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
15 views

Proving Equations (Complex Conjugates)

The question is: $z$ is a complex number given by $z$ $=$ $sin$$(\theta)$ $+$ $i(1-cos(\theta))$, $-\pi < \theta < \pi $ Show that if $w$ $=$ $\frac 1{z-i}$ then $w$ $=$ $z^* + i $ where ...
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2answers
27 views

Given diagonals, lower base, and height, find the legs and upper base of isosceles trapezoid

Given an the height, base, and diagonals of an isosceles trapezoid, how am I to find the upper base and the legs? I know I can find the area of the triangles made by the diagonals, but how is that ...
3
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1answer
83 views

What is the standard deviation of this random variable? (I want to check my calculation)

Consider the random variable (orientation angle) $0\le\theta\le 2\pi$ with the following PDF where $\theta_0$ is the mean orientation angle: $(n\in\mathbb Z , n\ge 0)$ ...
3
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7answers
66 views

Evaluating the following limit: $\lim _{x\to \frac{\pi }{4}}\left(\tan\left(2x\right)\tan\left(\frac{\pi }{4}-x\right)\right)$

I don't find the right identities for this $$\lim _{x\to \frac{\pi }{4}}\left(\tan\left(2x\right)\tan\left(\frac{\pi }{4}-x\right)\right)$$ Someone can help me ? Thanks.
2
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0answers
38 views

How to tell if a polynomial has exact trigonometric or logarithmic roots?

The polynomial $$ 64x^7 -112x^5 -8x^4 +56x^3 +8x^2 -7x - 1 = 0 $$ has seven roots, x = {1, $-\dfrac{1}{2}, \cos \dfrac{2n\pi}{11}$}, where n={1,2,3,4,5}. Is there any way to tell if an arbitrary ...
0
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1answer
25 views

Cartesian to Spherical coordinate conversion specific case when Φ is zero and θ is indeterminant

Following is the conversion for spherical to cartesian coordinate \begin{align} x &= r \cos\theta \sin\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\varphi \end{align} and we are ...
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1answer
30 views

Solve equation for t

$$s = 2 \ln|\tan(t) + \sec(t)|$$ I tried to solve it and got a quadratic equation which turned out to equal $arcsin(\dfrac{-2 \pm e^s}{2(1+e^s)})$ This doesn't seem right. Any thoughts?
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5answers
79 views

How to prove $\sin3θ=3\sinθ-4\sin^3θ$

I was solving an A Level past paper (November 2014 P32) when I stumbled upon this question. It first asks us to expand $\sin(2θ+θ)$ which is easy using the identity $\sin(A+B)=\sin A\cos B+\cos A\sin ...
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1answer
83 views

Which function to kill: Sine or Cos?

I got an equation which was a solution to some familiar Differential Equation I am solving, the solution takes the form of: $$V=Ce^{-ix}$$ but $$Ce^{-ix}=A\cos(x)+B\sin(x)$$ so ...
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2answers
22 views

Largest possible sphere is inscribed in a cube. What percentage is the volume of the sphere smaller than the volume of the cube?

Largest possible sphere is inscribed in a cube. What percentage is the volume of the sphere smaller than the volume of the cube? I have already found out: volume of the cube is $X^3$ volume of the ...
2
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3answers
55 views

Checking the result of integration, $\int\sinh^3x\cosh xdx$

I integrated $\int\sinh^3x\cosh xdx$ in the following way: \begin{align*} \int\sinh^3x\cosh xdx ={}& \int\sinh^2x\sinh x\cosh xdx = \frac12\int\sinh^2x\sinh(2x)dx ={} \\ {}={}& ...
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5answers
56 views

How to solve this limit involving sine and log?

