ-\sin42^\circ +\sin6^\circ-\sin66^\circ $$ $$ ...
0
votes
1answer
33 views

Trouble with series question from STEP past paper

I have answered all parts of this question but the last part. By using the identity, $\cot x - \tan x = 2\cot 2x$ ...
7
votes
2answers
44 views

Find the values of $\cos(\alpha+\beta) $ if the roots of an equation are given in terms of tan

It is given that $ \tan\frac{\alpha}{2} $ and $ \tan\frac{\beta}{2} $ are the zeroes of the equation $ 8x^2-26x+15=0$ then find the value of $\cos(\alpha+\beta$). I attempted to solve this but I ...
2
votes
2answers
68 views

Trig functions of complex numbers

I was studying complex numbers with the help of Boas textbook. I came about certain problems, which I solved only to find that the answers provided in the solution manual to be different. ...
1
vote
1answer
78 views

Inequality problem: $\tanh(\pi x)\sin(\pi x)\geq x^2(1-x^3)$ on $[0,1]$.

How to show that $$\tanh(\pi x)\sin(\pi x)\geq x^2(1-x^3)$$ on $[0,1]$? I tried to expand $\tanh(\pi x)\sin(\pi x)$ in a Taylor expansion: $$\tanh(\pi x)\sin(\pi x)=\pi^2 x^2 - \frac{\pi^4 ...
2
votes
3answers
91 views

Integrating using half angle formula

I am reading through my textbook and there is a part of the solution to an example that I do not understand... $$\int\sin^4x\cos^2x\,dx = \int(\sin^2x)^2\cos^2x\,dx$$ ...
0
votes
2answers
44 views

Find value of $t$ between the difference of 3D vectors.

Hint: The distance between $2$ vectors equals the magnitude of their difference. What is the value of $t$ for which the vector $\mathbf v = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + ...
6
votes
3answers
95 views

Integrating $\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}$ with respect to $\theta$

I'm having some issues with the following integral $$\int_{\frac{-\pi}{2}}^\frac{\pi}{2}\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}d\theta$$ My attempt is as follows, substitute ...
5
votes
5answers
54 views

Length of hypotenuse v/s change in height of the opposite

I have always struggled to understand mathematical concepts, and have a very different way of thinking about problems. I suspect this is a very simple problem, but its confusing me a great deal. I ...
1
vote
2answers
43 views

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n. from the series expansions of $\sin x$ and $\cos x$, I get that $\tan ...
1
vote
1answer
30 views

Solving $\cos(2x+\frac{\pi}{4})= -1/2 $

My suggestion: $$\cos\left(2x+\frac{\pi}{4}\right)= -\frac{1}{2}$$ $$ 2x+\frac{\pi}{4} = \frac{2\pi}{3} \pm 2\pi n, n\in\mathbb{Z}$$ $$ x= \frac{\left( \frac{2\pi}{3} - \frac{\pi}{4} \right)}{2} \pm ...
-2
votes
0answers
45 views

Overlap between two angles

Imagine that we have $4$ line segments in a plane. The starting point of all segments is the same and end point of them could be any arbitrary value . We call the angle between the segments $1,2 ...
2
votes
2answers
69 views

Trigonometric Integrals $\int \frac{1}{1+\sin^2(x)}\mathrm{d}x$ and $\int \frac{1-\tan(x)}{1+\tan(x)} \mathrm{d}x$

Any idea of calculating this two integrals $\int \frac{1}{1+\sin^2(x)}\,dx$ and $\int \frac{1-\tan(x)}{1+\tan(x)} \mathrm{d}x$? I found a solution online for the first one but it requires complex ...
0
votes
3answers
216 views

Trigonometry sine and cos problem

Knowing that $$ \sin a - \sin b = \frac{1}{2} \quad\quad\text{and}\quad\quad \cos a + \cos b = \frac{3}{2} $$ calculate $\cos (a+b)$. I have tried various methods but I can't seem to get ...