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

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0
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4answers
24 views

How to solve $12-\sin(\theta)=\cos(2\theta)$?

$$12-\sin(\theta)=\cos(2\theta)$$ What's the correct answer on the $[0,2\pi]$? I started with $12-\sin(\theta)=1-2\sin^2(\theta)$ and then i cant get anything sensible as i end up with ...
0
votes
3answers
33 views

Is there another way to solve this Trigo in series?

Find the value of $$\cos ^2\theta+\cos^2 (\theta+1^{\circ})+\cos^2(\theta+2^{\circ})+...... +\cos^2(\theta+179^{\circ})$$ Attempt, $$\cos x=-\cos(180^\circ-x),\sin x=\cos(90^\circ-x),\cos ...
-3
votes
1answer
31 views

Determine all numbers $x$ such that $\sin x = \sin a$

Let $a$ be a given number. Determine all numbers $x$ such that $\sin x = \sin a$. You may suppose that $0 \le a \lt 2\pi$, and distinguish the cases $a = \frac\pi2$, $a = \frac{-\pi}2$ and $a ...
1
vote
0answers
31 views

Trigonometric identity reduction

I want to be able to reduce some trigonometric expressions that have powers of sine and cosine. For example, for arbitrary real numbers $a$, $b$, and $c$, we can reduce the expression $$ a\cos^2\theta ...
5
votes
3answers
62 views

Given two points, how to find a circle through them that's also tangent to the $x$-axis?

A seemingly simple geometry problem that is surprisingly difficult. I want to find the radius of a circle that is tangent to the $x$-axis, but also must contain two given points. I understand there ...
0
votes
3answers
31 views

how to parameterize the ellipse $x^2 + xy + 3y^2 = 1$ with $\sin \theta$ and $\cos \theta$

I am trying draw the ellipse $x^2 + xy + 3y^2 = 1$ so I can draw it. Starting from the matrix: $$ \left[ \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 3 \end{array}\right]$$ I ...
0
votes
2answers
36 views

Integrating $\sin^3(x)/(2+\cos(x))$

I could use some help solving the following integral: $$\int \frac{\sin^3(x)}{2+\cos(x)} dx$$ So far I tried using the equality: $$\sin^3(x) = \frac{3}{4} \sin(x) - \frac{1}{4}\sin(3x)$$ which ...
1
vote
1answer
57 views

Evaluating $ \int_0^\theta \cosh(a\sin x) dx$

The integral below seems quite simple, but I couldn't find anywhere the result. $$ I = \int_0^\theta \cosh(a\sin x) dx$$ I tried to expand it into Taylor expansion series and successfully evaluate the ...
1
vote
1answer
24 views

Finding value (Trigo Series) [duplicate]

Find the value of $$\cos ^2\theta+\cos^2 (\theta+1^{\circ})+\cos^2(\theta+2^{\circ})+......+\cos^2(\theta+179^{\circ})$$ Can anyone teach me where to start with? I've no idea.
1
vote
0answers
19 views

Find $Z$ transform of given signal

Given the discrete signal $h(n)=r^n\frac{\sin{[(n+1)\theta]}}{\sin{\theta}}$ if $n \geq 0$ and $h(n)=0$ otherwise, find the $Z$ transform of $h(n)$. What I did: We know that ...
0
votes
3answers
51 views

How to solve the equations of the type $\sin a + \sin b = \sin x$?

I came across a question in my book that's like this: $$\sin20 + \sin40 = \sin x $$ I don't know if the values of the $a$ and $b$ make a difference (or in this case, the fact that $b = 2a$) but I'd ...
-2
votes
1answer
54 views

Eyebrow calculation [on hold]

Given a width of 71 and a height of 35, what are the following dimensions: left side, right side, radius, and base?
0
votes
2answers
101 views

Why is this true? $(\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1 \implies A+B+C=\pi)$ [on hold]

Why is this True? $$\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C = 1 \Rightarrow A+B+C = \pi$$ with this assumption that $$0\leq A,B,C<\frac{\pi}{2}$$
3
votes
1answer
51 views

Mandelbrot set of $c \cdot \cos(z)$

I'm given a task to write a program, that determines if a given point $c \in \mathbb{C}$ is in the Mandelbrot set of the function $$f_c(z) = c \cdot \cos (z)$$ That is if the set $\{z_n = f_c^n (0) : ...
0
votes
2answers
32 views

Find the value without using calculator (Trigo)

$\sin ^210+\cos ^240+\sin10\cos40$ How to find the value without using calculator and without the formula $\sin^2A-\sin^2B=\sin(A+B)\cdot \sin(A-B)$?
0
votes
0answers
15 views

Fitting a sinusoidal function to three known points

I have 3 points from a sine wave and I need to determine the sine function from this. There is a very similar question, but this question is with $-30°$, $0°$ and $+30°$: Fitting a sinusoidal ...
3
votes
3answers
48 views

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$ How do I simplify this limit? I tried multiplying dividing $\sin(\frac{x}{2^r})$ to use half angle formula but it doesnt give ...
-1
votes
2answers
95 views

Resolved! Thank you! [on hold]

Calculator says: $π/12±2πn,11π/12±2πn,−π/12±2πn,13π/12±2πn$
1
vote
3answers
85 views

Differentiation under the integral sign: Where is my mistake?

