Questions tagged [trigonometry]

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

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0answers
13 views

Finding missing coordinate of a rectangle given an angle

The vertices of a right triangle are A (1,3,1), B(5,2,1), C(2,3,k). Angle A is 90 degrees. (a) Find k. (b) Find the angle C rounded to the nearest degree.
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4answers
45 views

If $a\sin(x) +b\sin(2x)+c\sin(3x)=0$ for every real $x$, then $a=b=c=0$.

How to solve the following without calculus? In a trigonometric manner. If $$a\sin(x) +b\sin(2x)+c\sin(3x)=0$$ for every real $x$, then $a=b=c=0$. I solved the problem by integrating and using the ...
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1answer
65 views

How can one simplify $\sum_{n=1}^8\frac{\sin10n^\circ}{\cos5^\circ\cos10^\circ\cos20^\circ}$?

I have trouble solving the following question: Consider the following expression: $$\sum_{n=1}^8\frac{\sin10n^\circ}{\cos5^\circ\cos10^\circ\cos20^\circ}$$ The value of the above expression can be ...
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2answers
47 views

Why do we use the unit circle to solve for sin and cos

I know that in a unit circle where the radius is always one, sin is equal to y and cos is equal to x. But why do we use these values even when the radius or the hypothenuse of the triangle isn't equal ...
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1answer
50 views

Why doesn't this approximation work?

$$\cos{\left(\frac{2\pi}{9}\right)}$$ $$f(x) = \cos{(2\pi x)}$$ $$x = a + h$$ $$x = 0 + \frac{1}{9}$$ $$f(a+h) \quad\text{approximately is :} \quad \cos(0)-\sin(0)\cdot 2\pi \cdot \frac{1}{9}\quad \...
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1answer
43 views

Is there a more precise approximation to $\frac{\sin(\pi y/360)}{\sin(\pi x/360)}$ than $\sqrt[90^2]{\frac{\pi}{2\sqrt2}}^{(x^2-y^2)}\frac{y}{x}$?

If we have $s$ semicircle with the diameter $AB$ (with length $1$) and the center $O$, then we can approximate $$\frac{\operatorname{chord} AC}{\operatorname{chord} AD} \;=\; \sqrt[90^2]{\frac{{\pi}}{...
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2answers
64 views

$\tan \theta = -1 \pm \sqrt 6$ . What is the value of $\theta$ in degree?

I have got this : $\tan \theta = -1 \pm \sqrt 6$ in a problem. I have to find the value of $\theta$ in degree. Is there anyway we can find the angle in degree without using calculator ? I have no ...
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2answers
49 views

How do I find the length attained by a bullet when fired by a rifle?

The problem is as follows: The horizontal range of a bullet fired by a rifle from a certain height above sea level is given by the minimum value of the function presented below: $$f(x)=2\sec\left(\pi ...
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1answer
67 views

Determining the period of $ 3 \cos (A-B)x\;+\cos (2 A -B )x+3 \cos (A x)+2 \cos (B x)+3 $ when $\frac{A}{B}$ is (ir)rational?

I have this one-variable function $$3 \cos (A-B)x\;+\cos (2 A-B)x+3 \cos (A x)+2 \cos (B x)+3 $$ for $x>0$ where $A,B>0$ and $B>A$ are constants. How can I determine the period in both ...
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1answer
30 views

Find the value of θ if $\sinθ° = \cos(θ+40)°$

This is a $1$ mark question, so I assume I may be going about it wrong or in a longer than required way. Currently I am using $\cos(t) = \sin\left(\left(\frac{\pi}{2}\right) - t\right) $and then using ...
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0answers
21 views

Solving $2\tan(2x) = 3\cot(x)$ [duplicate]

Consider the equation $$2\tan(2x) = 3\cot(x)$$ What are the $x$ values (in degrees) that satisfy the equation for $0^\circ \leq x \leq 180^\circ$? When I try to solve it using the identity of $\tan(...
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1answer
44 views

