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

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2answers
70 views

Which do you prefer sine or cosine? [closed]

A soft question. Which do you prefer between sine function or cosine function? While I feel the sine funciton more clear and comfortable, but I prefer cosine, since $\Re{e^{ix}} = \cos{x}$, The ...
0
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1answer
25 views

Trigonometry for steradian angle

Polar coordinate system is very closely associated with trigonometry. For instance, given an angle in radian, we can find its corresponding 2-D cartesian coordinates using ...
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2answers
75 views

Establish a trigonometry-based $floor$ function

I have established the following function for calculating $floor$: $$f(x)=x-\frac{1}{2}-\frac{\arcsin(\sin(\pi(x-\frac{1}{2})))}{\pi}$$ It works correctly for all real values in the range ...
-1
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0answers
39 views

Trig identity relating angles in a triangle [duplicate]

If $\alpha,\beta$ and $\gamma$ are angles in a triangle show that $$\cos\alpha + \cos\beta + \cos\gamma -1=4\sin\frac{\alpha}{2}\sin\frac\beta2\sin\frac\gamma2$$
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2answers
24 views

Is it possible to get the angle between two vectors in a single direction?

I'd like to compute the angle between two vectors but always in a anticlockwise manner. Is this possible? I know the formula is arc cos (dot product of vectors / product of magnitudes of vectors) but ...
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1answer
62 views

How do I simplify a radical within a radical in this half-angle problem?

I don't understand how to simplify the following radicals and arrive at the final answer below. I can make it to this point: $$\sin\left(-\frac{3\pi}{8}\right)=\pm\sqrt{1+\frac{\sqrt2}{2}\over2}$$ ...
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1answer
25 views

Proof of half angle identity

Remember one of the step in proving it is to replace theta with 1/2 A. I am wondering how could you replace a variable to prove the identity. A is not the same as theta. Plus aren't we trying to find ...
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2answers
166 views

Fun Integral $ \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}$

$$ I\equiv \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}. $$ This integral does have a closed form. I am not sure where to start. We can factorize the denominator as $$ \cos^3 x+2\sin(2x)-5\cos ...
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3answers
36 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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3answers
130 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
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1answer
20 views

With which function I can replace $\arctan x^2$ in limit calculation?

I have read I can replace in limits $\arctan x$ with $x$. I conjectured that I can replace $\arctan x^2$ with $x^2$, however in following example it doesn't work: $$\frac{2x\left( 105x^4+150 x^2 ...
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2answers
54 views

Why are differential of $\sin^2(x)$ and integral of $\sin(2x)$ not the same?

I was working on a list of common integrals and differentials and I came across this question. If $${d\over d\theta}(\sin^2\theta) = \sin(2\theta)$$ Then why is $$\int \sin(2\theta) \space d\theta = ...
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1answer
17 views

Question about graph of sin function

There is question I had on my mind for 3-4 years. If you have a function:- f(x) = (sin(x))^n If you increase the value of n slowly the Value of the function at all points, except where it is one, will ...
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2answers
44 views

Proof for value of sum of sine and cosine

I have come across those sums of sine and cosine, while trying to show that windmills dont move without external force. Although it is clear that should be the case, i'm stuck in proofing it. I wonder ...
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2answers
29 views

If $y=\sec{^2θ} + \cos{^2θ}$ ,$θ$ isn't then.. [closed]

Cont......................... Y The given answer is $y ≥ 2$.. What I know is that value of $\sin$ and $\cos$ varies only from 1 to -1 which means that the answer should be $y ≤ 2$. What is the ...
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2answers
31 views

How to prove this part of a problem?

In triangle $ABC$, $\angle A=x$, $\angle B=2x$ and $AC/BC= 1/2000$ Can you prove that $x$ must be in between $0$ and $90$ degrees, and that $\sin (x)$ cannot be $0$. I can kind of explain this, ...
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2answers
29 views

The Distance Conundrum

I have sometimes wondered about a distance problem that involves travelling along the two triangular sides of distance between two points, then gradually shortcutting the distances into smaller and ...
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3answers
20 views

Find the exact value of the six trigonometric functions of 285° using a trig identity

I need help using the sum identity I have tried to use reference angles but I don't know how to start because the question just says 285° and not, for example sin285°
2
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4answers
45 views

Can someone explain this?