I've tried L'Hopital's Rule but the differentiated numerator involves cos(1/x) which does not exist when x approaches 0. $$ \lim_{x\to 0^+} \frac{x^2sin\frac{1}{x}}{\ln(1+2x)}$$
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2answers
39 views

Value of ratio of inverse trigonometric values

How to prove that the value of the following expression is $2$ manually? Please tell me the quickest method to deal with such problems. ...
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2answers
50 views

Find the value of $\left(\cos 52^{\circ}+\cos 68^{\circ}+\cos 172^{\circ}\right)$

Find $\left(\cos 52^{\circ}+\cos 68^{\circ}+\cos 172^{\circ}\right)$ $\color{green}{a.)\ 0 }\\ b.)\ 1 \\ c.)\ 2 \\ d.)\ \text{none of these} $ In exam I often fail to remember the formula's ...
3
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1answer
62 views

Summation of the given series

Is there anyway to find the sum of: $\cos(A)+\cos^2(2A)+\cos^3(3A)+....$ upto 'n' terms. Actually original question was to find sum of : $\cos(A)+\cos(2A)+\cos(3A)+...$ upto 'n' terms and I found it ...
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1answer
20 views

Calculating rotations required to

I have a situation similar to a question asked on StackOverflow with the following image: I have an object that rests at the tip of the green arrow (point XYZ), and I'm using the following formulas ...
2
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3answers
38 views

Find the maximum value of $(12\sin x-9\sin^{2} x)$

The maximum value of $(12\sin x-9\sin^{2} x)$ is equal to $a.)\ 3 \\ \color{green}{b.)\ 4} \\ c.)\ 5 \\ d.)\ \text{none of these}$ As $-1\leq \sin x\leq 1 ,\\ 12\sin x-9\sin^{2} x \\ ...
2
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3answers
54 views

If $\sin x+\sin^{2} x=1$ , Find $\cos^{12} x+3\cos^{10} x+3\cos^{8} x+\cos^{6} x+2\cos^{4} x+\cos^{2} x-2 $

If $\sin x+\sin^{2} x=1$, then the value of $\cos^{12} x+3\cos^{10} x+3\cos^{8} x+\cos^{6} x+2\cos^{4} x+\cos^{2} x-2 $ is equal to $a.)\ 0 \\ b.)\ 1 \\ c.)\ 2 \\ \color{green}{d.)\ ...
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2answers
29 views

Limit problem involving trig functions and radical

I am stuck on this limit and have no idea how to solve it and which trig identity to use. Any help would be appreciated. Thanks! $\lim\limits_{x \to 0^-} \frac{\sqrt{1+2\sin^2 ...
2
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3answers
49 views

Find the two other sides in a 15-30-135 triangle

A triangle has angle measures of 15, 30, and 135 degrees. The side opposite the 15 angle is x feet, the side opposite the 30 angle is y feet, and the side opposite the 135 angle is 2 feet. Find x and ...
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4answers
61 views

Integral $\int_0^{\pi/2} \sin(ax)\cos(x)\,dx$

I have to evaluate an integral $I(a) = \sin(ax)\cos(x)$ from $0$ to $\pi/2$.The variable of $a$ is not is greater than $1$: $$\int_0^{\pi/2} \sin(ax)\cos(x)\,dx$$ I attempted to change the function ...
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1answer
35 views

Page 72 of Courant and Hilbert's Methods of Mathematical Physics, Vol 1.

We have the following identities: $$ \beta_\nu = b_\nu -\frac{1}{2}(b_{\nu-1}+b_{\nu+1}),\ \ \ \ (\nu=2,3,4,\ldots)\\ \beta_1=b_1-1/2 b_2 $$ $$s_n(x)=\sum_{\nu=1}^n b_\nu \sin(\nu x) \\ ...
0
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1answer
22 views

In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$

In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$, Given $A+B+C = \pi$ $\bf{My\; Try::}$ Using $\bf{A.M\geq G.M}$ $$\frac{\cot^2 A+\cot^2 B}{2}\geq \cot ...
11
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1answer
232 views

What is the geometry behind $\frac{\tan 10^\circ}{\tan 20^\circ}=\frac{\tan 30^\circ}{\tan 50^\circ}$?

This identity is solvable by help of trigonometry identities , but I think there is an interesting and simple geometry interpretation behind this identity and I can't find it. I found it when I ...
0
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2answers
35 views

How to differentiate $ y=\sin^2(2x)\cos(x) $?

