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

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-3
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0answers
15 views

use De Moivre's theorem to show that: cos(3theta) = 4cos(3theta)-3cos(theta) and sin(3theta) = 3sin(theta) - 4sin(3theta)

let z = cos(theta) + isin(theta) a) find z^3 using binomial expansion b) use De Moivre's theorem to show that: cos(3theta) = 4cos(3theta)-3cos(theta) and sin(3theta) = 3sin(theta) - ...
2
votes
1answer
78 views

Integral with Logarithms

$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\log(\sin(x)) \ dx } = \dfrac { \pi { \ln}^{ A }(B) }{ C } -\dfrac { { \pi }^{ D } }{ E } $$ $$$$ This was one solution, but it went completely ...
0
votes
1answer
21 views

Cosine problem for use in fourier series

If $k$ is positive $\cos(k\pi) = (-1)^k$ If $k$ is negative $\cos(k\pi) = $ ?
1
vote
6answers
82 views

Evaluate $\lim_{x \to 0} \frac{\sin(x³)}{x}$ without L'hopital rule

I am trying to evaluate $$\lim_{x \to 0} \frac{\sin(x^3)}{x}$$ without L'hopital rule. I've tried Squeeze theorem but no luck.
1
vote
3answers
43 views

Consider the parametric curve: $x=6\cos^3(t), y=6\sin^3(t)$, write it in cartesian form.

Consider the parametric curve: $$x=6\cos^3(t), y=6\sin^3(t)$$ Write it in Cartesian form. I am really struggling with the solution for this. I've been trying to find $t$ from $x$, and then ...
0
votes
1answer
30 views

General expression that represents a combined period of 2 sine functions

How to find the general expression that represents the combined period of $y=5\sin(\pi x/6)$ and $y=3\sin(\pi x/4)$? what are the limitations of this model?
-3
votes
3answers
29 views

Finding the length of the two equal sides of an isosceles triangle only given the area!!! [closed]

The question is~ The triangle has the area of 40m squared and has a 90• angle, what is the length of the two equal sides (rounded to the nearest metre) Please help i don't even know where to start ...
0
votes
1answer
24 views

Eigenfunction and their orthogonality with respect to the weight function

The Eigenfunction and their orthogonality with respect to the weight function $$\sigma$$ is defined as $$\int _a{}^b\phi _n\text{(x)}\phi _m\text{(x)$\sigma $(x)dx=0}$$. Given that I have some ...
0
votes
1answer
34 views

Solve, for $0 \leq x < 360$, $5\sin 2x = 2\cos 2x$. What about $\cos \theta = 0$?

Solve $5\sin 2x = 2\cos 2x$ for $0 \leq x < 360^\circ$. Let $\theta = 2x$. Then $(5\tan \theta -2)\cos \theta = 0.$ So $\tan \theta = \dfrac 2 5$ or $\cos \theta = 0.$ Calculating the ...
1
vote
4answers
56 views

How to solve $ \int \tan{\theta}\sec^4{\theta}d\theta $

I got the answer of: $$\large{\frac {\tan^2{\theta}}{2} + \frac {\tan^4{\theta}}{4} + C}$$ Wolfram Alpha got: $\large{\frac {sec^4{\theta}}{4}}$. I don't know what I did wrong: I set $\large{u = ...
2
votes
2answers
32 views

Calculating difference of sin and cos

Need to calculate $\sin 15-\cos 15$? In degrees. I got zero, but it is wrong. Though, it seems to me that I was solving correct, I was doing this was: $\sin (45-30) - \cos (45-30) = ...
0
votes
0answers
8 views

Finding common roots to a variable number of functions

I am trying to solve the following problem. Given $a\in\mathbb R^n$, $u\in\mathbb{R}^n$, $m\in\mathbb{N}^\star$, Find the/some common roots $(t_1,...,t_m)$ of the $\frac{m(m-1)}{2}$ ...
6
votes
3answers
139 views

A curious equation containing an integral $\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}$

I came across an interesting problem that I do not know how to solve: Find $x>0$ such that $$\int_0^{\pi/4}\arctan\left(\tan^x\theta\right)d\theta=\frac{\ln2\cdot\ln x}{16}.$$ Could you ...
2
votes
4answers
71 views

How to determine negative values of arc functions?