$\sec(x/2) = \cos(x/2)$ I worked on this and got here... (Let (x/2) = u) $\cos u - \sec u = 0$ $\cos u(1 - \sec^2u) = 0$ $\cos u[ -1(-1 + \sec^2u)] = 0$ $\cos u(-\tan^2u) = 0$ So, the solutions ...
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2answers
37 views

Is there a way to integrate $\cos^{2} {3x}$ with a different technique than integration by parts?

The question is just as it is on the title: Is there a way to integrate $\cos^{2} {3x}$ with a different technique than integration by parts? And in case there is, how can I do it?
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2answers
51 views

Calculate for $(1+\tan 20^\circ)(1+\tan 25^\circ)$. Help me with my works

I have no idea what I am doing here, I started with $\tan 20^\circ=\tan(45^\circ-25^\circ)=(1-\tan 25^\circ)/(1+\tan 25^\circ)$ I am sure the work I have shown so far are ok, but how do you get ...
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1answer
10 views

Why aren't these solutions valid for an equivalent form of the terms in this equation?

Sec(x/2) = cos(x/2) I worked on this and got here... (Let (x/2) = u) Cosu - secu = 0 Cosu(1 - Sec^2u) = 0 Cosu[ -1(-1 + sec^2u)] = 0 Cosu(-tan^2u) = 0 So, the solutions would be: x = pi + ...
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2answers
34 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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0answers
23 views

Geometrical interpretation of tan(x) > x [duplicate]

Is there a geometrical interpretation for $tan(x) > x$ when $x \in (0, \frac{\pi}{2})$ in the unit circle? I can't picture it since $x$ is the angle (although it makes sense when you graph the two ...
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1answer
25 views

General solution of trigonmetric equations

Is there any formula for general solution of equations like $|\cos \alpha| \leq \beta$ ? Specifically, $|\cos x| \leq \frac{1}{2}, x \in [0,1]$ ? I want all the values of $x$ satisfying the above in ...
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2answers
37 views

Limit of composite trigonometric function

I am trying to find an easy way to compute the limit as $x \to 0$ of $$f(x) = \frac{\sqrt{1+\tan(x)} - \sqrt{1+\sin(x)}}{x^3}$$ from first principles (i.e. without using l'Hôspital's rule). I have ...
0
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1answer
24 views

How do I use limit laws to evaluate $\lim\limits_{ x \to \pi/2} [\tan(x) (\sin^2(x)-1)]$?

I'm having trouble with the limit laws.. especially when it comes to anything that has trig in it.
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2answers
28 views

Using trigonometric ratios to express area of regular polygons

I am very confused. My book just asked me to use trigonometric ratios to express the area of a regular polygon with 9 sides and lengths of 8. I don't even know what this means. So far I have ...
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1answer
41 views

Angle measurement - radian and degree

I want to present my reasoning here about angle measurement and to get some opinions about its correctness and usage. About radians: One radian denotes the angle that has the center of the circle as ...
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2answers
40 views

Is this a typo, or am I missing something?

I have a handout for my precalc II class. It says $\sinh(-x) = -\sin(x)$ It should be $\sinh(-x) = -\sinh(x)$ right? I don't see how a negative input could make a hyperbolic function circular.
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0answers
14 views

Solution to Tanθ = -3/4 in converting to cylindrical coordinates

I am attempting to convert (8, -6, 7) from rectangular coordinates into cylindrical coordinates. We have r = 10, but then I end up with tanθ = -3/4 and I am not sure how to get an exact answer for ...
4
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2answers
337 views

How can I prove this trigonometric statement true?

$$ {1+\sin^{2}\left(x\right) \over \cos^{2}\left(x\right)} = 1 + 2\tan^{2}\left(x\right)$$ This statement is part of a larger problem, but I need to prove that this is true before moving on. I'm ...
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1answer
30 views

Integral about lengh o f an arc

I can't find a way to solve this: $$ \int_{\pi/2}^{\pi} \sqrt{8sen^2(t)cos^2(t)}dt $$ The integral is to calculate the length of an arc, by parametric equations. The answer is $\sqrt{2}$, but i'm ...
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5answers
35 views

I have a trigonometry question that needs to be answered

Can someone please answer the question: If $\tan{2\alpha} = \frac{1}{2}$, find $\sin{2\alpha} $ and $\cos{2\alpha}$? Thank you.
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0answers
40 views

Formula for nth derivative of $\arcsin^k(x/2)$

I need to find formula for $n$-th derivative of $\arcsin^k(\frac{x}{2})$. I have found formula for ...
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4answers
77 views

What is $\cos(k \pi)$?

