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
votes
2answers
28 views

What is the y-cooridinate for the point on the curve with x-cooridante 20?

What is the y-coordinate for the point on the curve with x-coordinate 20? $F. 160$ $G. 162$ $H. 164$ $J. 166$ $K. 168$ The explanation says "The correct answer is G. If the x-coordinate is 20, ...
-2
votes
3answers
67 views

$\cos 27^{\circ}~$? not using calculator .trigonometry [on hold]

$\cos 27^{\circ}~$? Answer should be in square root form without using calculator. Don't use the complex form. Thanks!
4
votes
3answers
50 views

How to find the integral $\int_0^{70 \pi} |\cos^{2}x\sin x|\,dx$?

I need help with this problem: $$\int_0^{70 \pi} \left|\cos^{2}\!\left(x\right)\sin\!\left(x\right)\right| dx$$ My friend says it's 140/3 but I don't see how.
0
votes
1answer
33 views

Why does the $\tan$ reduction formula have a restriction?

My book says the reduction formula is only valid for an integer $n > 1$. Why? This derivation doesn't require $n$ to be an integer or greater than $1$.
0
votes
2answers
45 views

Trigonometric equation for double angle

Can anyone direct me to a basic method solving this equation: $$ [\sin(2x)+(\sqrt{3} \cos2x)]^2 \ - \ 5 \ = \ \cos(30º-2x) $$
2
votes
1answer
45 views

Skewed Trigonometric Function

What would be an expression for a periodic function (period $2\pi$) that essentially behaves just like a negative sine function, but it has the following quirk: It's $0$s lie on the usual places ...
1
vote
2answers
45 views

Solving for $\theta$ in a circle

Let's say you have a pendulum hanging straight down and touching the ground at the lowest point. The pendulum has length $l$. If you pull the pendulum back so that the end is height $h$ above the ...
2
votes
1answer
17 views

Eliminating parameter to get Cartesian equation

$x = \sin(t/2)$ $y = \cos(t/2)$ $-\pi \le t \le \pi$ How would I go about getting the Cartesian equation of these?
5
votes
1answer
75 views

Show $\sin(x+h) \cdot \cos x - \cos(x+h) \cdot \sin x = \sin h$ (without limits please - straight trigonometry only).

I've tried an algebraic approach using the identity $\sin(x) = \sin(x+h-h) = \sin(x+h)\cos(h) - \cos(x+h)\sin(h)$, leading to a complicated expression I'm having trouble simplifying: ...
2
votes
0answers
35 views

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = ...
6
votes
4answers
87 views

Integral of Sinc Function Squared Over The Real Line

I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$ Would a contour work? I have tried using a contour but had no success. Thanks. Edit: About 5 minutes after posting this ...
3
votes
4answers
44 views

How can I bring $\sin(x)$ to the following form?

What steps do we take for the following? $$\sin x = \frac{{2\tan\frac{x}{2}}}{1+\tan^2\frac{x}{2}}$$
4
votes
4answers
98 views

Evaluate $\int{\sin^3(x)\cos^2(x)}dx$

I'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!
0
votes
1answer
31 views

What trig. identity would help solve $2 + \cos(2x) = 3\cos(x)$?

I need help with a homework question that has me puzzled. I need to solve the following equation: $$2 + \cos(2x) = 3\cos(x)$$ I don't see a good trig identity to apply. I tried $\cos(2x) = ...
15
votes
4answers
2k views

Found an odd relationship! Could someone help me to prove or debunk it?

I finished up in hospital which typically means that one has A LOT of spare time to kill and after using electronic devices so much that it makes you sorry I flinched into doodling and and ...
3
votes
1answer
60 views

How should I prove that: $\sum_{i=1} ^{n}(\sin(\frac{i\pi}{n}))^2=\frac{n}{2}$

$$\sum_{i=1} ^{n}\Big(\sin\big(\frac{i\pi}{n}\big)\Big)^2=\frac{n}{2}$$ An interesting conclusion and checked for validity...holds for $n\geq 2$, but yet do not know how to prove it. Are there any ...
0
votes
1answer
34 views

Find the measure of the angle c [on hold]

abc is a triangle (a^3+b^3+c^3)/(a+b+c) = c^2 Find the measure of the angle C with steps
2
votes
2answers
54 views

How to prove that $\tan^2(\frac\theta2)= \tan^2(\frac\alpha2)\tan^2(\frac\beta2)$?

I'm unable to solve this question: $\cos(\theta)=\dfrac{\cos(\alpha)+cos(\beta)}{1+\cos(\alpha) \cos(\beta)}$ Prove: $\tan^2\left(\frac\theta2\right)= ...
0
votes
0answers
29 views

When do the values of trigonometric function not change when pi is added?

