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

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1answer
30 views

How to appraoch solving this series?

I am given the following series and asked to solve it. $\sum\limits_{n=1}^\infty \dfrac{(-2)^n}{(2n+1)!}$ I recognize that this series is somewhat similar to the Taylor series for $sinx$ which is ...
1
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2answers
39 views

Calculus I Integral

$\int(\sqrt x + \sec x \tan x)\, dx$ using $u$-substitution. So far as this class has been taught... the class knows that in order to use $u$-substitution, a $u$ and its derivative must be present ...
4
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1answer
45 views

If $\frac{\sin\alpha}{\sin\beta} \le 1+\epsilon$, then $\frac{\alpha}{\beta} \le 1+\sqrt\epsilon\;\;$ (for acute $\alpha$ and $\beta$)

Prove the following: For $0 \le \alpha,\beta \le \frac{\pi}{2}$, if $$\frac{\sin \alpha}{\sin \beta} \le 1+\epsilon$$ then $$\frac{\alpha}{\beta} \le 1+\sqrt\epsilon$$ This is from this ...
2
votes
2answers
34 views

Induction proof of the identity $\cos x+\cos(2x)+\cdots+\cos (nx) = \frac{\sin(\frac{nx}{2})\cos\frac{(n+1)x}{2}}{\sin(\frac{x}{2})}$ [duplicate]

Prove that:$$\cos x+\cos(2x)+\cdots+\cos (nx)=\frac{\sin(\frac{nx}{2})\cos\frac{(n+1)x}{2}}{\sin(\frac{x}{2})}.\ (1)$$ My ...
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2answers
59 views

Show that the trigonometric integral $\frac{\pi}{2}-\sin (x) \ll (1+x)^{-1}$.

How can I show that for non-negative $x$ we have $$ \int_x^{\infty} \frac{\sin(t)}{t} dt \ll (1+x)^{-1}. $$ I think this should be an easy task but still I'm unable to solve it. I tried to estimate ...
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0answers
14 views

Maxima and minima of a specific function

For $0\leq x\leq360$ ($x$ in degrees), hat are the maxima and minima of $f(x)=\frac{sin(x)+cos(x)}{cos(x)^2+1}$ in surd form?
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1answer
28 views

Show that $\sin \theta = \frac{\cos(0.5\theta)-\cos(1.5\theta)}{2\sin (0.5\theta)}$ [closed]

How to show that $$\sin \theta = \frac{\cos(0.5\theta)-\cos(1.5\theta)}{2\sin (0.5\theta)}?$$
2
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0answers
25 views

Perpendicular Vectors in 3D space

I was wondering whether given two Vector's v0 and v1 whether I could find the two perpendicular vectors at a given distance, d, from v1, perpendicular to the v0/v1 line. I know that v0 and v1 will ...
2
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2answers
65 views

Calculate $\sin\dfrac{3\pi}{14}-\sin\dfrac{\pi}{14}-\sin\dfrac{5\pi}{14}$

I have interesting trigonometric expression for professionals in mathematical science. So, here it is: $$\sin\dfrac{3\pi}{14}-\sin\dfrac{\pi}{14}-\sin\dfrac{5\pi}{14};$$ Okay! I attempt calculate it: ...
2
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2answers
64 views

Evaluate: $\csc^2\left(\frac{\pi}{9}\right)+\csc^2\left(\frac{2\pi}{9}\right)+\csc^2\left(\frac{4\pi}{9}\right)$

$$\csc^2\left(\frac{\pi}{9}\right)+\csc^2\left(\frac{2\pi}{9}\right)+\csc^2\left(\frac{4\pi}{9}\right) \;=\; \text{???}$$ $\bf{My\; Try::}$ Let $\displaystyle \frac{\pi}{9} = \theta\;,$ Then ...
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1answer
62 views

Can someone please help me solve this equation?

Can someone please help me solve this equation for a, b, and c, when u, v, x, y, ia, ib, ja, jb, ka, kb, f, g, and h are known. u+(cos(a)*f)+(cos(b)*g)+(cos(c)*h)=x and ...
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9answers
2k views

Why are the Trig functions defined by the counterclockwise path of a circle?