Can someone explain me how do they work? I memorized the table of values for sin/cos/tan/cot and radian-degreee values. But, how come, when it is $\arcsin -1/2$ this is $-\pi/6$. So the same thing as ...
0
votes
2answers
38 views

How to solve trig equation?

I know it must be really simple, but I really do need some help before exams, so, please: I need to find $\sin x + \cos x$ if $\tan(x) = 4/3$ and $0<x<90$
0
votes
0answers
33 views

Need proof of integration of sine parametrized functions [duplicate]

Yesterday, i encountered an integral formula (actually it's a generalization, i think). This : $$\int_0^\pi x f(\sin x)\,dx = \frac{\pi}{2} \int_0^\pi f(\sin x)\,dx$$ For simple functions like ...
1
vote
1answer
35 views

Confused on the argument of this function?

So say I wish to go from $$12\sin (t)+4\cos(t)$$ to the form $$A\cos (t+k)$$ by using the double angle formula I can get that $$\cos(k)=4$$ and $$\sin(k)=-12$$ and so we can find ...
0
votes
0answers
12 views

How to determine the period of the following functions?

How would the following make a difference to the period of a function? $$ \cos(t)~~~~ (1)$$ $$ \cos(\omega t)~~~~ (2)$$ $$ \cos(\omega t + \phi)~~~~ (3)$$ Would this be right, $(1)$ has period ...
4
votes
4answers
52 views

Show that $ \frac{\cos5x + \cos4x} {1-2\cos3x} = -\cos2x -\cos x $

The Question reads - $$ \frac{\cos5x + \cos4x} {1-2\cos3x} = -\cos2x -\cos x $$ I tried using the obvious approach by converting $5x , 4x $ and $ 3x$ to either $2x$ or $x$ but all that seemed to do ...
3
votes
3answers
28 views

Proving simple trigonometric identity

I need help with this one $$ \frac{\sin^2 \alpha}{\sin\alpha - \cos\alpha} + \frac{\sin\alpha + \cos \alpha}{1- \mathrm{tan}^2\alpha} - \cos\alpha = \sin \alpha $$ I tried moving sin a on the other ...
0
votes
4answers
32 views

How deep is the water? [closed]

A bowl is in the shape of a hemisphere (half sphere) with radius 10 cm. The surface of the water in the container has a radius of 7 cm. How deep is the water?
3
votes
2answers
83 views

How to prove the the addition of tangent is the same as the multiplication? [duplicate]

If A,B,C are angles of a triangle show that: $$\tan A+ \tan B+\tan C = \tan A \tan B \tan C $$ I've tried this many times but I cannot seem to prove it, can someone show me how to solve this ...
0
votes
3answers
34 views

How to go from this equation in terms of cosine and sine to this one in terms of only one. [duplicate]

$$x=A\cos(\omega t)+B\sin(\omega t )\equiv\mu \cos(\omega t+\phi)$$ I'm thinking it must have something to do with the double angle forumla. Any help?
4
votes
3answers
189 views

How to find cotangent?

Need to find a $3\cot(x+y)$ if $\tan(x)$ and $\tan(y)$ are the solutions of $x^2-3\sqrt{5}\,x +2 = 0$. I tried to solve this and got $3\sqrt{5}\cdot1/2$, but the answer is $-\sqrt{5}/5$
7
votes
2answers
75 views

Roots of a polynomial whose coefficients are ratios of binomial coefficients

Prove that $\left\{\cot^2\left(\dfrac{k\pi}{2n+1}\right)\right\}_{k=1}^{n}$ are the roots of the equation $$x^n-\dfrac{\dbinom{2n+1}{3}}{\dbinom{2n+1}{1}}x^{n-1} + ...
0
votes
0answers
33 views

As the Moon revolves around the Earth, the side of the Moon that faces the E..