I've often seen expressions like: and I'm not sure in which all conditions this is applicable (maybe which trigonometric functions). I know when 2$\pi$ is added, it always comes the same value, ...
-3
votes
1answer
17 views

Find the measures of the angle [on hold]

abc is a triangle where a=283 cm b= 317 cm c= 428 cm Find the measures of the angles with steps Use law of cosines c^2 = a^2 + b^2 - 2ab cos(C)
0
votes
1answer
32 views

Prove that $\sum_{k=1}^{\frac{n-1}{2}}\cos\left(\frac{2\pi k}{n}\right)=-\frac{1}{2}$ if $n=1\mod 2$

I found out that this equality holds by accident,$$\sum_{k=1}^{\frac{n-1}{2}}\cos\left(\frac{2\pi k}{n}\right)=-\frac{1}{2}$$ if $n=1\mod 2$. However, I am not able to prove this directly with rules ...
0
votes
1answer
51 views

find the sum of the series

If $a_1, a_2, \ldots, a_n$ are in arithmetic progression whose common difference is $d$,then find the sum: $$\sin(d) \cdot \left(\csc(a_1)\csc (a_2)+\csc(a_2)\csc (a_3)+\ldots+\csc(a_{n-1})\csc(a_n) ...
0
votes
0answers
31 views

Is my proof of trigonometric equality correct?

I had a solved example in my book whose proof actually turned out to be different from what I did. I was just wondering whether whatever I have done is correct or I've missed something: Thanks, and ...
1
vote
4answers
58 views

Trigonometry Identity: $\tan \theta\sin \theta + \cos \theta = \sec \theta$

Sorry if my question seems too simple. I cannot find a proof and my text book does not provide one either. I am supposed to prove: $$\tan \theta \times \sin \theta + \cos \theta = \sec \theta$$ I ...
2
votes
1answer
39 views

Finding the value of trigonometric function of any angle?

Whenever I have to calculate the value of a given trigonometric function for an angle, I always refer to a table similar to this: But what if I want to find the value for sin$\theta$, where ...
2
votes
2answers
56 views

Prove that $\frac{{-\cos(x-y)-\cos(x+y)}}{-\cos(x-y)+\cos(x+y)} = \cot x \cot y$

I solved this from my implicit differentiation, and i end up with this answer, they say it's right but not simplified, I tried to simply it but I get $\cot(x)\cot(y)-\tan(x)\tan(y)$ ...
3
votes
3answers
99 views

Problems with trigonometry getting the power of this complex expression

I'm here because I can't finish this problem, that comes from a Russian book: Calculate $z^{40}$ where $z = \dfrac{1+i\sqrt{3}}{1-i}$ Here $i=\sqrt{-1}$. All I know right now is I need to use ...
2
votes
5answers
65 views

Minimize $\cos(t)\cos(t-\alpha)$

How can I minimize $f(t)=\cos(t)\cos(t-\alpha)$? I guessed that the minimum is precisely halfway between the adjacent roots $\pi/2$ and $\pi/2+\alpha$. However, I'm not sure how to prove this. Is ...
3
votes
2answers
68 views

Derivatives of trig polynomials do not increase degree?

Let $c = \cos x$ and $s = \sin x$, and consider a trigonometric polynomial $p(x)$ in $c$ and $s$. The degree of $p(x)$ is the maximum of $n+m$ in terms $c^n s^m$. Is it the case that repeated ...
1
vote
3answers
78 views

Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let $$ \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} $$ I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
1
vote
1answer
21 views

Proof using vectors - trigonometric formulas

Question: If two vectors a and b make angle $\alpha$ and $\beta$ with the x-axis, prove, using vectors, that: $$\cos(\beta - \alpha) = (\cos \alpha) (\cos\beta) + (\sin\alpha) (\sin\beta)$$ I tried ...
2
votes
2answers
79 views

Integration of $1/\sin^3 x$

I need a explanation of this problem: $$ \int \frac{1}{\sin^3 x}\,dx $$ Change the variable $$ t = \tan (x/2) $$ With use of $\tan$, $\cos$, $\sin$ and $\cot$, only. So how do I ...
-2
votes
1answer
41 views

Problem involving Basic Trigonometry [closed]

If $\sqrt{2} \cos{A}=\cos{B} +\cos^3{B}$, $\sqrt{2} \sin{A}=\sin{B} -\sin^3{B}$, then find the value of $\sin(A-B)$.
2
votes
1answer
48 views

Ranges in trigonometry

How to find the range of the sum or difference of two trigonometric functions? $2\sin x-3\cos x$ Before this whenever the question of range i have solved they were either single trigonometric ...
1
vote
2answers
41 views

Proving that the volume of a pyramid is one-third that of its corresponding prism.

