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
35 views

Inverse Trigonometric Function: Find the Exact Value of $\sin^{-1}\left(\sin\left(\frac{7\pi}{3}\right)\right)$

$$\arcsin\left(\sin\left(\frac{7\pi}{3}\right)\right)$$ I cannot use this formula, correct? $f(f^{-1}(x))=x$ The answer in the book is $\frac{\pi}{3}$ How do I approach solving a problem such as ...
1
vote
1answer
36 views

$\int_{0}^{x_{x>0}} \left(\sin^2(x)\cos(x)\right) \text{d}x=\frac{\text{d}}{\text{d}x}\left(\sin^2(x)\cos(x)\right)$

The original problem is: $$\int_{0}^{x_{x>0}} \left(\sin^2(x)\cos(x)\right) \text{d}x=\frac{\text{d}}{\text{d}x}\left(\sin^2(x)\cos(x)\right)$$ With the work I've done I'm only left to this ...
3
votes
2answers
33 views

Area of shaded region circle help

Find the area of the shaded region Area of the sector is $240^\circ$ or $\frac{4\pi}{3}$ Next find $\frac{b\cdot h}{2}$ which is $\frac{2\cdot2}{2}$ which is $2$. Then subtract the former ...
2
votes
2answers
58 views

a integration with two constants

I am trying to solve below integration $$\int\frac{dx}{(x^2+a^2)(x^2+a^2+b^2)^{\frac12}}$$ I tried substituting $x=a \,tan\,u$. Then I ended up with ...
1
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2answers
16 views

How would I find a point on a sphere with a UV coordinate?

I'd like to do the opposite of the example specified here: https://en.wikipedia.org/wiki/UV_mapping Can somebody explain to me how to do it? Thanks, For any point $P$ on the sphere, calculate $\hat ...
0
votes
3answers
75 views

Show $\tan(x)+\tan(y)+\tan(z) = \tan(x) \tan(y) \tan(z)$ [duplicate]

I am not able to show that: If $x+y+z=\pi$, show that $\tan(x) + \tan(y) + \tan(z) = \tan(x) \tan(y) \tan(z)$.
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3answers
47 views

How to prove the trigonometric identity $\frac{\cot x}{1- \tan x} + \frac{\tan x}{1 - \cot x} - 1 = \sec x \csc x$

I am doing some practice questions for a Math class and I was told that similar questions would be in the exam. So I need to learn this but I have no idea where to even start with this question: ...
0
votes
1answer
36 views

Find triangle $ABC$ satisfies $1+2\sqrt{2}\sin\frac{B}{2}\sin\frac{C}{2}=\cos B+\cos C$

$\color{Red}{\texttt{Find all the triangle ABC}}$ whose angles satisfies $$2\left (1+tan^2\frac{C}{2} \right )\left [ cos^2\left (\frac{13\pi }{2}+\frac{B}{2} \right ...
2
votes
1answer
42 views

Proof of the law of reflection without calculus

I am working on some optimization problems, and I am aware of the method of proving that the "angle of incidence equals the angle of reflection" using Fermat's principle and calculus. However, my ...
1
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2answers
29 views

Intersection between two three-dimensional planes

The intersection of the planes defined by $x \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$ and $x \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$ is a line. Find an equation of ...
0
votes
1answer
23 views

A triangle has an area of $12$ in$^2$, and two of the sides of the triangle have lengths $5 in.$ and $7 in$

A triangle has an area of $12$ in$^2$, and two of the sides of the triangle have lengths $5 in.$ and $7 in$. Find the angle included by these two sides. (Assume the angle is acute) How do I do this?? ...
1
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2answers
76 views

Why does $\int \sin y\;dx = x \sin y + C$?

I've just started to learn calculus on my own. I don't get how $$\int \sin y\;dx = x \sin y + C$$ I've tried to search on Google, but I couldn't find a clear answer.
2
votes
1answer
71 views

How to show that $1/\cosh(x) < \sinh(x)/x < x/\sinh(x)$?