I was solving some A Level past papers and I came across this question. We have the equation of the line $ y=\sin^2(2x)\cos(x) $ for $ 0\leq x \leq \frac{\pi}{2} $ and there is a maximum point M. We ...
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4answers
53 views

If $\boxed{\tan A+\tan B+\tan C=6, \\ \tan A\tan B=2} $ in $\triangle ABC$ then find the type of triangle.

In $\triangle ABC$, $\tan A+\tan B+\tan C=6 \\ \tan A\tan B=2 $ Then the triangle is $a.)\text{Right-angled isosceles} \\ b.) \text{Acute-angled isosceles}\\ ...
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3answers
56 views

If $A=\sin^{20}\theta +\cos^{48}\theta $ then identify the correct option.

If $A=\sin^{20}\theta +\cos^{48}\theta $, then for all values $\theta$ a) $A\geq 1$ b) $ 0< A\leq 1$ c) $1<A< 3$ d) None of these $0 \leq \sin^{20}\theta \leq 1$ $0 \leq ...
0
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3answers
57 views

Integration of $\arctan$

This question seems so silly, yet I want to know if this is true! I know for a fact that the following is true$$\int \frac{1}{1+x^2} \,\text{d}x = \arctan(x) + C$$ But does this mean this is true ...
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2answers
26 views

Finding roots of cubic (trig)

The question is By putting $x$ $=$ $\frac 23 cos (\theta)$ Find the exact roots of the equation in terms of $\pi$ $$ 27x^3 - 9x = 1 $$ What I have attempted: $$ ...
2
votes
2answers
63 views

Sin(x): surjective and non-surjective with different codomain?

Statement that $\operatorname{sin}(x)$ not surjective with codomain $\mathbb R$ and surjective with codomain $[-1,1]$ found here: Non-surjective: $\mathbb{R}\rightarrow\mathbb{R}: ...
1
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1answer
27 views

Inverse trigonometric expansion related question

I know expansions for $\sin^{-1}(x)+\sin^{-1}(y)$, but does there exists any expansion for $\sin^{-1}(x \pm y)$ if not then what is the reason?
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1answer
43 views

How sin(90°+ θ) is equal to M'P'/OP' or Cos θ?

I'm learning Trigonometry right now with myself and at current I'm understanding how to find the trigonometric ratio of the angle (90°+ θ) in those of θ. I'm little bit confused right now in the ...
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3answers
49 views

$\sin A \in [-1,1]$ and $\cos A \in [-1,1]$. Then why is $\tan A =$ more than $1$ or less than $-1$ [closed]

$\sin A \in [-1,1]$ and $\cos A \in [-1,1]$. Then why is $\tan A $ outside of $[-1,1]$?
5
votes
2answers
51 views

Solve $\sin x = x - 2 \pi/3$

What is $x$ if $\sin x = x - 2 \pi/3$? The answer is $x \approx 2.61$ but how do I work that out (without Taylor series - this is homework for 10th grade)? Thanks.
1
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1answer
46 views

Integral with simple fractions

I have a problem with this integral $$\int_\ \frac{\cos x }{\sin x \sqrt{1+\cos^2x}} \, dx$$ Using substitution $u = \sin x $ we get $$\int_\ \frac{1 }{\ u \sqrt{2-u^2}} \, du$$ I think the ...
0
votes
4answers
73 views

Calculate the Limit as x approaches 0

I am asked to calculate the following limit $$ \lim_{x\to0}\frac{\ln(1+\sin x)}{\sin(2x)} $$ First, I tried expressing $1+\sin x=t$, then express $x$ from that equation but my equation seemed to just ...
1
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1answer
44 views

Evaluate $\int_{\pi}^{3 \pi/2}\frac{1}{1-\rho \sin{2 \theta }} d\theta$

I would like to evaluate $$\int_{\pi}^{3 \pi/2}\frac{1}{1-\rho \sin{2 \theta }} d\theta$$ For $-1<\rho <1$. Unfortunately nothing I have tried has got me very far so I would appreciate ...
0
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1answer
46 views

