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

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Taking the sin of arccos

When solving for the value of x in the equation $$\sin^{-1}{(\sqrt{2x})}=\cos^{-1}(\sqrt{x})$$ one would take the sin of both sides of the equation cancelling out the arcsin leaving ...
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1answer
59 views

Finding parametric equations for the curved path of a particle around a half-circle

I have a question about parametric equations. So far I've learned how to find the parametric equations for a straight line, I know about replacing $x^2$ and $y^2$ in the equation of the unit circle, ...
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1answer
21 views

Measuring Diagonals (part 2)

Check out part 1 here: Measuring diaognals without Sine Law - this is an equivalent construct. Suppose we construct a triangle with lengths $5,7,\sqrt{32}$ on the cartesian plane. Using Heron's Area ...
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3answers
46 views

Solving for $y_3$ in a complicated expression

I need to solve this equation for $y_3$: $$y_3=z_1+(z_1+z_2)\sin\left(\arctan\left(\frac{y_3-z_1}{\sqrt{(z_1+z_2)^2–(y_3-z_1)^2}}\right)\right)$$ Is there a solution to this equation in terms of ...
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2answers
62 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
41 views

How do I calculate the inverse of these matrices?

In learning how to rotate vertices about an arbitrary axis in 3D space, I came across the following matrices, which I need to calculate the inverse of to properly "undo" any rotation caused by them: ...
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1answer
48 views

Logarithms with trigonometric inequality

My class is going to have an exam tomorrow, but we can't figure out how to solve such equations. $$\log_{\ \large tg(x)} \sqrt{\sin(x)^2 - 5/12} < 1 $$ We tried to transform $1$ to $\log_{\ ...
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2answers
83 views

Calc- Trig Identity Help!

I have a few questions on trig/calc stuff I am having trouble with, for some reason I am just not getting the concept. 1: What happens if you take $B=2\pi$ in the addition formula? Do the results ...
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0answers
43 views

Is there a function to standardize the resultant when a point vector is reflected by a cone?

In a previous question Reflect a point vector in a conical surface and determine average resultant vector. an expression:- $V1_x = P_x (1 - 2\sin^2(\gamma))$ $V1_y = - P_y (1 -\sin^2(\gamma))$ ...
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5answers
144 views

Did I do something wrong in this problem?

Consider this: "A boat sails $6$ km West, then $5$ km Northwest. Use trigonometry to find the boat's distance and bearing from its starting point." To solve this, I first drew on paper a shape sort ...
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3answers
129 views

Trigonometric equation: $2(\sin^6 x+\cos^6 x)-3(\sin^4 x+\cos^4 x)+1=0$

I'm new here, but I need your help so much to solve an equation: $$2(\sin^6 x+\cos^6 x)-3(\sin^4 x+\cos^4 x)+1=0$$ I tried a lot like making $2[(\sin^2 x)^3 + (\cos^2 x)^3$
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1answer
366 views

Equation of a line maintaining equal ratio distance between two points.

Say I have two points. One at $A = (0, 0)$ and another at $B = (0, 10)$. I wish to derive an equation for a line that, for any point $P$ on the line, would equal a set ratio for the distance $P$ to ...
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1answer
162 views

Finding the height wherein the given are only 2 angles of depression .

Can someone please help me with this problem? I really can't get it right. * From the top of the tower, the angle of depression of the base of the flagpole is 51 degrees, while from 2 floors ...
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2answers
54 views

Solve for x, Trig question.

Solve the following for x: $\sin(2x)=\cos(x)$ with $0\le x \le 2 \pi$ Not sure how to do this, all help appreciated.
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1answer
82 views

How are solutions are there for $\cos(97x) = x$?

How are solutions are there for $\cos(97x) = x$? Could anyone please tell me how to start?
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2answers
163 views

Solving for A when $f(x) = A\sin x$

Let A be a positive constant. If the graph of the function $f(x) = A\sin x$ intersects the graph of its derivative perpendicularly, then what does A equal? (This problem was provided by a professor ...
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3answers
159 views

What is the theta in this cartesian to polar coordinates problem?

If x=1 and y=1, what is the theta and why? I know for a fact that the answer is pi/4 but I do not get why.
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4answers
1k views

How to prove $\cot ^2x+\sec ^2x=\tan ^2x+\csc ^2x$?

