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

learn more… | top users | synonyms (2)

8
votes
3answers
143 views

solve for $x$: $\frac{\sin(x)}{x}=\frac{5}{6}$

Is it possible to solve for x the following equation without root finding: $$\frac{\sin(x)}{x}=\frac{5}{6}$$
1
vote
1answer
35 views

Where i am going wrong in solving the inequality?

If $\cos x \left(\cos x+\frac12\right) >0$ then where should $x$ lie in the interval $(0,\pi)$ What I tried When i made two cases i got correct answer but when i used wavy-curve method. I am not ...
2
votes
1answer
88 views

Find $\int_{\pi /4}^{65\pi /4} \frac{dx}{(1+2^{\cos x})(1+2^{\sin x})}$

Find the value of: $$I=\int_{\pi /4}^{65\pi /4} \frac{dx}{(1+2^{\cos x})(1+2^{\sin x})}$$ First, I rewrote the limits as the function goes from $\frac{\pi}{4}$ to $\frac{9\pi}{4}$. Now the integral ...
1
vote
3answers
168 views

Arcsin estimation

How do I prove that $$\arcsin (x)>\frac{3}{1+2\sqrt{1-x^2}}\text{ ?}$$ We received this example while we are learning integration so it must have something to do with it. But I can't seem to ...
2
votes
1answer
39 views

Put the numbers $\cos a=a; \sin \cos b=b; \cos \sin c=c$ in ascending order

Let $a,b,c \in \left[0;\frac{\pi}2\right]$ such that $$\cos a=a; \sin \cos b=b; \cos \sin c=c.$$ Put the numbers $a, b, c$ in ascending order. My work so far: If $x>0$, than $\sin ...
1
vote
1answer
37 views

Find the area bounded by $r=6\sin(2\theta)$

Winplot plot: I tried this: $$A = 4 \cdot \frac 1 2 \int_0^{\pi/2} (6\sin(2\theta))^2 d\theta$$ Is that right? How about $$A = 8 \cdot \frac 1 2 \int_0^{\pi/4} (6\sin(2\theta))^2 d\theta$$
0
votes
1answer
24 views

Find the area inside $r=2+2\sin(\theta)$ but outside $r=4\sin(\theta)$.

Winplot plot: I tried this: $$A = 2 \frac 1 2 \int_0^{\pi/2} (2+2\sin(\theta))^2 - (4\sin(\theta))^2 d\theta + 2 \frac 1 2 \int_{\pi}^{3\pi/2} (2+2\sin(\theta))^2 d\theta$$ Is that right? How ...
3
votes
2answers
79 views

Evaluation of $\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}$

Evaluation of $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} = $$ $\bf{My\; Try::}$ I have solved Using Direct formula:: $$\sin \frac{\pi}{n}\cdot \sin ...
-2
votes
1answer
31 views

Multi angle formula of an irrational multiple of theta

Given that the double angle formula allows you to take $cos(2\theta)$ and put into terms of $cos(\theta)$ and $sin(\theta)$, would it be possible to do the same to an irrational multiple of \theta ...
0
votes
1answer
26 views

Need help solving the equation $\cos a-\sqrt{2}\sin a\cos a=0$ in the interval $[-\pi, \pi]$ [closed]

What are the solutions to $\cos a-\sqrt{2}\sin a\cos a=0$? I did not learn how to do this, and thanks! The answer needs to be between $[-\pi,\pi]$
0
votes
2answers
20 views

Finding all angles for a quadratic equation.

What are the infinite angles in notation for $2\cos^2(a)+\cos(a)=1$ ? Can you please show me how to do this?
0
votes
2answers
58 views

What is the trigonometric form of the complex variable $z=0+0i $?

I'm confused how do i determine the trigonometric form of the complex variable $z=0+0i$ , it has modula such that is 0 but what about it's argument ? Note : At a least i would like to know it's ...
-1
votes
2answers
51 views

Solving Trig Values [closed]

So given the size of the hypotenuse being $2$, the angle is $24°$, slope $\frac{3}{7}$. I am looking for the opposite and the adjacent numbers.
0
votes
1answer
45 views

Trigonometric Word Problem in 3D

The question I am having trouble on is as follows: "As an Expert Mathematics Witness, you have been presented with a Ballistics Report, and a Police Report as your evidence. Use the information ...
1
vote
1answer
52 views

Trig Formula for distance

I found a formula for calculating the distance to the sun for any given day of the year. The formula uses the cosine function and I am not able to calculate the distances correctly. I fear it is my ...
-1
votes
3answers
42 views

Obtaining $\frac{\sin x}{x}$ via integral of Bessel function of the first kind

How can I prove that, $$\frac{\sin x}{x} = \int_0^{\pi/2} J_0(x\cos\theta)\cos\theta \hspace{0.11cm} \mathrm{d}\theta$$
1
vote
3answers
120 views

Is there a purely algebraic proof to show that $-1\leq\sin x\leq1?$

I have to prove the boundedness of $\sin x$ (strict inequality) ie. $-1\leq\sin x\leq1$. I know a geometric proof using trianglesbut I am not too satisfied with it as it does not prove that ...
0
votes
1answer
36 views

How to calculate $\cos\left(\frac{k \pi}{n}\right)$ : simplification to real numbers.