My understanding is that $\cos$ is defined by the value of $x$ as you trace the graph of a circle counterclockwise, starting at the point $(1, 0)$. Similarly, $\sin$ traces the $y$ value. I understand ...
0
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1answer
24 views

$\sin(z)$ in polar coordinates

the following formula can be found in the literature: $\vert \sin(z) \vert^2 = \sin(x)^2 + \sinh(y)^2$, $z=x+iy;$ $x,y\in\mathbb{R}$. I am wondering if there is a similar formular in polar ...
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2answers
24 views

How to prove this trig identity turning $\tan$ into $\cot$?

$$\frac{\tan u - \tan v}{1 + \tan u \cdot \tan v} = \frac{\cot v - \cot u}{\cot u \cdot \cot v+1}$$ I've been trying to prove this for a while, no luck (I do know it's true). I've attempted to turn ...
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1answer
27 views

Vector addition and Pythagorean theorem

Finding length or magnitude using vector addition and the Pythagorean theorem. I am trying to understand why vector addition and the Pythagorean theorem are giving different results? Vector ...
5
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4answers
222 views

A String Tied Around The Earth

Say you're standing on the equator and you have a string below you tied around the equator (40,075 km) that is the length of the equator + 1 meter (40,075.001 km). What is the maximum height you can ...
0
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1answer
26 views

Proof of an inequality with a trigonometric expression

I am trying to show that for every $\lambda \in (-\pi,\pi)\setminus\{0\}$ and every $n \in \mathbb{Z}_{>0}$ $$\frac{1}{n}\sqrt{\frac{1-\cos(n-1)\lambda}{1-\cos\lambda}} \leq 1$$ This inequality ...
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0answers
32 views

Simplify this trigonometric identity $(1-\cos^2)(1+\cos^2)$ [closed]

Simplify the trigonometric identity : $$(1-\cos^2)(1+\cos^2)$$
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2answers
27 views

Why does $\tan\left(\frac{\pi}{2} - \beta\right) = \frac{t}{s}$ imply that $\tan\beta = \frac{s}{t}$?

I have been preparing for the SAT on KhanAcademy. For one of the trigonometry problems, the following conversion is made: $$\tan\left(\frac{\pi}{2} − \beta\right)= \frac{t}{s}$$ $$\tan(\beta) = ...
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1answer
28 views

If in a $\triangle ABC,b=3$ cm,$c=4$ cm and the length of the perpendicular from $A$ to the side $BC$ is $2$ cm.How many such triangles are possible?

If in a $\triangle ABC,b=3$ cm,$c=4$ cm and the length of the perpendicular from $A$ to the side $BC$ is $2$ cm.How many such triangles are possible? I found $CD=\sqrt{AC^2-AD^2}=\sqrt5$ I found ...
2
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3answers
42 views

About the nature of continuity of trigonometric functions and equality

I was recently somewhat confused by the result of an exercise from a textbook that read: Question How many solutions are there to the equation $(\tan x)\sin^2(2x)=\cos x$, $-2\pi \leq x \leq ...
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votes
1answer
35 views

Trigonometric identity related to unit circle

I really need help for this Given that $\displaystyle\sin A = \frac35$, $\displaystyle\tan B = -\frac{5}{12}$ and that $A$ and $B$ lie in the same quadrant, find $\sin B$.
2
votes
2answers
47 views

Help solving $\int_{0}^{\frac{\pi}{6}} \frac{1}{\cos(u)^9}du$

My calculus final is coming up. My teacher was nice enough to give old exams as study material, which I have been doing. One questions asks: $$\int_{0}^{\frac{1}{2}} \frac{1}{(1-x^2)^5}dx$$ Using ...
2
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2answers
30 views

Multiple Angle formulas, alternate forms

Relatively simple question, that might not be simple to answer: I have noticed that there are ways of expressing every double angle formula of a given trigonometric function using only that function ...
1
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1answer
45 views

Find A and B for $\cos A = \cos B$ and $\sin A = \sin B$ simultaneously?

