Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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136 views

Bound for $\left|\sin(x) +\cos(x)\right|$

I'm taking a numerical analysis class and i'm needing to bound $\left|\sin(x) + \cos(x)\right|$ quite often. So far i've been putting that this is always $\leq |1 + 1| = 2$. Is this the minimal bound? ...
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2answers
174 views

How to integrate $\int\frac{\sqrt{16x^2-9}}{x}dx$?

I have the integer; $\int\frac{\sqrt{16x^2-9}}{x}dx$, and I am having trouble doing the trigonometric substitution. So for integrals in the from of $\sqrt{x^2-a^2}$ where $a$ is a constant is by ...
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1answer
73 views

How to solve this trigonometry equation?

Show that $3(\sin θ - \cos θ)^4 + 6(\sin θ + \cos θ)^2 + 4(\sin^6 θ + \cos^6 θ)$ is independent $θ$. This question was there in one of the cbse sample papers for class x . Tried many methods but ...
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6answers
283 views

Range of a trigonometric function

Question: Prove that: $$0 \leq \frac{1 + \cos\theta}{2 + \sin\theta}\leq \frac{4}{3}$$ I have absolutely no idea how to proceed in this question. Please help me!
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0answers
34 views

Finding the widest angle to shoot a soccer ball from the sideline using optimization!! [duplicate]

I'm trying to do an independent project for my Math class, but I was stuck and couldn't figure out how to use optimization to find position along the sideline that gives the widest angle to shoot. ...
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1answer
58 views

$\renewcommand{\tan}{\sin}\renewcommand{\arctan}{\arcsin}$Simple trig problem

I have this triangle, I need to find the angle at the $C$ corner. What I tried: $$\operatorname{tan} C=\frac23$$ $$C=\operatorname{arctan}\frac23$$ $$C\approx 33.69$$ However this is the wrong ...
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2answers
63 views

Establishing the identity

$$\csc(x)- \cot(x)= \frac{\sin(x)}{ 1+ \cos(x)}$$ I'm completely stumped. There are a few examples with the signs reversed but this is just different enough that none of the examples work. Is this a ...
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1answer
43 views

Finding Angle of Depression

Finding angle of depression but i dont have idea that what i should mark as hyportenus or opposite? Please help!!!! So, here is the question that Chris is standing on the top of the cliff of 70 m and ...
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1answer
76 views

How to solve $x = \arccos(x)$ [duplicate]

How does one solve $$ x = \arccos(x) $$ for $x$? Is there an exact solution achievable by hand? I tried wolfram|alpha, but it only spits out the solution $$ x = \text{ root of }\quad x-\cos^{-1}(x) ...
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1answer
34 views

How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
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71 views

Trigonometrical Solve

There are 2 different values of $ \ \theta \ $. They are $ \ a \ $ and $ \ b \ $, such that $ \ 0 \ < \ a,b \ < \ 360^\circ \ $. If $ \ \sin(\theta+\phi) = \frac{1}{2} \sin2\phi \ $ , prove ...
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61 views

Periodicity of a triginometric function

I have a trigonometric function and I'm interested to know whether or not it has a period. At this stage I'm fairly certain that it is not periodic. However, I don't know how to prove it. Can anyone ...
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2answers
82 views

Intersection of $y=a\sin2x$ and $y=\sin x$

Two graphs $y=a\sin2x$ and $y=\sin x$ are intersecting on $[0,\pi$]. Find a. What I did: $a\sin2x=\sin x$ $\sin x(2a\cos x-1)=0$ $\sin x=0$ (doesn't fit the criteria) $2a\cos x-1=0$ $\cos x= ...
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2answers
63 views

Halving a tiny angle by doubling a side of a triangle.

You have a triangle $ABC$ with a right angle $\angle CAB$. The line stretch between $A$ and $B$ shall be very long in comparison to the other sides. Now you are going to measure the angle $\angle ABC$ ...
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1answer
411 views

Find the value of: tan(100) + tan(125) + tan(100)•tan(125)

The question given was - solve all the equations that are possible to solve without a calculator. (This is a part of it) Please try to solve this question (obviously without a calculator) if it's ...
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2answers
75 views

No. of real solutions of the equation $2 \cos (\frac{x^2 + x}{6}) = 2^x + 2^{-x} $

How many real solutions are there of the equation $2 \cos (\frac{x^2 + x}{6}) = 2^x + 2^{-x} $? Please illustrate.
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2answers
77 views

Inequality: $\tan(x) > 1$

So far, I've not come very... far. It ends up with me trying to solve it more intuitively than mathematically. I figured, first I'll find the place of equality, which is at $x = \arctan 1 = ...
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3answers
214 views

Prove that $\sec^2{\theta}=(4xy)/(x+y)^2$ only when $x=y$

Show that the equation below is only possible when $x=y$ $$ \sec^2{\theta}=\frac{4xy}{(x+y)^2}$$ The only way I can think of doing this is by rewriting it as $$ ...
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1answer
69 views