I was going through an old exam paper and I saw this question. How to show that for $0 < x < \pi/2$, $$\frac{1}{\cosh(x)} < \frac{\sin(x)}{x} < \frac{x}{\sinh(x)}\;?$$ I can see ...
0
votes
2answers
41 views

Proof of the identity: $c\sin \frac{A-B}{2} \equiv (a-b) \cos \frac{C}{2}$

Trigs is not my strongest apparently... I need to prove $c\sin \frac{A-B}{2} = (a-b) \cos \frac{C}{2}$ for a general triangle $ABC$. Here is what I do, or rather, here is how I fail at proving it: ...
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2answers
40 views

Prove that the following question based on trignometric identities [closed]

Prove that: $$\frac{1 + \cos A + \sin A}{1 + \cos A - \sin A}=\frac{1 + \sin A}{\cos A}$$
3
votes
3answers
27 views

Alternative area of a triangle formula

The problem is as follows: There is a triangle $ABC$ and I need to show that it's area is: $$\frac{1}{2} c^2 \frac{\sin A \sin B}{\sin (A+B)}$$ Since there is a half in front I decided that ...
1
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0answers
24 views

Cosine formula to show if an angle is obtuse or acute

Keeping in mind the cosine formula: $a^2 = b^2 + c^2 -2bc\cos A$, or rearranging $\displaystyle{\cos A = \frac{b^2+c^2-a^2}{2bc}}$, I need to show when $A$ is acute and when it is obtuse. Consider ...
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2answers
41 views

How to solve trigonometry question [closed]

How do I solve: Solve for $0° \leq \theta \leq 360°$. $$ \tan{\theta} = -\frac{1}{\sqrt{3}} $$
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4answers
49 views

Solve trigonometric inequality $\cos x \geq \sin^2 x - \cos^2 x $

Solve trigonometric inequality $$\cos x \geq \sin^2 x - \cos^2 x $$ My incorrect solution: $$\cos^2 x-\sin^2 x \geq -\cos x $$ $$\cos 2x \geq \cos (\pi - x) $$ which means: $$ 2x \geq -(\pi + x)$$ ...
0
votes
2answers
39 views

Solving the equation $\tan(x)=\cos(x+33.44)$

Please show a method of solving the equation $\tan(x)=\cos(x+33.44)$. I tried several methods (half-angle, cosine of sum, multiply cosines,etc...), but nothing worked. How should one solve such ...
5
votes
3answers
35 views

Solve trigonometric equation $ \cot x + \cos x = 1 + \cot x \cos x $

Solve trigonometric equation: $$ \cot (x) + \cos (x) = 1 + \cot (x) \cos (x) $$ I tried to multiply both sides with $\sin x$ (which I'm not sure if I can multiply with sin).
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votes
1answer
39 views

Writing an expression for a change in angular velocity of an angle

Let $AB$ is rotating at $\omega_{AB}=4$ rad/s. Find $\omega_{CD}$ when $\theta=\pi/6$. So the first thing I did was wrote an express for $CD$ call it $r$. $\phi$ is Angle $CAB$ for reference. By ...
0
votes
1answer
36 views

What's wrong with my version of this integral?

This is the steps to find the correct answer:[][]2 I used trig identities instead and got a different answer. I checked it multiple times, and I'm not sure what I did wrong:
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2answers
64 views

Find the value in the following question [closed]

Find the value of $$\left(\cos \frac{5\pi}{14}-\frac{1}{2016}\right) \left(\cos \frac{15\pi}{14}-\frac{1}{2016}\right) \left(\cos \frac{ 45\pi}{14}-\frac{1}{2016}\right)$$
2
votes
1answer
65 views

Calculate angle on bent bar based on height

I'm writing a small piece of software that shows a preview of a bent rebar. I am however unable to figure out how to calculate the angle so the shape fits within given height $(B)$ requirements. $A, ...
2
votes
1answer
40 views

Equation of the form $tan(\alpha)=cos(\alpha+C)$ where $C\in\mathbb{R}$

I have seen the following math problem posed online by a high school student (knowing their material, most likely it wasn't given as an exercise): Find the solutions for the equation ...
0
votes
1answer
7 views