Sampling the Sine Function

Consider the sampled sine function, $f(n)=\sin(\omega n)$, where $n$ is an integer. If $\omega_2 = 3\pi/2$, does there exist an $0 \leq \omega_1 \leq \pi$ such that $\sin(\omega_1 ...
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2answers
44 views

De Moivres Theorem question and complex numbers

Question is: Find the cube root of $27 (\cos 30° + i \sin 30°)$ that, when represented graphically, lies in the second quadrant. I did this: ...
1
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1answer
37 views

Solving $\int\sec^3(x) \, dx$ example, trouble getting from step to step

I'm doing an example problem that is solving the integral of $\sec^3(x) \, dx$. The first step shown in the problem takes it from that to: $$\sec(x)\tan(x) - \int \sec(x)\tan^2(x) \, dx$$ The book ...
0
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5answers
64 views

If $\tan^2(\theta)+2\sec^2(\theta)=5$. Find the value of $\sin^2(\theta)$

I have a trig problem which i can't really understand where to start. It says If $$\tan^2(θ)+2\sec^2(θ)=5.$$ Find the value of $$\sin^2(θ).$$ I think it has something with to do with Pythagorean ...
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2answers
30 views

Given the real number $t = - \frac 5 4 \pi$, give the values of the sine, cosine and tangent.

I've found that these problems are quite hard for me, such as this: given the real number $t = - \frac 5 4 \pi$, give the values of the sine, cosine and tangent. Is there any good explanation on HOW ...
0
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1answer
47 views

As x approaches infinity, why does $ \lim_{x \to \infty}\arctan \left(\frac{x-2}{2}\right) = \frac{\pi}{2} $ [closed]

Just wondering why the following is true! $$\lim_{x\to \infty } \arctan\left(\frac{x-2}{2}\right) = \frac{\pi}{2} $$ Thanks!
3
votes
3answers
68 views

Proving whether the series $\frac{\cos(n)}{n}$ is absolutely convergent

I have the infinite sum $$\sum_{n=1}^\infty \frac{\cos(n)}{n}$$ and I am able to show that it is conditionally convergent by using the Dirichlet Test (and the Lagrange Trig Identity to show the ...
2
votes
3answers
24 views

How Angle AOP' is equal to (90° - θ) in the second figure?

I'm learning Trigonometry right now with myself and at current I'm understanding how to find the trigonometric ratio of the angle (90°- θ) in those of θ. I'm little bit confused right now in second ...
3
votes
0answers
29 views

Is it always possible to find the roots of $P(z)=az^4+bz^3+cz^2+bz+a$, where $a,b,c \in \mathbb{R}^*$, by first dividing both sides by $z^2$?

A classic way to solve quartics in the form $P(z)=az^4+bz^3+cz^2+bz+a$, if we know that the roots lie on the unit circle, is to divide both sides by $z^2$ and then use the fact that if $$z=\cos \theta ...
0
votes
1answer
28 views

Find $\sinh^{-1}x$

The hyperbolic sine function, $\sinh(x)$ , is defined by the equation: $$ \sinh(x) = \frac {e^x-e^{-x}} {2}$$ Find a formula for its inverse, $$ \sinh^{-1}(x) $$
0
votes
1answer
27 views

Sum of transformations of continuous uniform random variable

Let $X$ be uniformly distributed on $(a,b)$. I want to find the cdf of $$ \sin^2(X) + \cos^2(X) $$ My feeling is that since $\sin^2(X) + \cos^2(X) = 1$, the cdf will be: $$F(1 \le x)= \begin{cases} ...
0
votes
2answers
37 views

Find all solutions to the equation $3 \sec^2(x) - 4 = 0$

I know this is possible using the quadratic formula, but I want to find a cleaner way (If possible) to solve this problem. Any help is appreciated, Thanks.
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votes
1answer
29 views

A trigonometry based triangle problem

In the triangle ABC below, side a is 10 units, and side b is 12 units. cos(angleACB) = 1/5. Find the value of cos(angleCBA). I'm pretty sure that I should use the law of sines, or the law of cosines, ...