What is being represented by this 2 images?

image 1 image 2 It's possible that image 1 is showing some kind of methods for building polygons out of trigonometric functions ? It's also possible that image 2 is a quadratic bezier curve ?
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1answer
47 views

Finding the angle

The center of the bottom square rotates so that a line cast from the highlighted corner perpendicular to the right face will intersect with the highlighted corner of the other square. I'm trying to ...
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1answer
80 views

Calculating arbitrary sines/cosines

Can you calculate arbitrary sines/cosines by using angle addition and double angle formulas? I thought that Taylor Series was the standard for calculating the sine of arbitrary angles. What is I mean ...
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4answers
5k views

Calculus: Find the Limit of (f(x+h)-f(x))/h as h approaches 0 for f(x)=cos(2x)

Find the limit: $$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ Given that $f(x) = \cos(2x)$ Tried many ways, but I kept on getting an indeterminate form. I can't find a way to cancel out terms on the ...
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52 views

Determining the location of a point in a triangle under the given constraints

ABC is a triangle with AC = 1, AB = c/b and BC = a/b. Q is a variable point on AC such that CQ = x and QA = 1 – x. The perpendiculars from A and C to BQ (extended if necessary) are $d_2$ and $d_1$ ...
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1answer
367 views

$\sin^2 \alpha + \sin^2 \beta - \cos \gamma < M$ given that the sum of the angles is $\pi$

Question: Find the least real value of $M$ such that the following inequality holds: $$\sin^2 \alpha + \sin^2 \beta - \cos \gamma < M$$ Given that $\alpha, \beta, \gamma \in \mathbb{R}^+$, ...
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1answer
75 views

solving a trigonometric equation 1-cos(180°-x)+[(sin180°+x)/2]=0 .

$1-\cos(180^{\circ}-x)+\left(\frac{sin(180°+x)}{2}\right)=0$ Can someone help me on solvinf it. I did $1-(-\cos x) + \left(\frac{sin(180°+x)}{2}\right)=0 $
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1answer
49 views

$AB$ is a chord of a circle $C$. Let there be another point $P$ on the circumference of the circle, optimize $PA.PB$ and $PA+PB$

$AB$ is a chord of a circle $C$. (a) Find a point $P$ on the circumference of $C$ such that $PA.PB$ is the maximum. (b) Find a point $P$ on the circumference of $C$ which maximizes $PA+PB$. My ...
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1answer
631 views

Coordinate of intersection between line and square

TL;DR given a square and a point $p$, I need the intersection between the perimeter of the square and a ray cast from the center of the square through point $p$. This is my approach so far, but I will ...
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4answers
214 views

Using the Law of Sines to find all triangles with given values of two sides and an angle

Our teacher skimmed over this and we have homework over it. Textbook is mostly unhelpful. I'm confused on how ambiguous case works, and everything I see online just confuses me more. I'm not quite ...
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5answers
241 views

Is $\cos(x^2)$ the same as $\cos^2(x)$?

I want to know something about trigonometrical functions, is $\cos(x^2)$ the same as $\cos^2(x)$ ?
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5answers
83 views

Solving a trigonometric equation: $2 \sin(3a)=\sqrt{2}$

I have the following equation : $2 \sin(3a)=\sqrt{2}$ Not sure how to solve it (Because it's a transformed sin function, meaning 6 solution with 3 cycles in $2\pi$) after a moment I finally found ...
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6answers
511 views

How do you derive this trig identity from the common ones? $\cos^2x=\frac{1+\cos2x}{2}$

$$\cos^2x=\frac{1+\cos2x}{2}$$ Just came across this identity one today. Where does this come from? Is this an easy derivation from the more popular identities, or is this one you just take it at ...
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1answer
198 views

Prove that $\sin{(\pi 2x)}\left(\,\csc{(\pi x)}+\csc{(\pi (0.5-x))}\,\right)$ is an increasing function

Could anybody show that $f(x)=\sin{(\pi 2x)}\left(\,\csc{(\pi x)}+\csc{(\pi (0.5-x))}\,\right)$ is increasing on the interval $x\in[0, 0.25]$? Of course, $\csc{(x)}=\frac{1}{\sin{(x)}}$. Here is ...
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1answer
61 views

How find this value of $\prod_{1\le i<j\le n}(w^i-w^j)^2$

give the positive integer number $n$, and $w=\cos{\dfrac{2\pi}{n}}+i\sin{\dfrac{2\pi}{n}}$ where $i^2=-1$ find the vaule $$\prod_{1\le i<j\le n}(w^i-w^j)^2$$ My try:note $$w^n=1$$ ...
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5answers
362 views

integrate $ \frac {(x^3 + 36)} {(x^2 + 36)}$

I know I have to use long division first, but I don't really know how to do it in this case $$\int \frac{x^3 + 36}{x^2 + 36}dx$$
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1answer
223 views

Tan Binomial formulas from a set S and its k-subsets

Working around, I found some Tan Binomial formulas. Let's $S$ be a set such that: $$ S=\left\{\text{ }\tan ^2\left(\frac{1\pi }{n}\right), \tan^2\left(\frac{2\pi }{n}\right), ...
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1answer
419 views

Is sine of angles greater than 90 degrees a convention?