As the Moon revolves around the Earth, the side of the Moon that faces the Earth is partially illuminated by the Sun. These "phases" of the moon are described by a fraction F of a lunar disc. For ...
0
votes
4answers
35 views

How to find maximum value of trig function?

How to find maximum value of this: $$y = 5\sin x - 12\cos x$$ And I am more intrested in solving process, rather than answer. I know the answer. I am familiar with derivatives, not so good, but as I ...
4
votes
1answer
44 views

Trigonometric root of a polynomial

If $4\cos^2 \left(\dfrac{k\pi}{j}\right)$ is the greatest root of the equation $$x^3-7x^2+14x-7=0$$ where $\gcd(k,j)=1$ Evaluate $k+j$ I tried factorizing the equation but it wasn't ...
4
votes
2answers
78 views

Find the value of : $\cos x \cos 2x…\cos 999x$ given that $x=\frac {2\pi}{1999}$

Given $x=\frac {2\pi}{1999}$ Find the value of $$\cos x \cos 2x \cos 3x ...\cos 999x$$ So I tried expanding $\sin {2000x}=2\sin 1000x \cos 1000x$ Then rewriting $\cos 1000x= \cos {(999x+x)}$ No ...
1
vote
2answers
38 views

If $\frac {\sin^4\theta}a +\frac {\cos^4\theta}b=\frac 1 {a+b}$ P.T $\frac {\sin^8\theta}{a^3} + \frac {\cos^8\theta} {b^3} = \frac1{(a+b)^3}$ [duplicate]

The question reads if $$\frac {\sin^4\theta} {a} + \frac {\cos^4\theta} {b} = \frac 1 {a+b}$$ Then prove that $$\frac {\sin^8\theta} {a^3} + \frac {\cos^8\theta} {b^3} = \frac 1 {(a+b)^3}$$ I ...
6
votes
1answer
71 views

Proving the existence of $b$ such that $\prod_{k=1}^n(1-\cos(a_k-b))=\frac{1}{2^n}$

Let $n>0$ and $a_1,\ldots,a_n\in \mathbb R$. Prove there is some $b$ such that $\prod_{k=1}^n(1-\cos(a_k-b))=\frac{1}{2^n}$ This is motivated by this question Finding a point on the unit ...
1
vote
0answers
42 views

proof of inequality perhaps using trigonometric identity

I need help on the following problem. Let $x,y,z$ be the positive real numbers and satisfy $x+y+z=xyz$ then, ...
0
votes
1answer
33 views

Clarifications on hyperbolic identity

On one of my workings in a tutorial sheet, It is stated that $$\left.\text{ae}^{\left(\frac{n\ \pi \ x}{H}\right)}+\text{be}^{-\left(\frac{\text{n$\pi ...
0
votes
0answers
37 views

How to calculate a person's Latitudinal and Longitudinal location based off of Sun and time

INTRO I remembered hearing about it being possible to calculate a person's position or the position a picture was taken, based on time of day and the position of the Sun, position meaning latitudinal ...
1
vote
2answers
25 views

find the solutions to the equation $4\sin^2\theta + 1 = 6\sin\theta$ in the interval $0^\circ \leq \theta < 360^\circ$

Find the solutions to the following equation for $0^\circ \leq θ < 360^\circ$: $$4\sin^2 θ + 1 = 6\sin θ$$ My work: $$4\sin^2\theta - 6\sin\theta + 1 = 0$$ Factor $$\sin\theta= \frac{1}{4}(3+ ...
2
votes
0answers
44 views

area of rotated squares on top of each other

I've got a problem of calculating 'visible' areas of rotated squares on top of each other. Say I have two squares $s_0$ and $s_1$ on top of each other. They have the same side length and same center. ...
0
votes
0answers
42 views

Is there such thing as a Co-theta?