The logarithmic form of $\text{arcsin}(x)$ and its implications

Background I recently set out to derive the exponential forms of the inverse trigonemtric functions using eulers identity and demoivres theorem, deciding to start with $arcsin(x)$ I first got that: $$...
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0answers
62 views

Determine $k$ such that:$\lim\limits_{n\to\infty}(n(2kn+1)\frac{\pi}{2}-nx_n)=\frac{1}{2005\pi}$

Let $x_n$ be a solution of the equation: $\tan{x}=x$ from the interval $\left((2kn-1)\frac{\pi}{2},(2kn+1)\frac{\pi}{2}\right)$, where $k\in\mathbb{N}$. Determine $k$ such that:$$\lim\limits_{n\to\...
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1answer
34 views

Finding proof for trigonometric ratio of 90+theta

When finding trigonometric ratios for 90+theta. Why don’t we make the diagram like this ? I see we are not getting an angle of 90 degree. So , can’t we say there is no values of trigonometric ratio ...
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20 views

How do we prove to say the value of trigonometric angles

We say $ cos 0 ^ 0$ = 1 and sin 0 degree = 0. Why it it like that ? What is the proof for it? I’m not able to get it. I thought of using the formula x=r$\theta$. But not understand what to put. If cos ...
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1answer
18 views

How do we find the signs of trigonometric ratios

This is the image my textbook uses for finding whether the ratios are +ve or -ve. Here , i feel they mean to say that tan (+x) = tan x since b and a are +ve . We can tell it by looking at the x and y ...
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1answer
27 views

If line PQ is tangent to circle R at point Q, and line PS is tangent to ⊙R at point S, what is the perimeter of quadrilateral PQRS?

Assuming that 4 refers to line $T$, would that not make this question impossible since $T$ is the hypotenuse of $\overline{PQ}$ and $\overline{RQ}$? If $4$ were to be the length of $\overline{PQ}$, ...
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3answers
81 views

Why is the definite integral of $\int_{1}^{\infty} \frac {\ln (1+x^2)}{x^2}$ equal $\frac{\pi}{2} + \ln(2)$?

I currently have a question on why when you partially integrate the definite integral. $$\int_{1}^{\infty} \frac {\ln (1+x^2)}{x^2} = \frac{\pi}{2} + \ln(2) $$ I obtained this result via plugging it ...
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1answer
38 views

Find $\cos(t)$ if $\sin(t)=\frac{x}{x+1}$

I know that this is a very elementary trigonometry question, but for some reason I can't understand what it is asking. Any advice is appreciated.
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1answer
50 views

Solving $x = \frac13\sec(a)$ for $a$ (after integrating via trig substitution)

For the integral $\dfrac{1}{x\sqrt{9x^2-1}}$, I decided to use $x = \frac13\sec(a)$. The expression simplifies to the integral of $1$, simply becoming $a$. The issue is making $a$ the subject of $x = \...
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1answer
44 views

How do I construct this in Latex? [closed]

I need to find out the easiest way to draw this triangle using tikz. I have tried different approaches from my colleagues but it seems that I am not getting an acceptable drawing
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1answer
64 views

What are the circular-to-hyperbolic trig identities (eg, $\cos(ix)=\cosh(x)$) trying to tell me?

I have been wondering about the utility of $$\begin{align} \cos(ix)&=\cosh (x) \\ \sin(ix)&=i\sinh(x) \\ \cosh(ix)&=\cos(x) \\ \sinh(ix)&=i\sin(x) \end{align}$$ I feel like these are ...
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1answer
23 views

Angle between arbitrary rectangle and horizontal plane

This seems to me like a fairly simple problem but I'm constantly re-thinking it because something seems wrong about how I'm solving it. I feel like there may be perhaps an easier solution or something ...
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0answers
31 views

How to solve $\sin(45^{\circ}-x) = -\frac{1}{3}, 45^{\circ}<x<90^{\circ}$

how does one solve $$\sin(45^{\circ}-x) = -\frac{1}{3}$$ for $45^{\circ}<x<90^{\circ}$ Thank you!
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0answers
19 views
+50