How many pair solution satisfy both conditions [equation and inequation]

How many pair $(x,y)$ solutions satisfy these conditions: $|\tan(\pi . y)|+\sin^2(\pi x) = 0$ and $x^2+y^2\le 2$? Answer: 9.
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2answers
32 views

Zeros of Trigonometric Equation

I'm studying the function $$ f(x) = \log(x + 1) + \cos(x)/2 $$ The first derivative is: $$ f'(x) = 1/(x + 1) − \sin(x)/2. $$ To find the first two positive critical points (without Wolfram and the ...
0
votes
0answers
20 views

Trigonometric functions on set of co-primes to $n$

E.g. I plotted the function values of $y=\tan(x\frac{\pi}{n})$ for integer $x$-values in the range of $0,\cdots,n$ where $n$ is a given odd integer ( to prevent the case of undefined ...
3
votes
4answers
120 views

If $x\cos(\theta)-\sin(\theta)=1$ then what is the value of $x^2+(1+x^2)\sin(\theta)=1$

The question given is, If $x\cos(\theta)-\sin(\theta)=1$ then find the value of $x^2+(1+x^2)\sin(\theta)$. There are four options given $1$, $-1$, $0$ and $2$. I tried using $\sin^2+\cos^2=1$. I ...
2
votes
3answers
101 views

How do I integrate $\frac {\sin^3x}{\cos^2x}$

How do I integrate $\frac {\sin^{3}x}{\cos^{2}x}$. I have tried to convert to $\tan$, but I could not reach to conclusion. Any help will be appreciated. Thanks.
2
votes
4answers
34 views

Calculating Rotation from centroid

I have a polygon as such: where the green polygon is the rotated polygon and the purple is the extent of the polygon. Is there a way to calculate the angle of rotation of the green polygon from the ...
1
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0answers
32 views

Solve:$\int_{0}^{t}{{\left(\cos({…})+\sin({…})\right)} \frac{\lambda^2 e^{(…)}}{\sqrt{\pi (t-r)}} \text{Erfc}{\left(… \right)} }~\mathrm{d}r$

I have another nasty integral to solve as follow: $$ I(t)=\int_{0}^{t}{{\left(\cos({\frac{\gamma}{4(t-r)}})+\sin({\frac{\gamma}{4(t-r)}})\right)} \frac{{\lambda^2} e^{2 \lambda^2 r+ \lambda ...
8
votes
3answers
966 views

Why I am getting different answer?

I have just started learning single variable calculus. I'm confused in a problem from sometime. I didn't get why my answer is different from the book. $$ \require{cancel} \begin{align} &\int\sin ...
6
votes
2answers
136 views
1
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2answers
31 views

Trigonometric Alternate Form Problem for Electrical 3 Phase Proof

Looking for a strictly trigonometric solution for three-phase systems. Trying to find alternate form for: $$\sin(x)-\sin(x-120^{\circ})$$ From using WolframAlpha for the expansion: ...
0
votes
0answers
48 views

If $A=\frac{\pi}{7},$ Then $\tan A\cdot \tan 2A+\tan 2A\cdot \tan 4A+\tan A\cdot \tan 4A$

The value of $\displaystyle \tan \left(\frac{\pi}{7}\right)\cdot \tan \left(\frac{2\pi}{7}\right)+\tan \left(\frac{2\pi}{7}\right)\cdot \tan \left(\frac{4\pi}{7}\right)+\tan ...
-1
votes
1answer
36 views

Trying to simplify this expression using trig identities. [closed]

Simplify $$\frac{1- \tan^2 x}{\tan x+1}.$$ The answer is $1 - \tan x$ but I can't seem to figure it out. I tried all the identities. (Edited the bottom from tan(x)-1 to tan(x)+1)
3
votes
3answers
40 views