So I'm trying to find $\int_0^\infty \sin(x^2)\,dx$ by the method of differentiation under the integral sign. The idea is to use differentiation with respect to t on A(t) -- defined below -- and then ...
2
votes
5answers
101 views

Determine whether $f(x)$ is increasing or decreasing

Let $f(x) = -x + (x^3/3!) + \sin(x)$ How do I determine if $f(x)$ is increasing or decreasing? I have already found the derivative of this function which is: $f'(x) = -1 + (x^2/2) + \cos(x)$ And I ...
-1
votes
1answer
63 views

Write $20 \sin \theta + 17 \cos \theta$ as a single cosine

Write $20 \sin \theta + 17 \cos \theta$ as a single cosine with phase displacement. I don't know how to start this one. If somebody could give me the formula or a sample that would be amazing!
0
votes
2answers
24 views

solving trigonometry equation $90$ for $ x$

*Solve each equation for all values of $x$: $3\sin x+3=\cos^2 x$ I've tried changing trig values but I don't think its right.
0
votes
1answer
20 views

What is the optimal way to detect a collision between an AABB figure and a non-AABB figure?

Background I'm looking to do this programmatically in Java, but if desired you can post solutions in C/C++ or plain English instructions if you're not a programmer, but I would appreciate an ...
2
votes
1answer
45 views

Trigonometry Identity (Proving)

How to prove this identity? $$\frac{\cos 2\alpha+\cos 2\beta}{1+\cos 2(\alpha+\beta)}=\frac{\cos (\alpha-\beta)}{\cos (\alpha+\beta)}$$ I've tried solving from L.H.S and R.H.S. But failed. Anyone ...
0
votes
4answers
78 views

Exact value of sin (θ/2) if cos θ = 3/5

Exact value of $\sin\frac{\theta}{2}$ if $cos θ = \frac{3}{5}$ and $360° < θ < 450°$: Okay, so I put this into the half argument property and got: $$\sin\frac{\theta}{2} = \pm ...
1
vote
2answers
29 views

Prove trigonometric identity, hence or otherwise find the general solution

The following question requires one to prove the below trigonometric identity $$\cos 3x = 4\cos ^3 x - 3\cos x$$ Hence, or otherwise, find the general solution of the following equation $$(4\cos ^2 x ...
3
votes
2answers
103 views

Guessing the other root to a quadratic equation

I just attempted to do the question below, but it seems that even after seeing the answer I'm not sure I understand the motivation for the solution. Let $\alpha ...
0
votes
4answers
53 views

Problem Verifying Two Challenging Trig Identities

My math teacher gave us an equality involving trigonometric functions and told us to "verify" them. I tried making the two sides equal something simple such as "1 = 1" but kept getting stuck. I would ...
0
votes
1answer
17 views

uniform angular distribution-change of origin

Given a variable which is uniformly distributed for $0<\theta<\pi$ on, let's say, a circle around the origin $O$ with radius $R$($\theta$ starting on the positive x-axis and turning ...
2
votes
2answers
37 views

Finding all values of $\theta$ which describes a straight line

I am having quite a bit of trouble understanding the below question; my assumption is that I should bring the right-hand side in terms of $\sin \theta$ or $\cos \theta$ however am not able to proceed ...
0
votes
0answers
21 views

Making a metric out of distance measure

I'm working with a pseudo-distance measure that is not a metric since it does not hold the triangle inequality. It is called Dynamic Time Warping. The problem is - I need to perform some projections, ...
3
votes
3answers
63 views

Finding the period of $f(x) = \sin 2x + \cos 3x$

I want to find the period of the function $f(x) = \sin 2x + \cos 3x$. I tried to rewrite it using the double angle formula and addition formula for cosine. However, I did not obtain an easy function. ...
0
votes
0answers
15 views

Detect when two edges make a “inner” angle or an “outer” angle

So, given three points, a direction of movement and the side of the movement, find out the "external" or "internal" angle value. In the left pic, I'm above the red line, moving from edge 1 to edge ...
0
votes
3answers
19 views

Trig algebra problems, taking out a factor of tan

$$ \sin\theta-\cos\theta=0 $$ ${\sin\theta\over\cos\theta}=\tan\theta $ $$ \cos\theta (\tan\theta-1)=0$$ $$\tan\theta=1$$ $$\cos\theta=0$$ $$\theta=45, 90$$ However the second solution is not true ...
2
votes
1answer
47 views

Trigonometry express $4\cos x+3\sin x$ in the form $R \cos (x+a)$.