Fourier Arbitrary-Phase Sinusoid Series

A Fourier Cosine Series uses an infinite sum of only cosine waves to represent a target function whose left endpoint is $0$, by considering its even extension. The even extension seems to amount to ...
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3answers
109 views

what function follows $\int_{a}^{2a} f(z) dz = \cos(a)$ where $a$ and $z \ne 0$ and $f(z)$

what functions follows $\int_{a}^{2a} f(z)dz = cos(a)$ where $a$ and $z \ne 0$ and $f(z)$ is differentiable everywhere besides the point at $0$. $$\int_{a}^{2a} f(z)dz = \cos(a)$$ What steps would I ...
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0answers
25 views

Lebesgue Integrability of $(\sin x)^p(\cos x)^q$ on $[0,\pi /2]$

I’m trying to understand the solution to this question. Why does the solution consider the function over the intervals $(0, \pi/4)$ and $(\pi/4,\pi/2)$ separately ? Would it not be easier to say the ...
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1answer
54 views

Angles determining a point interior to a triangle

Given a triangle $ABC$, a point $P$ interior to the triangle can be determined by two angles, for example the angle $\alpha = \angle PAC$ and the angle $\beta = \angle PBA$ (See diagram below). In ...
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28 views

Can you define sine and cosine in the complex numbers?

So I have been reading the paper "Thermalization for Perturbations of Dynamical Systems" by G. Barrera and M. Jara (you can find the paper here) and there is a step that I don't quite ...
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0answers
11 views

Inequality for distortion coefficients (optimal transport in Riemannian geometry)?

Define the so-called modified distortion coefficients by \begin{align*} \sigma_{K,N}^{(t)}(\theta)=\begin{cases}\frac{\sin\big(\sqrt{\frac{K}{N}}\theta t\big)}{\sin\big(\sqrt{\frac{K}{N}}\theta\big)}\...
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0answers
40 views

Computing $\sin 1^\circ\sin 2^\circ\sin 3^\circ\cdots\sin 178^\circ\sin 179^\circ$

Compute $$\sin 1^\circ\sin 2^\circ\sin 3^\circ\cdots\sin 178^\circ\sin 179^\circ$$
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19 views

trigonometric equations (find minimum value of alpha with the degree of measure in terms of prime numbers)

I did find it difficult to find a appropriate way just waiting for that one step or identity to used to solve this question The least positive angle $\alpha$ for which $$\left(\frac{3}{4}-\sin^{2}(\...
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1answer
14 views

Cosine modeling of a spot of paint on the wheel of a paddle-steamer

A paint spot $X$ lies on the outer rim of the wheel of a paddle-steamer. The wheel has radius $3$m and as it rotates at a constant rate, ${X}$ is seen entering the water every $4$ seconds. $H$ is the ...
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1answer
104 views

$\sum_{k=1}^{180^\circ}\cos\left(\frac{2\pi}{k}\right)$

How to compute $$\sum_{k=1}^{180}\cos\left(\frac{2\pi}{k}\right)$$ I have tried several ways to do it (including euler formula, etc), but all failed. The answer given by wolframalpha appeared to be a ...
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3answers
46 views

Solve limit for a

Determine a so that: $\lim_{x\to0} \frac{\tan(ax)}{\sin(x)} = 2$ So far, I have used the L'hopital rule: $\frac{\frac{1}{a \cos(x)}}{\cos(x)} = \frac{1}{a \cos^3(x)} = 2$ But I am not sure if this is ...
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1answer
51 views

Not sure if this idea for calculating the square roots of a value has any merit

Okay, so I was thinking about square roots the other day and I spent some time thinking of how to visualize them: Say we have 10,000. The square root would be 100. So 100 of 100 equals 10,000. Now ...
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2answers
52 views

Generalizing Solutions for Trigonometric functions (Secant)

I have been working on the problem: $3\sec^2(2x)-5=1$ Where you have to solve for x and then enter the generalized solutions in ascending order. This is using the Acrobatiq platform, but I am ...
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1answer
37 views

How do we solve the system of trigonometry equations?