How can I prove the following equation? \begin{eqnarray} \cot ^2x+\sec ^2x &=& \tan ^2x+\csc ^2x\\ {{1}\over{\tan^2x}}+{{1}\over{\cos^2x}} &=& ...
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3answers
86 views

Trigonometry, angles of depression.

A news helicopter hovers at a height of 500m. The angles of depression of a fire moving in the direction of the helicopter are first 10(deg) and then 15(deg), How far has the fire moved between thee ...
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3answers
70 views

Trigonometry: Law of Sines

How can this be solved using law of sines? I get a different answer then when I solve it using law of cosines..
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1answer
91 views

Prove this proprety of $f(x)$

I've asked this question before a long time ago, but I didn't get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$? Let ...
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2answers
441 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
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1answer
45 views

$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta$ ,$\theta=$?

This question is a follow up question to this answer. In the equation: $$(a - b \cot \theta) \cos^2 \theta = -\frac{b}{2} \cot \theta.$$ $a$ and $b$ are given. What is the best way to solve for ...
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2answers
224 views

Finding all rational $p,q,r$ satisfying $p\cos{\frac{\pi}{7}}+q\cos{\frac{2\pi}{7}}+r\cos{\frac{3\pi}{7}}=1$

Find all rational numbers $p,q,r$ such that $$p\cos{\dfrac{\pi}{7}}+q\cos{\dfrac{2\pi}{7}}+r\cos{\dfrac{3\pi}{7}}=1.$$ My idea: we can find $x=\cos{\dfrac{\pi}{7}}$ the equation root? Because ...
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4answers
115 views

For $\sqrt[3]{-1+i}$, is $r$ (when put in polar form) $\sqrt[6]{2}$?

And when you put that into the nth root form... It becomes $2^{1/18}\cos\theta + 2^{1/18}\sin\theta$? $n$th root form given is: $\sqrt[n]r\cdot\cos(\theta+2\pi k)n$
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1answer
124 views

there are two docks

There are two docks, dock A and Dock B, on a large lake. The distance between the two docks is 72.5 km. Dock B is directly east of dock A. One day, a steam boat leaves from dock A at noon, and heads ...
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3answers
143 views

Minimize the sum of tangents when sum of angles are constants

Minimize $\sum_{1≤ i≤ n} \tan(A_i)$ where $A_i>0$ and $\sum_{1≤ i≤ n} A_i=C$ where $\frac{\pi}{2}>C>0$ is some constant. I tried to use $\tan(\sum_{1≤ i≤ n} ...
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2answers
185 views

How to solve $\tan2x-\sin4x = 0$? [duplicate]

Possible Duplicate: How to find $x$ in some trigonometric equations How to solve these trigonometric equations? $$\tan2x-\sin4x = 0$$
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5answers
552 views

Prove $\tan(A+B+Y)=\frac{\tan A+\tan B+\tan Y-\tan A\tan B\tan Y}{1-\tan A \tan B-\tan B\tan Y-\tan Y\tan A}$

Fernando begs for help on this most difficult trig identity. $$\tan(A+B+Y)=\frac{\tan A+\tan B+\tan Y-\tan A\tan B\tan Y}{1-\tan A \tan B-\tan B\tan Y-\tan Y\tan A}.$$ I know $$\tan(A+B)=\frac{\tan ...
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1answer
1k views

Solving triangles and quadratic equations

When calculating the pieces in a triangle with only two sides and an intermediate angle is known, one must solve a quadratic equation. By solving the equation are 2, 1 or 0 solutions, as ...
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3answers
112 views

Inequality for $\cot$

How can I prove that for all $t\in[0,\frac{\pi}{2}], \cot^2t\leq\frac{1}{t^2}\leq1+\cot^2t$, with $\cot$ the cotangent function ? Thank you
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1answer
126 views

The graph of $\cot$ is the image of the graph of $\tan$ by a simple transformation

How can I justify that the graph of the function cotangent : $\cot$ is the image of the graph of the function $\tan$ by a simple transformation.
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1answer
197 views

Solution for following trigonometric equation?