Solve trigonometric inequality $ \sin x \sin 2x - \cos x \cos 2x > \sin 6x $

Solve this trigonometric inequality: $$ \sin x \sin 2x - \cos x \cos 2x > \sin 6x $$ My steps: $$ \cos x \cos 2x - \sin x \sin 2x < - \sin 6x $$ $$ \cos 3x < \sin (-6x)$$ $$ \cos 3x < ...
3
votes
4answers
69 views

Solve trigonometric equation $ 3 \cos x + 2\sin x=1 $

Solve trigonometric equation: $$ 3 \cos x + 2\sin x=1 $$ I tried to substitue $\cos x = \dfrac{1-t^2}{1+t^2}, \sin x = \dfrac{2t}{1+t^2}$. Yet with no results.
2
votes
0answers
56 views

Find the value of a trigonometric expression in terms of a given quantity

Given $\tan^2 2^\circ +\tan^2 4^\circ + \cdots + \tan^2 88^\circ=a$, find the sum $$\sum_{x=1}^{89} \tan^2 x+\cot^2 x$$ in terms of $a$. Since, the given $a$ has only even terms, I am not able to get ...
0
votes
3answers
86 views

Solve trigonometric inequality $ \sin x + \sin^2 x+ \sin^3 x > 0 $ [closed]

I have no idea how to start to solve this trigonometric inequality: $$ \sin x + \sin^2 x+ \sin^3 x > 0 $$
2
votes
1answer
44 views

Help with trig identities to solve an AIME geometry question

Quadrilateral $ABCD$ has side lengths $AB = 20$, $BC = 15$, $CD = 7$, and $AD = 24$, with diagonal length $AC = 25$. If we write $\angle ACB = \alpha$ and $\angle ABD = \beta$, then $\tan (\alpha + ...
-1
votes
1answer
54 views

Find the minimum width of the indicated lane [closed]

A lane runs perpendicular to a road 64 feet wide. If it is just possible to carry a pole 125 feet long from the road into the lane, keeping it horizontal then what will be minimum width of the lane?
1
vote
1answer
33 views

Small angles tangent approximation

Please ignore my scribbles. From the picture we can say: $$\frac{1}{2}r^2\sin\theta < \frac{1}{2}r^2\theta <\frac{1}{2}r^2\tan \theta$$ When we divide out the inequalities we get: $$\sin ...
2
votes
2answers
78 views

Computing the limit of a series involving trigonometric identities

Let $x_n\in(n\pi,(n+1)\pi)$ with $\tan(x_n)=x_n$. Can $$s:=\sum_{n=1}^\infty \frac{1}{x_n^2}$$ be determined explicitly? My ideas so far: First, I wrote $x_n$ as $$x_n= \pi (n+z_n)$$ with ...
2
votes
4answers
101 views

How compute $\cos(5\theta)$ and $\sin(5\theta)$?

I would like to compute $\cos(5\theta)$ and $\sin(5\theta)$. I can use the formula $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ and $\sin(a+b)=\sin(a)\cos(b)+cos(a)\sin(b)$ but it's a little bit to long. ...
2
votes
1answer
84 views

solving integral

How can I solve this integral to get the result as follow: $${\sqrt{\alpha} \over 2\pi} \int_{0}^{t} {1\over \sqrt{r^{3}(t-r)}}[\sin({\alpha\over2r})+\cos({\alpha\over2r})] \mathrm{d}r= ...
6
votes
1answer
103 views

Find $\cos{A}+\cos{B}$

In $\Delta ABC$,if $$\cos{C}\cdot(\sin{A}+\sin{B})=\sin{C}\cdot\cos{(A-B)}$$ Find $\cos{A}+\cos{B}$ Thus ...
3
votes
3answers
62 views

How to solve the trigonometric equation $\sin x-\cos x-2(2)^{\frac 1 2}\sin x\cos x=0$

the question is: Find the solutions of the equation: $\sin x-\cos x-2(2)^{\frac 1 2}\sin x\cos x=0$. Let $\sin x+\cos x=u \text{ and } \sin x \cos x=v \implies \sin^2x+\cos^2x+2\sin x\cos x=u^2 ...