I am unable to solve the following system of trig equations. My intention is to find the values of $x$ and $x_o$. $$6\cdot \sin(x) = 8\cdot \sin(x_o)$$ $$6\cdot \cos(x) + 8\cdot \cos(x_o) = 10$$ Here'...
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2answers
37 views

Calculate $\operatorname{tg}( \alpha), $ if $\frac{\pi}{2} < \alpha<\pi$ and $\sin( \alpha)= \frac{2\sqrt{29}}{29}$.

Calculate $\operatorname{tg}( \alpha), $ if $\frac{\pi}{2} < \alpha<\pi$ and $\sin( \alpha)= \frac{2\sqrt{29}}{29}$. Please provide a hint. I know that $\operatorname{tg}( \alpha)=\frac{\sin( \...
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3answers
40 views

How do I find the value of cotangent when cosecant is given? [closed]

Find the value of cot(θ), given that csc(θ)=4.
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3answers
62 views

Solve trigonometric equation $2x=\sin(2x)+\pi/2$

I would appreciate help in solving this equation: $$2x =\sin 2x + \frac{\pi}{2}$$ I am aware that instead of $2x$ in $\sin(2x)$ I could put the whole right part of the equation, and then again and ...
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2answers
37 views

Uncertain how to solve this trigonometric equation

I am currently attempting to discover how to find the general solutions to $$\sqrt 3\tan^2x=2\tan x+\sqrt 3$$ The given solutions are $x= \frac{\pi}{3}+ \pi k$ , $\frac{5\pi}{6} + \pi k$ To solve this ...
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1answer
31 views

How do you find the least value when an inverse trigonometric function is in the denominator?

The function: $f(x)=\frac{4\pi^2}{3\arccos{(x^4-2x^2)}}+\frac{5\pi}{3}$ If $B=\frac{m}{\pi}$ where $m$ is the minimum value which $f(x)$ can take. Find the value of $B$. The choices given in my book ...
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2answers
27 views

How to find the domain in $f(x)=\frac{\arcsin x}{| \arcsin |x||}+\frac{|\arccos|x||}{\arccos|x|}+1$?

The problem is as follows: Find the range in the following function: $f(x)=\frac{\arcsin x}{| \arcsin |x||}+\frac{|\arccos|x||}{\arccos|x|}+1$ $\begin{array}{ll} 1.&\{3,1\}\\ 2.&\left\{\frac{...
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1answer
55 views

arg($1-e^{i\theta}$)

Im trying to calculate arg($ 1-e^{i\theta} $) for a problem im trying to solve but I have run across a slight problem: In this problem I have defined a branch cut to be $\mathbb{C}\setminus [1,\infty)$...
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1answer
44 views

Geometry(length of an angle bisector) [closed]

The length of two sides of a triangle are $b$ and $c$. Let $s$ be the length of the angle bisector of the angle between the two given sides. The length of the third side of the triangle is:
2
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1answer
43 views

How to solve $x\cos(t)+y\sin(t)=1$ for $t$

When trying to find the points, $P_1$ and $P_2$, on a circle of radius $R$ such that the tangent line to those points passes through the point $P_0$, it was all simple geometry until I ran into $x\cos(...
2
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1answer
55 views

Bessel function approach as a trigonometric function

I was trying to find the eigenvalues, which is the positive roots of the equation bellow: $$J_{1}(a\lambda)Y_{1}(c\lambda) -J_{1}(c\lambda)Y_{1}(a\lambda) =0$$ I was presented with this trigonometric ...
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2answers
30 views

How do I rewrite cos(-150) using a positive acute angle? [closed]

I need help to solve this. All of my answers I input are answered as incorrect. What am I doing wrong?
-4
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0answers
24 views

How do I solve this question? (Trig) [closed]

I need help solving this, I solve it and the program marks it as wrong. question

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