Find relationships between A and B, in $\cos A = \cos B$ and $\sin A = \sin B$ simultaneously. I don't understand what the question is asking from me? I know for $\cos A = \cos B$ to be true, $A = ...
1
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1answer
17 views

how to take out oscillations of a sine wave and leave only amplitude?

There is a maths trick to take out the oscillations of a sine signal of varying amplitude, to keep only it's maximum transient amplitude in between peaks. It's something like the the square root of: ...
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3answers
73 views

why $\cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\beta - \alpha)$?

I'm studying linear algebra and there is a chapter in a book that says about unit vector and it says this $$ \cos\alpha \cos\beta + \sin\alpha \sin\beta = \cos(\beta - \alpha) $$ Why?? I'm newbie and ...
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0answers
14 views

Relative velocity- Finding the direction of wind.

An aircraft is flying due south at $350~\text{kmh}^{-1}$. The wind is blowing at $70~\text{kmh}^{-1}$ from the direction of $\theta$, where $\theta$ is acute. Given that the pilot is steering the ...
1
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1answer
22 views

Find a right triangle section of a larger one

I'm programming a game, and find myself stumped. I know the target ball (dx, dy) and (cue x, cue y) and the x value of the camera. It's been 20 years and I'm sad to say I've lost some ground here. ...
0
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1answer
19 views

Tangent parallel to the initial line for polar equation =, can r^2 be used instead?

Given a formula for a polar equation: $$\ r^2 = a^2 \cos^22 \theta $$ It could be said that to find the points parallel to the initial line, $$\frac{dy}{dx} = \frac{d (r\sin\theta)}{d\theta} = 0$$ ...
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4answers
84 views

In a $\triangle ABC,a^2+b^2+c^2=ac+ab\sqrt3$,then the triangle is

In a $\triangle ABC,a^2+b^2+c^2=ac+ab\sqrt3$,then the triangle is $(A)$equilateral $(B)$isosceles $(C)$right angled $(D)$none of these The given condition is $a^2+b^2+c^2=ac+ab\sqrt3$. Using sine ...
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1answer
30 views

Solve:$ 45=-21.9*cos(\frac {\pi}{6}(t-1))+51.6$

I'm interested in learning how to solve this without a graphing calculator. For $t = 0$ to $12$, $$ 45=-21.9\cdot\cos(\frac {\pi}6(t-1))+51.6$$
0
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1answer
112 views

How do I find similar relationships between $x$ and $y$ in the following cases (Trig)? [duplicate]

If $\cos(x)=\cos(y)$, then $x = 2k\pi \pm y$ for $k$ in $\mathbb Z$. How do i find similar relationships between $x$ and $y$ for: a) $\sin (x) = \sin(y)$ b) $\cos (x) = \cos(y)$ and $\sin(x) = \sin ...
2
votes
2answers
37 views

Show that $\tan \alpha \tan \beta + \tan \beta\tan \gamma +\tan \gamma\tan \alpha =1$

I would appreciate if somebody could help me with the following problem: Q: Show that If $\alpha + \beta + \gamma = {\frac{ \pi}{2}}, \alpha,\beta, \gamma>0 $ then $$\tan \alpha \tan ...
0
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1answer
29 views

How to calculate a point between two angled lines based on distance from the lines?

Please take a look at the picture below for the diagram reference: I am trying to calculate the point where it is perfectly 3.3 cm vertically from the 44.52 cm line AND 5.5 cm horizontally from the ...
0
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2answers
36 views

Proof of $\sin (\frac{3 \pi}{2}-A)=-\cos (A)$ geometrically

We already know that $\sin (\frac{3 \pi}{2}-A)=-\cos (A)$ but is there any method to prove it geometrically? Could someone suggest something?
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0answers
14 views

How to find the third point of a triangle in a 3D space (arm rig)

I am attempting to create a system that will replicate arm movement, so far I have mastered this in a 2D plane however I am having trouble adding the third dimension. Here is what is given, You know ...
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1answer
26 views

How do I calculate the third point of a triangle in a 3Dimensional Plane

I am attempting to create a system that will replicate arm movement, so far I have mastered this in a 2D plane however I am having trouble adding the third dimension. Here is what is given, You know ...
0
votes
1answer
41 views

How to find a point equidistant between two points on a sphere.