I have been asked to express $4\cos x+3\sin x$ in the form $R \cos (x+a)$. I know that the formula to express it in that form is $a \cos x+b\sin x=R \cos (x-a)$. But as the question is asking me to ...
1
vote
1answer
61 views

Integration a trigonometric expression

How would you evaluate the following indefinite integral? $$ \int \frac {\ln{(x)} \cdot \cos{(x)}}{\sin^2 {(x)}} dx $$
8
votes
7answers
681 views

Squaring a trigonometric inequality

A very, very basic question. We know $$-1 \leq \cos x \leq 1$$ However, if we square all sides we obtain $$1 \leq \cos^2(x) \leq 1$$ which is only true for some $x$. The result desired is $$0 \leq ...
5
votes
3answers
70 views

$\sin(x^2)$ in terms of $\sin(x)$ and $\cos(x)$

One of my students asked me "Can you write $\sin(x^2)$ in terms of $\sin(x)$"? I said I'd think about it. Having thought about it for a while, I now know that I definitely don't know the answer! Lets ...
0
votes
2answers
28 views

If $\sin s=-1/3$ and $s$ is in the $4$-th quadrant, find the exact value of $\sin (2s)$ [on hold]

Could someone solve this step by step so I can wrap my head around the process?? If $\sin s=-1/3$ and $s$ is in the $4$-th quadrant, find the exact value of $\sin (2s)$.
-7
votes
0answers
23 views

Trigonometry related question [on hold]

Using the parent function, explain how we can graph $g(x) = -2\cos(\pi{x} + \frac{2\pi}3) $using transformation. Specify the amplitude, period and phase shift for $g(x) = -2\cos(\pi{x} + ...
2
votes
1answer
39 views

Prove that $1/(\sin x + 1) - 1/(\sin x - 1) = 2 \sec^2 (x)$

Can anyone solve this for me? Prove that $\frac1{\sin x + 1} - \frac1{\sin x - 1} = 2 \sec^2 (x)$. This is as far as I went: $$\frac{(sin x - 1) - (sin x + 1)}{(sin x + 1)(sin x - 1)}$$ ...
-6
votes
0answers
23 views

trigonometry related problem [on hold]

Explain a scenario where we can apply cosine rule to solve triangle instead of sine rule. Give examples if there are situations where we can apply only sine rule but not cosine rule. When does the ...
0
votes
0answers
28 views

Solutions of trigonometric equation $a\sin(x) + b\cos(x) = n$

Is there a solution of the equation $a\sin(x) + b\cos(x) = n$ in rational numbers (i.e. $a,b,n,x$ are rational) where $x$ is not of the form $90n^\circ$? (This question was also there on Integer ...
0
votes
3answers
47 views

If a = 3i + 2j and b = -7i + 4j, find a + b as…

"Trig functions enable you to make mathematical models of vector quantities:" If $\vec{a}$ = 3$\vec{i}$ + 2$\vec{j}$ and $\vec{b}$ = -7$\vec{i}$ + 4$\vec{j}$, find a + b as: A) a sum of two ...
1
vote
1answer
39 views

the roots & the limit of $2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$

If $$2^{(2\pi)^{\cos(2\pi)}}\sqrt{\cos(2\pi)}=2^{2\pi}$$ Can you obtain or is it plausible to find the roots and the limit of $$2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$$ if $0 < \cos(x)$ and $0 < ...
1
vote
1answer
31 views

A variant on Laplace's method $\int_{-\pi}^{\pi} \cos^{2n} (x/2) \cos( m x) dx \sim \frac{1}{\sqrt{n} }$

I am trying to estimate some probability using the inversion formula of characteristics of some discrete random variable and it finally boils down to the following integration which is very similar to ...
0
votes
2answers
40 views

Trigonometric double angle formulas problem

I want to simplify the answer to an equation I had to compute, namely, simplifying $\sin^2 (2y) + \cos^2 (2y)$. I know that $\sin^2 (y) + \cos^2 (y) = 1$ but is there anything like that I can use at ...
-1
votes
1answer
37 views

If $\sin{x}+\sin{y}+\sin{z}= \cos{x}+\cos{y}+\cos{z}=0$, find the value of $\cos{2x}+\cos{2y}+\cos{2z}$. [on hold]

Is there any way to solve this question using complex numbers? I am trying the general way too but I am unable to solve the question.
-2
votes
0answers
39 views

Integer solutions of equation sin x + cos x = n [on hold]

If I have the equation $a.sin(x) + b.cos(x) = n$ , is there a solution where all of a,b,x,n are rational and where x is not of the form $90n$ (degrees)
0
votes
1answer
23 views

Find a possible polynomial under certain conditions

The polynomial of degree $5$, $P(x)$ has leading coefficient $1$, has roots of multiplicity $2$ at $x=5$ and $x=0$, and a root of multiplicity $1$ at $x=-5$. Find a possible formula for $P(x)$.