I plan to make a notation for trigonometry called co$\theta$ An example is that co$\theta$ is equal to 45 and $\theta$ is equal to 45. Co$\theta$+$\theta$=90. In this equation, ...
0
votes
1answer
44 views

$x+y = π/2$ and $ \sin x +\cos y =1$. We should find $\sin x =$?

Maybe I rushed a bit and thought that the answer was $\sin x = 1$, but still after watching it closely I can not understand why this is not a possible answer. (The correct answer in my book is $0.5$)
-1
votes
2answers
40 views

Use the quadratic formula to find all degree solutions and θ if 0° ≤ θ < 360°

Use the quadratic formula to find all degree solutions and $\theta$ if $0° \le \theta < 360°$. Use a calculator to approximate all answers to the nearest tenth of a degree. (Enter your answers as ...
1
vote
0answers
53 views

3D surface intersections

I tried to look at 3D Hypersurface intersections of 4D this way based on four Mathematica (circular) trigonometric parametrization combination selections. No hyperbolic functions are directly ...
2
votes
2answers
90 views

Distance between two lines, passing through origin

How could I find the distance between two lines if I want that distance to be measured through a point (such as the origin)? The two lines would be straight lines, such as x=20, y=-15. The solution ...
2
votes
0answers
19 views

Calculate new pitch and roll after rotating about the z axis

I am wanting to know how to find out the new pitch and roll values when rotating around a circle. I have become a little stuck on how to achieve this, but hopefully someone will be able to point me in ...
2
votes
5answers
70 views

Purely algebraic proof of the trigonometric inequalities

While calculating various limits of trigonometric functions, one must resort to the squeeze theorem which is founded on the inequalities $$1 > \frac{\sin x}{x} > \cos x$$ for some "small" $x$. ...
0
votes
2answers
32 views

Calculating the right angled triangle's cathetus

We just started learning the Pythagorean theorem at school and we got a pretty difficult assignment. 5 meter tall bamboo broke and the top of it touched the floor 2 meters from the base of the ...
0
votes
2answers
56 views

If $\sin(\pi \cos(a)) = \cos(\pi \sin(a))$ then show that $\sin(2a) = 3/4$ [closed]

If $$\sin(\pi \cos(a)) = \cos(\pi \sin(a))$$ then show that $ \sin(2a) = \frac{3}{4}.$ Please tell me how to calculate the value step by step.
3
votes
1answer
49 views

Why does $a\cdot \cos(kx)+b\cdot\sin(kx)=\sqrt{a^2+b^2}\sin(kx+\phi)$ hold?

I've just bought a book to learn how Laplace, Fourier- and z-transformation works and stumbled over \begin{align} a_k\cos(kx)+b_k\sin(kx)&=A_k \sin(kx+\varphi_k)\\ &= ...
1
vote
1answer
22 views

Find all degree solutions in the interval 0° ≤ θ < 360°

Need help with a math problem. Any help is greatly appreciated. Find all degree solutions in the interval $0° ≤ θ < 360°$. If rounding is necessary, round to the nearest tenth of a degree. Use ...
3
votes
2answers
34 views

Write the complex number in trigonometric form (homework question)

Write the complex number in trigonometric form, once using degrees and once using radians. Begin by sketching the graph to help find the argument θ. (Do not use cis form.) $$−1 + i$$ My work: I ...
1
vote
2answers
33 views

Potentially incorrect answer key to trig difference identity practice

I am trying to answer this question: Given cos(x) = 5/13 and sin(x) is negative, find: sin(x) sin(x-π) cos(x-π) sin(x-π/2) cos(x-π/2) The answers that I got were: sin(x)=-sqrt(13^2-5^2)=-12 ...
-2
votes
1answer
42 views

Evaluating a trigonometric integral

Show that $$ \int \frac{y^2\, dx - x^2\, dy}{x^2 + y^2} = -\frac{4a}{3} $$ where Life is a semi circle at $x = a\cos t$ and $y = a\sin t$ from $t = 0$ to $t = \pi$. I tried it but this is where I ...