I have the following trigonometric equation in $\theta$: $$0=G_{\omega}(1/r^2)({\csc^2}\theta){(r\cos\theta-x)}^2+(\cot\theta)(r\cos\theta-x)+r\sin\theta-y.$$ Is there an analytical solution for ...
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2answers
573 views

How do you simplify the addition of square root

$2\sqrt{x} + \sqrt{3}$ How do I simplify this?
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2answers
306 views

A complex number in the power of another complex number

I saw this question, and found a formula: $$=\cos \left( d\log |a+bi|+c\arctan \frac{d}{c}\right)+i\sin \left( d\log |a+bi|+c\arctan \frac{d}{c}\right).$$ Which I later translated to Microsoft Math ...
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4answers
344 views

tan=sec? test questions

If $\theta$ is in quadrant 1 and $\tan(\theta) = .6$ then $\sec(\theta) = $? This seems pretty easy to me: $\tan^2(\theta)-\sec^2(\theta)=1$ $-\sec^2(\theta)=.64$ $\sec(\theta)=8$ Another one, ...
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2answers
185 views

solving $\sec (3 \beta + 10) = \csc (\beta + 8)$

$\sec (3 \beta + 10) = \csc (\beta + 8)$ (in degrees) I am supposed to find one solution, and the angles are acute. I do not know the answer or how to get the answer. It is confusing for me ...
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1answer
4k views

angle theta in standard position

I have no idea how to start this homework. Here is the question. Sketch a angle theta in standard position such that theta has the least possible positive measure, and the given point is on the ...
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2answers
509 views

How to solve trigonometric equation $\sin(x)+x\cdot \cos(x)=0$?

I'm facing the problem of solving $$\sin(x)+x \cdot \cos(x)=0$$ using $$\tan(x)=\sin(x)/\cos(x)$$ I end at $$x+\tan(x)=0$$ on the other hand, I also tried $\cos(x)= \pm ...
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1answer
125 views

How to draw arrows by rotating lines in 3d space?

I am trying to figure out direction vectors of the arrowheads of an arrow. Basically I'm given a normalized direction vector ...
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3answers
436 views

The limit of $\sin(1/x)$ as $x\to 0$ does not exists

Prove that the following limit does not exist. $$ \lim_{x\to 0} \sin\left(1 \over x\right) $$ Our definition of a limit: Let $L$ be a number and let ${\rm f}\left(x\right)$ be a function which is ...
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4answers
298 views

Geometry: Auxiliary Lines

Geometry: Auxiliary Lines Any idea about this problem? Would be very interesting to see a geometric solution.
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1answer
535 views

Where is zero degrees on a graph

I am using the following formula to calculate the position of a point after rotation in my web application. x' = xcos(0) - ysin(0) y' = xsin(0) + ycos(0) But ...
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1answer
194 views

I am so confused!

Alright, so I have a triangle that is part of a mesh (3d model) and I need to get the rotation of it's x,y and z axis. Did I say all that right? rotation of y axis ...
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3answers
422 views

Limit of infinite loops of sin x as n tends to infinity [duplicate]

Show that $$lim_{n\to\infty} \text {sin sin ... sin x} = 0 $$ for all x. Note that the n here refers to the number of sin in the expression above.
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1answer
69 views

Is the series convergent? If convergent what will be the limit? [closed]

Is the series $$\sum\tan^{-1} \left(\frac1{2k^2}\right)$$ convergent? If convergent what will be the limit?
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3answers
338 views

Simplification of Sines and Cosines raised to a power - Fourier Series

I need urgent help simplifying a few sines and cosines into expressions involved sine and cosine to a maximum power of 1. e.g. $\sin^2x = 1/2 + 1/2\cos(2x)$ - this one is easy. $\sin^3x = \sin^2x ...
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3answers
2k views

Finding the values of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.

i know that the values of $\cos n\pi=(-1)^{n}$ and $\sin n\pi=0$. Now i want to know that what is the general expressions of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.
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5answers
713 views

Solving the equation $\sin t = -\sqrt{2}/2$

Solving the equation $$ \sin t = -\frac{\sqrt{2} }{2} .$$ I know the solution is $1.25$ and $1.75$, but I do not know how to get there. An explanation would be GREATLY appreciated, thanks!
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1answer
226 views

Finding two sides of a triangle [duplicate]

Possible Duplicate: calculate sides of the right triangle if I know 1 side and all the angles I'm not sure how to do this with only 1 side given, but I have a right triangle with a 30 ...