Is there any way to prove that for any isosceles triangle, the volume of a solid created when that triangle is projected to a point determining the height above the angle opposite the hypotenuse is ...
-2
votes
1answer
61 views

Law of sines Prove that [closed]

abc is a triangle , b = 12 cm , prove that : the Area of abc = (3ac/r) that r = radius abc is a triangle , b = 12 cm , prove that : the Area of abc = (3ac/r) that r = radius
2
votes
1answer
38 views

How to evaluate $\sum_{k=1}^n\ln\left(2\cos\left(\frac{2\pi\cdot3^k}{3^n+1}\right)+1\right)$

By using wolfram alpha, it seems like that $$\sum_{k=1}^n\ln\left(2\cos\left(\frac{2\pi\cdot3^k}{3^n+1}\right)+1\right)=0 \text{ for all }n\in\mathbb{N}.$$ But I don't know how to prove this ...
1
vote
3answers
26 views

Find the measure of angle E.

http://static.k12.com/eli/bb/811/7537/0/2_36640_44211/7537/cfcbab7622b25115e3996826ebe54350776a6601/media/a0fb44a9ac3761c0d89bd1c3ffa513c508eb78bf/mediaasset_650483_1.gif help please i still mix the ...
-1
votes
3answers
82 views

What is the value of $ \sin 10^\circ + \sin 20^\circ + \sin 30^\circ + \ldots + \sin 360^\circ $? [closed]

What will be the answer $ \sin 10 + \sin 20 + \sin 30 + \cdots + \sin 360 \, ? $
0
votes
1answer
34 views

Is $\sin \theta_{xy}\leq \sin \theta_{xz}+\sin\theta_{yz}$, where $\theta_{ab}$ is angle between unit vectors $a$ and $b$?

Suppose $x,y,z\in\mathbb{R}^n$ are unit vectors. The angle between unit vectors $a$ and $b$ is $\theta_{ab}=\arccos(a\cdot b)$ where $a\cdot b$ is the dot-product. Is $\sin \theta_{xy}\leq \sin ...
0
votes
2answers
44 views

Find the value of $a$.

please help I'm lost on what numbers to add or what formula to use
1
vote
1answer
49 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
5
votes
1answer
59 views

Prove $\sin^{2m}\alpha\cdot\cos^{2n}\alpha\leq\frac{m^m n^n}{(m+n)^{(m+n)}}$

If $n$ and $m$ are natural numbers, Prove: $$\sin^{2m}\alpha\cos^{2n}\alpha\leq\frac{m^mn^n}{(m+n)^{(m+n)}}$$ Additional info:We should only use AM-GM inequality.We can use Trigonometry ...
2
votes
3answers
42 views

Trigonometric formula simplifies to $\sin x\cos x[\tan x+\cot x]$

Again, I have a little trouble figuring out how we got from the first step to the next one. It would be really appreciated if someone could help me out. $$ \begin{split}LHS &= ...
1
vote
2answers
42 views

Help showing equality involving $\tan$ function

$$\Large\frac{\left(\frac{\tan \frac \pi 4+\tan x}{1-\tan\frac \pi 4\tan x}\right)}{\left(\frac{\tan \frac \pi 4-\tan x}{1+\tan\frac \pi 4\tan x}\right)}=\frac{\left(\frac{1+\tan x}{1-\tan ...
1
vote
2answers
24 views

Geometry, finding the possible values of 'a'

$P (a,4)$, $Q (2,3)$, $R (3,-1)$ and $S (-2,4)$ are four points. If $|PQ| = |RS|$, find the possible values of a I know this is a pretty basic problem but I'm having a lot of trouble with it, here is ...
2
votes
1answer
38 views

$\tan(x), \cot(x)$ function properties

Does $\tan(x)$ and $\cot(x)$ has symmetry axis? (like e.g $\cos(x)$ at $\pi k$ for $k \in \mathbb{Z}$), I tried think in the direction that $\sin(x)/\cos(x) = \tan(x)$ and both of them have symmetry ...
0
votes
3answers
54 views

$\cos (x)= -0.7, 2π ≤ x < 4π$

I have the answer for this, but my teacher hadn't taught the whole "when cosine is an even, the value of $-\arccos (-0.7)$ is a solution too." Please: -tell me when a $\cos$/$\sin$ function is ...
1
vote
3answers
29 views

Calculating the angle for a path between two nodes in a graph

I want to (programatically) draw an edge between two nodes in a graph, starting on the outside of the nodes. Below is an illustration of what I'm (poorly) trying to describe: I have the $(x,y)$ ...
0
votes
2answers
52 views

Principal period of $\sin\frac{3x}{4}+\cos\frac{2x}{5}$ [duplicate]

Find the principal period of $$\sin\frac{3x}{4}+\cos\frac{2x}{5}$$ It was easy to find principle when single trigonometric function is given, but i don't know how to find principal period of sum of ...