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

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82 views

Please solve this equation

I think I'm missing something here. Please include steps solving this equation to $x$. $$0=\cos^2{x}+\cos{x}-\sin^2{x}$$
2
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2answers
66 views

On proving an identity given a system of trig equations

We are given the following: $$a^2 + b^2 + 2ab\cos\theta = 1 \tag1$$ $$d^2 + c^2 + 2cd\cos\theta = 1 \tag2$$ $$ac + bd + (ad + bc)\cos\theta = 0\tag3$$ It is required to prove that: $$a^2 + c^2 = ...
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0answers
33 views

what does secant equal 2 mean? [duplicate]

I need your help, I am a little confused. My question is, what does the value of secant mean? I asked this question previously but unfortunately I did not understand the answers so I am trying to ...
0
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0answers
25 views

Vector Magnitude during rotation

Probably something I should now already but this is confusing me no end! Lets say we have a force which is directed at 69 degrees inclination (from the X axis) with a magnitude of 500, shown below: ...
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2answers
33 views

Circle area and lim

I was trying to show how to find $\pi$ value from formula $\pi R^2$, but I dont understand where is my mistake. So i am calculating area using $n$ triangles 1 let $R=1$, then one triangle area is ...
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2answers
32 views

Very simple question regarding sum/difference identity

If I have $\sin(0.7x-47t+C)$ where do I carry my constant $C$? The same with my sum-to-product identities. This problem is showing up for me because I'm studying mechanical waves at the moment. I ...
3
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9answers
161 views

Find the exact value of $\sin (\theta)$ and $\cos (\theta)$ when $\tan (\theta)=\frac{12}{5}$

So I've been asked to find $\sin(\theta)$ and $\cos(\theta)$ when $\tan(\theta)=\cfrac{12}{5}$; my question is if $\tan (\theta)=\cfrac{\sin (\theta) }{\cos (\theta)}$ does this mean that because ...
2
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2answers
42 views

How to resolve multiply differentiation function algorithms?

My simple function is $f(x)=\frac{1}{2}e^{-x}\sin(2x)$; Can I resolve for multiply differentiation $f^{(n)}=?$ algorithm? Thx for answer.
3
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4answers
126 views

Picture/intuitive proof of $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$?

Is there a nice geometric, intuitive or picture proof as to why the easily algebraically provable identity $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$ is true? Note I'm not looking for a ...
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0answers
16 views

Direct to indirect angle

I have a system which it computes angle in direct way and an other system in indirect. What is the formula to convert angles from one to other? Thx
2
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3answers
66 views

Finding an area of a triangle inside of a triangle, given certain areas of other triangles, and area ratios.

I'm studying for the Waterloo Math Contest (Galois, Gr. 10) taking place in April of 2015 and I am preparing by looking at previous problems and solving them. This is question 4(c) on the 2010 Galois ...
2
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2answers
31 views

Calculate coordinate of any point on triangle in 3D plane

I am really stuck and can't find right way to write a formula(s) that will calculate Z coordinate of point on triangle plane in 3D plane. I know all coordinates of triangle points ( Ax, Ay, Az, Bx, ...
3
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1answer
184 views

Can this be simplified?

$$ e^{-i\frac43\pi n} - e^{-i\frac23\pi n}, n\in \mathbb{N} $$ I am trying to simplify this but cant. Any ideas appreciated.
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5answers
53 views

Finding the range and domain of $f(x)=\tan (x)$

I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I ...
1
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2answers
43 views

How do I go about solving this derivative of inverse tangent?

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

How to find length of the sides of a triangle given the ratio of the sines of the sides?

Consider $\triangle ABC$. Let $\dfrac{\sin A}{\sin B} = \dfrac56$ and $\dfrac{\sin B}{\sin C} = \dfrac45$. Find $\dfrac{\vert AC\vert\cdot \vert AB\vert}{\vert BC\vert}$. If there is no definite ...
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0answers
16 views

need help understanding sin(bsin(x)

can someone explain why assuming $b<<1$ $\cos(\beta \sin(2\pi f_mt))\approx 1$ and $\sin(\beta \sin(2\pi f_mt))\approx \beta \sin(2\pi f_mt) $ the equations are part of a fm narow band ...
4
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3answers
72 views

Find the minimum of $\displaystyle \frac{1}{\sin^2(\angle A)} + \frac{1}{\sin^2(\angle B)} + \frac{1}{\sin^2(\angle C)}$