So I have this problem involving astronomy, but because astronomy uses all sorts of fancy words I'm going to make it more simple by using an analogy of the earth. The process, mathematically would be ...
0
votes
2answers
20 views

When solving for $\sin$ or $\cos(2t)$ given $\sin$ or $\cos(t)$ is the quadrant relevant?

I notice some of the homework problems in my book ask me to find the sine or cosine of twice an angle given the sine or cosine of the angle. They also mention $P(t)$ is in some given quadrant. My ...
2
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0answers
15 views

Question related to trigonometric identities and ratios [duplicate]

Prove that $$\tan \frac{\pi}{30} \tan \frac{7\pi}{30} \tan \frac{11\pi}{30} \tan \frac{13\pi}{30}=1$$ I tried this and i was able to reduce it to $$\cot \frac{\pi}{30} \cot \frac{2\pi}{30} \cot ...
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3answers
52 views

Proving a trigonometric identity: [duplicate]

Prove that $\sin \frac{{2\pi }}{7} + \sin \frac{{4\pi }}{7} + \sin \frac{{8\pi }}{7} = \frac{{\sqrt 7 }}{2}$. I have tried to square both side and got ${\sin ^2}\frac{{2\pi }}{7} + {\sin ...
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0answers
21 views

Infinite number of properties of the sine function graph

Since we can differentiate the sine function an unlimited number of times, does this imply that the graph at every point have an infinite amount of properties like slope, curvature, change in ...
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2answers
57 views

When is $\sin\colon\mathbb{C}\to\mathbb{C}$ purely real/imaginary?

Sketch the sets on which $\sin\colon\mathbb{C}\to\mathbb{C}$ is purely real/imaginary. My current result is that for purely real numbers $\sin$ is purely real and for purely imaginary numbers ...
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0answers
62 views

Did the book make a mistake in the identity$\frac{1}{\cos 2x+\tan x} = \sin 2x$?

EDIT: OP says that they misread the text from which this question was drawn. It actually said $$\frac{1}{\cot(2x) + \tan(x)} = \sin(2x)$$ where $\cos$ has been replaced by $\cot$; that identity is ...
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2answers
27 views

show that $y = \cos x$, has a maximum turning point at $(0, 1)$ and a minimum turning point at $(\pi, -1)$

show that $y = \cos x$, has a maximum turning point at $(0, 1)$ and a minimum turning point at $(\pi, -1)$ Turning points occur when the gradient is 0 or $\frac{dy}{dx} = 0$ $f(x) = \cos x$ ...
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votes
1answer
44 views

Interpreting a “sunrise function”: $t=-1.4\sin\left(\frac{2\pi}{365} (d-75)\right) + 7$ [closed]

In Africa, the time for sunrise any day during the year can be shown as the formula: $$t=-1.4\sin\left(\frac{2\pi}{365} (d-75)\right) + 7$$ where $t$ is the time in hours after midnight and ...
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votes
1answer
63 views

The value of $\tan \frac \pi {-12}$ [closed]

Use a compound angle formula to determine the exact value of $\tan \frac \pi {-12}$.
0
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2answers
37 views

Finding the max/min turning points of $y=\sin x$

Use differentiation to show that $$y = \sin x$$ has a maximum turning point at $\left(\frac{\pi}2, 1\right)$ and a minimum turning point of $\left(\frac{3\pi}2, -1\right)$. I know that the ...
1
vote
2answers
56 views

Evaluate $\lim _{x\to \pi }\left(\frac{\sin x}{\pi ^2-x^2}\right)$ without L'Hopital or Taylor series

I want to evaluate $$\lim _{x\to \pi }\left(\frac{\sin x}{\pi ^2-x^2}\right)$$ without using L'Hopital's rule or Taylor series. My thinking process was something like this: in order to get rid of ...