I want to ask question for which I have been finding answer for. Please could anyone explain me why $\cos(k \pi) = (-1)^k$ and also explain me same for $\sin(k \pi)$?
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2answers
29 views

Sin & Cos Equation/Relation

If sin(x) = 0.3, find cos(pi-x) how i would solve this: let x = sin-1(0.3) solve for cos(pi-[sin-1(0.3)]) Is there a way to solve this by hand? Is the above method wrong?
0
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1answer
47 views

Why do different trig functions sum differently?

Why does the $\sum_{n=1}^{\infty} \sin (\frac 1 {n^2})$ converge but the $\sum_{n=1}^{\infty} \cos (\frac 1 {n^2})$ diverge?
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1answer
42 views

Help with this trigonometry problem?

Is there an easier way of doing this problem: A square tower stands upon a horizontal plane. From a point in this place from which three of its upper corners are visible their angular elevations ...
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0answers
39 views

An isosceles triangle with a measure of the sides abc of $5,5$ and $4$.

Find the angles of the triangle in an isosceles triangle of length 5 as the hypotenuse and $\sqrt{21}$ as the height of the triangle as well as the angle bisector, and measure the angles and find ...
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0answers
34 views

Polygones inscribed with in a circle

Let's say that there is a circle in two dimension and the diameter of the circle is 1.First start with an equilateral triangle inscribed with in the circle and the measure of the angles are equal to ...
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0answers
15 views

Trig Star problem

I need to find an angle where I have a radius of 50. The Radius starts at point A which Forms a 90 degree angle CAF the distance between C & F is 70.71. The angles For ACF and AFC are both 45 ...
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2answers
57 views

Finding the lengths of the sides of a triangle given 3 angles only.

If a right triangle ABC with an angle A at 90 degree, B 45 degree, C 45 degree is their a way of finding the length of the sides abc without knowing any of their lengths. Normally we use ...
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2answers
35 views

Unusual result when comparing trigonometry and Pythagoras in triangles.

I'm a Scottish Higher maths student. I was looking over some old textbooks, and came across a seemingly easy question, involving a circle within a triangle. I used the expected method to solve it; ...
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2answers
28 views

How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$

How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$ I tried to factor and I got $2\cos^4(x)+(-2\sin^2(x)-3)(\cos^4(x)+\sin^4(x))$ but that doesn't lead me to my goal. ...
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1answer
21 views

Trigonometry, rewriting an expression

How do I rewrite $((\cos t)^3) - 2((\cos t)\cdot((\sin t)^2))$ to $3(\cos t)^3 - 2 \cos t$?
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0answers
23 views

Multiple Waves all in phase (Wave packets)

Suppose there are seven waves of slightly different wavelengths and amplitudes and we superimpose them (textbook is talking about wave packets). The wavelengths range from $\lambda _9 = 1/9$ to ...
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2answers
24 views

Trig Reduction With Pythagoras #2

If $\sin 10^\circ = p$, then determine $$\sin 280^\circ$$ in terms of $p$. $$\sin 280^\circ=\sin(180^\circ+100^\circ) = -\sin 100^\circ$$ $$-\sin 100^\circ = -\sin(90^\circ+10^\circ) = -\sin ...
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1answer
19 views

Trig Reduction with Pythagoras

If $\sin 10 = p$, then determine $$\tan^2 30^\circ \times \tan^2190^\circ$$ in terms of p.
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2answers
51 views

Trigonometry and Quadratic Equations

If $\tan x+\tan^2 x+\tan^3 x=1$ Then, find the value of $2\cos^6 x-2\cos^4 x+\cos^2 x$.