Is it possible to find the minimum value of $E$ where $$E = \frac{1}{\sin^2(\angle A)} + \frac{1}{\sin^2(\angle B)} + \frac{1}{\sin^2(\angle C)}$$for any $\triangle ABC$. I've got the feeling that ...
0
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0answers
35 views

inverse of function with sine mooculus

I'm trying to do a calculus course on line: mooculus and I'm trying to answer this question: The height in meters of a person off the ground as they ride a Ferris Wheel can be modeled by h(t) = ...
2
votes
1answer
41 views

Fourier series of oscillation in form $\cos(2 \pi \frac{k}{T}+\phi)$

I would like to calculate the fourier coefficients of $\cos(2 \pi \frac{k}{T}+\phi)$ where $T \in \mathbb{N}$ is the period and is arbitrary but fixed, $k \in [1, N-1]$ is the number of oscillations ...
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1answer
31 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 ...
3
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2answers
230 views

I am having problems figuring out how to derive this.

I have the function $$\tag{1} f(x)=\ln\sqrt{8+\cos^2x}$$ So we derive it as follows: $$\tag{2} f(x)=\ln(8+\cos^2x)^\frac{1}{2}$$ $$\tag{3} f(x)=\frac{1}{2}\ln(8+\cos^2x)$$ $$\tag{4} ...
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1answer
34 views

$f(x)=sec(x)$ inequality inconsistency\trouble

I'm currently attempting to find the range of $f(x)=\sec(x)$ by considering $\cos(x)$ in the intervals of $0<\cos(x)\leqslant 1$ and $-1\leqslant \cos(x)<0$ (as $\sec(x)$ is undefined for ...
1
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2answers
93 views

How do I go about solving this derivative?

I have the function $$f(x)=\ln\sqrt{8+\cos^2x}$$ so $$1.f(x)=\ln(8+\cos^2x)^\frac{1}{2}$$so$$2.f(x)=\frac{1}{2}\ln(8+\cos^2x)$$so $$3.f'(x)=\frac{1}{2}\left[\frac{-2 \cos x^{\sin x}}{8+\cos ...
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3answers
83 views

Calculate $\frac{2\cos40^\circ-\cos20^\circ}{\sin20^\circ}$

I am trying to solve this task i.e. calculate this expression without using calculator, in terms of known values for angles such as 30,60,90,180 degrees :). ...
0
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1answer
33 views

Finding the range and domain of $h(x) = \sec (x)$

I am attempting to show how to find the range and domain of $h(x) = \sec (x)$. Here's my working so far. Consider $h(x) = \sec (x)$, which is defined as $h(x) = \sec (x)=\frac{1}{\cos(x)}$. We know ...
0
votes
1answer
38 views

How to solve: $0 = -\sin \space 3x \cdot3, \left({\pi\over 12}, {7\pi \over12}\right)$

While working on some Rolle's Theorem problems I came to: $$f(x) = \cos 3x$$ This is both continuous on the given interval (and everywhere really) $[{\pi\over 12}, {7\pi \over12}]$ and ...
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1answer
31 views

Word Problem sun, mars, moon

The sun has a diameter of $8.65 \times 10^5 \text{mi}$. Mars is $1.42 \times 10^8 \text{mi}$ from the sun. Its moon, Phobos, has a diameter of $17.4 \text{mi}$. What is the maximum distance that ...
4
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1answer
44 views

Points for which $AX^2-BX^2$ is constant

My problem is from Israel Gelfand's Trigonometry textbook. Page 9. Exercise 8: Two points, A and B, are given in the plane. Describe the set of points for which $AX^2-BX^2$ is constant. I would ...
0
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2answers
47 views

double partial differentiation

I'm having troubles with solving problems with partial differentiations... and this one is double. I don't thing we've even learned this in class... Question: If $z=f(x,y)$, where $x=r\cos(\theta), ...
4
votes
2answers
49 views

Elementary Trigonometry problem

My problem is from Israel Gelfand's Trigonometry textbook. Page 9. Exercise 7: Two points, $A$ and $B$, are given in the plane. Describe the set of points $X$ such that $AX^2+BX^2=AB^2.$ The ...
0
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2answers
74 views

What is the meaning of calculating sine of a number?