The sine function is defined as the opposite side of the angle in question over the hypotenuse of the $90^\circ$ triangle. $$\sin(â) = \frac{opposite}{hypotenuse} \tag{$0^\circ<â<90^\circ$}$$ ...
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2answers
98 views

Finding $\cos(A+B)$ from $\sin(A+B)$

The question states to start with the identity: $$ \sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$$ It then asks to use the above to prove that: $$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$$ All I can think ...
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2answers
107 views

Find the magnitdue of a vector along a specific heading

I've gotten some seemingly conflicting answers about this, so I'm hoping someone can help straighten this out for me. Given a vector of components East/West (u) and North/South (v), how do I find the ...
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4answers
1k views

Integral of $\sin^2 \pi x$

Evaluate $$\int_0^{1/4} \sin^2 \pi x \; dx$$ Can someone please explain what to do if theres a power and how to do it in general thanks
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281 views

Determining similarity between paths (sets of ordered coordinates).

With limited knowledge of mathematics, I am not sure what tags to use for this question. I have a path on a 2D surface called $(p1)$. A path consists of a set of ordered $(x,y)$ coordinates. By ...
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1answer
90 views

For the slope of the line at a point, why am I getting a different result by using the calculus method?

I am evaluating the slope of the secant as it approaches $f(30)$ for the function $f(x) = 2\sin(x) - 2$. Using calculus I can easily find that the derivative is $f'(x) = 2\cos(x)$. If I sub in $30$ ...
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2answers
375 views

If $\dfrac{\cos^4\theta}{\cos^2\phi}+\dfrac{\sin^4\theta}{\sin^2\phi}=1$, show $\dfrac{\cos^4\phi}{\cos^2\theta} +\dfrac{\sin^4\phi}{\sin^2\theta}=1$

If $\dfrac{\cos^4\theta}{\cos^2\phi}+\dfrac{\sin^4\theta}{\sin^2\phi}=1$, prove that $\dfrac{\cos^4\phi}{\cos^2\theta} +\dfrac{\sin^4\phi}{\sin^2\theta}=1$. Unable to move further ...request you ...
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2answers
795 views

Figuring the volume of a partially filled cone without the radius of the material inside the cone

Say I have a cone. For simplicities sake, this cone is at the bottom of a storage silo. Is has a flat bottom, flat top and angled sides. I know the height of the material inside the cone but don't ...
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3answers
138 views

How to find $\cos A$ from $\sec$ and $\tan$?

It's given that $$\sec A + \tan A = 4$$ How would you find $\cos A$ from this?
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160 views

Why is this solution to $2\sin^2(x) + \sin(x) - 1 = 0$ incorrect?

$$2\sin^2(x) + \sin(x) - 1 = 0$$ $$\sin(x) = \cos (2x)$$ $$ \sin(x) = \sin (2x + \dfrac{1}{2} \pi)$$ $$ x = -\dfrac{1}{2}\pi - k \cdot 2\pi$$ or $$ x = \dfrac{1}{6} \pi + k \cdot \dfrac{2}{3} ...
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4answers
85 views

Solving a set of 3 Nonlinear Equations

In the following 3 equations: $$ k_1\cos^2(\theta)+k_2\sin^2(\theta) = c_1 $$ $$ 2(k_2-k_1)\cos(\theta)\sin(\theta)=c_2 $$ $$ k_1\sin^2(\theta)+k_2\cos^2(\theta) = c_3 $$ $c_1$, $c_2$ and $c_3$ are ...
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1answer
53 views

inverse trigonometry derivation

Prove that : $sin^{-1}x+sin^{-1}y = sin^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^2}]$ If -1 $\leq x \leq 1; -1 \leq y \leq 1 $ and $x^2+y^2\leq 1$ or if $xy <0 $ and $x^2+y^2 > 1$ solution : Let ...
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1answer
185 views

A finite sum of trigonometric functions

By taking real and imaginary parts in a suitable exponential equation, prove that $$\begin{align*} \frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases} 1&\text{if } k ...
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3answers
287 views

Order of operations in rotation matrix notation.

I'm trying to convert this equation to C# but I'm not a mathematician and I find math notation ambiguous: See the first matrix in this article: http://mathworld.wolfram.com/RotationMatrix.html ...
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1answer
101 views

Trigonometry Identities

Consider a collection of five points evenly spaced around a circle to form a regular pentagon. Assume the figure is scaled so that the sides of the pentagon have length 1. Question: Use Ptolemy’s ...