When we calculate sine/cos/tan etc. of a number what exactly are we doing in terms of elementary mathematical concept, please try to explain in an intuitive and theoretical manner and as much as ...
2
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2answers
138 views

solve $\tan(x) = \sqrt{1-x^2}$

I am not sure if you should be deriving it or converting tan into $\sin(x)/\cos(x)$. Even then, I do not know what to do from there
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1answer
58 views

Find the 6th root of $-3+4i$ and plot on complex plane

So I have a rough idea on how to get the answer but I'm getting stuck on the angle or argument for the equation. The question is: Find the 6th root of $-3+4i$. I first find the $r$ value which ...
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2answers
453 views

Solving for Cos Exactly

How to solve $\cos(\dfrac{5\pi}{6})$ and $\cos(\dfrac{7\pi}{6})$ exactly? I couldn't use special triangles to solve this either.
2
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2answers
32 views

Get the angle in a circle using center, radius and one point in a circle.

There is a circle and i know Point1 this is fixed and i know another point Point2 which can be anywhere in the circle. and i want to know the angle which is made at center. Thanks Your help will be ...
1
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2answers
57 views

How to solve equations containing trigonometric functions and powers

I mean, is there a way to solve analytically something like this: $$ \tan(x) - x = 0 $$ or like this equation $$ \tan(x) - x^2 = 0 $$ I know this will produce infinite number of roots but could ...
0
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1answer
57 views

Value of $\frac{\cos 45}{\sec 30 + \operatorname{cosec} 30}$

I just put the values from the trignometric table to solve, but the answer is different in the answer book. $$\frac{\cos 45}{\sec 30 + \operatorname{cosec} 30}$$
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4answers
61 views

Prove that $1+\tan^2 x=\sec^2 x$ [duplicate]

I have no idea how to prove this. Does anyone know where to start? We're allowed to use other trigonometric identities but i'm not sure why these are useful.
1
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2answers
83 views

Basic trigonometry identities question

$$\eqalign{\tan^2\theta-\sec^2\theta &=\tan^2\theta-\dfrac1{\cos^2\theta}\\&=\dfrac{\sin^2\theta}{\cos^2\theta}-\dfrac1{\cos^2\theta}\\&=-\dfrac{\cos^2\theta}{\cos^2\theta}\\&=-1.}$$ ...
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3answers
34 views

Solve for Radian Exactly

$$\tan(A) = \frac{\sqrt{3}}{-3}$$ I've tried using special triangles but couldn't find a matching faction using sohcahtoa.
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2answers
23 views

Half tangent representation

If $x$ is defined by the interval $\pi/2>x>0$, and $\tan(x)=A$, what is $\tan(x/2)$? This is a multiple choice question on a test, and I don't have a approach because all the answer choices are ...
1
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1answer
17 views

Trig expression simplification

Could someone explain how to simplify $\dfrac{\sin(2x)}{2-2\cos^2(x)}$? I've had tried the power reduction identity but the result did not seem much more simple. Any help would be appreciated.
2
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1answer
47 views

Do sine and cosine of complex numbers have anything to do with right-triangles or circles?

I've recently been working on a web application that draws iterating function generated fractals. I've noticed that the sine and cosine functions can be used to draw exquisite plots using an ...
1
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1answer
57 views

finding exact value of $\sec^{-1} 5$

Find the exact value of $\sec^{-1} 5$ (decimal answer). I know that $\sec^{-1}5=\cos^{-1}\dfrac{1}{5}$, but I don't know how to proceed from here. I drew a right triangle with sides $1$ and $5$ ...
0
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2answers
45 views

Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
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2answers
46 views

Could someone explain me this “ownership” of the arctangent [duplicate]

someone could explain to me this: $$\int { \arctan { \left( \frac { 1 }{ { u }^{ 2 } } \right) } } \,du=\int { \frac { \pi }{ 2 } } -\arctan { \left( { u }^{ 2 } \right) } \, du$$
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2answers
35 views

Substitution of an implicit variable

I wasn't sure how to title this question: I want to manipulate the integral $$I(a,b) = \int_0^{\frac{\pi}{2}} \frac{d \phi}{\sqrt{a^2\cos^2 \phi + b^2 \sin^2 \phi}}$$ with this subsitution: $$\sin ...
1
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1answer
31 views

Trigonometry / Obtuse angle

If $\cos A = 4/5$ and $\sin B = 5/13$, where $A$ is a acute and $B$ is obtuse, find, without evaluating the angles $A$ and $B$, the values of a) $\sin (A-B)$ b) $\cos (A+B)$ I'm stuck figuring out ...
1
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2answers
31 views

Trigonometry / Finding the exact value

Given that $\cos \theta = \dfrac{-4}{5}$ and $\sin \theta$ is positive, obtain the exact values of $\cos (6\pi+\theta)$ i don't understand this question.