I don't know if there is a general rule to the expression $\cos(k \pi /n)$. However write $\cos(k \pi /n)$ in terms of real numbers like square root numbers and something like this? As an example : ...
2
votes
2answers
44 views

Resolving $\sec{x}\left(\sin^3x + \sin x \cos^2x\right)=\tan{x}$

Going steadily through my book, I found this exercise to resolve $$ \sec{x}\left(\sin^3x + \sin x \cos^2x\right)=\tan{x}$$ Here's how I resolve it ($LHS$) and again bear with me as I truly ...
0
votes
1answer
9 views

Find a Weight Function with specific characteristics

I need to build a weight function and I want to understand how you would do that. The reasoning you would use to define it. My function has to be something like: $f(\alpha)$ which is: $0$ if ...
-1
votes
2answers
45 views

Solving $\tan 2 \theta = -\sqrt{3}$

I'm preparing for upcoming exam, Can anybody give me some hint to solve this? $$\tan2 \theta = -\sqrt{3}$$
0
votes
3answers
56 views

How to solve this Trigonometric equation $\tan^{2}\theta + \sec(2\theta)=1$?

What is the general solution of this trigonometric equation $$\tan^{2}\theta + \sec(2\theta)=1$$ from the following options: a) $m\pi$ b) $n\pi\pm \frac{\pi}{3}$ c) ...
1
vote
0answers
79 views

What did i do wrong? Trigonmetry and algebra

I have a question where I need to find perpendicular gradient and new points on a line. What I have is a triangle. I'm given no angles or lengths. I had to draw another triangle around it. First I ...
1
vote
1answer
52 views

Please help us diagram this trig problem.

We are not looking for a solution. We are confused by the wording and need help with the diagram. Then we can solve the problem on our own. Here is the exercise in the text. A surveyor is ...
1
vote
2answers
43 views

Inverse of $\tan^{2}\theta$?

I re-arranged: $$3\tan^{2} x -1=0$$ to get $\tan^{2}\theta = \frac{1}3$. I noticed the inverse of the $cos, sin$ and $tan$ functions are written as $\cos^{-1}\theta, \sin^{-1}\theta$ and ...
0
votes
1answer
25 views

Converting Polar Coordinates to Regular Coordinates

If you guys didn't know, I have my quiz tomorrow and I have one last thing to ask to this Community! I am completely confused on how to convert polar coordinates to regular coordinates. The teacher ...
0
votes
5answers
25 views

Labeling Negative Polar Coordinates

i have a quiz coming up tomorow and i have this major question.. its a question about how to plot it exactly when it is negative. Let me go through the whole set: 1) ($4$ , $60^\circ$) 2)($-4$ , ...
1
vote
1answer
30 views

Sum of powers of primitive root of unity- Trig Proof

I'm trying to prove that if $z=\operatorname{cis}(2\pi/n) = \cos(2\pi/n) + i\sin(2\pi/n)$, that is, $z$ is a primitive $n$-th root of unity, for any integer $n\geq 2$, $1+z+z^2+\cdots+z^{n-1}=0$. I've ...
0
votes
1answer
29 views

Triple Integration of gravitational potential

Integrate $\int_{0}^{2\pi } \int_{0}^{a} \int_{0}^{\pi/2} \frac{ G\rho r^2 sin \theta}{ {(r^2-2rt cos \theta + t^2 )}^{\frac{1}{2}}} d \theta dr d \phi$, where $\rho, t $ are constants. Sorry ...
0
votes
0answers
30 views

In $a\sin x + b \cos x =R\sin(x+\theta)$, why is $R$ always taken to be non-negative?

The expression $$a\sin x+b\cos x $$ may be rewritten in the form $R\sin(x+\theta)$ where $$R^2=a^2+b^2; \quad R=\sqrt{a^2+b^2}$$ and $$\frac{R\sin\theta}{R\cos\theta}=\tan\theta=\frac{b}{a};\quad ...
0
votes
0answers
4 views

Calculate Universal Time for when an object in orbit reaches a given radius / altitude?

Assuming that an object in orbit WILL reach a given radius / altitude at some point in the future, how can I work out the exact time it will reach that point? Assume that the object is a Satellite in ...
0
votes
2answers
49 views

Find A such that $A\cos(t+\theta)=\frac{1}{4}\cos(t)+\frac{1}{2}\sin(t)$

I need to find A such that $$A\cos(t+\theta)=\frac{1}{4}\cos(t)+\frac{1}{2}\sin(t)$$ This is how I proceed, $$A\cos(t+\theta) = A\cos(t)\cos(\theta)-A\sin(t)sin(\theta),$$ Hence ...
2
votes
1answer
64 views

Area of Convex hull

For every point set $A \subset R^2$, prove that in general the sum of the coordinates of $\phi(T)$ is independent of a triangulation T and is associated to the area of the Convexv_Hull(A). We ...
0
votes
1answer
19 views

Why is $\omega $ the natural/angular frequency?

Pardon me cause I'm a little confused. If we have something like: $y=A\sin \left( \omega t-\delta \right)$ why would $\omega$ be considered the natural frequency? I always thought the frequency of a ...
0
votes
3answers
141 views

Why $\frac{\pi}{12}$ equals to $\frac{\pi}{3} - \frac{\pi}{4}$

I'm going back to basic trigo for the sake of being able to help my kids and also being bad younger at it, I want to be able to overcome that lack of understanding and honestly, I hate unfinish ...
0
votes
1answer
36 views

Linear Combination of two trig functions

1 Please click this 1 to see the attached picture. I am adding two trig function but don't know how to show my working for linear combination of the two. We are expected to use $r\cos(x-\alpha)$. The ...
0
votes
0answers
33 views

How do i work out this question?

Hi This is a question i have to answer: Tom steers his boat in a North Easterly direction for 15 seconds, East for 6 seconds and then stops to avoid hitting a duck. If his boat travels at a constant ...
0
votes
2answers
43 views

Find the Maclaurin series for $\cos^2(x)$

I am given this as a hint: $\cos^2(x) = \frac{1 + \cos(2x)}{2} \\$ I am not really sure how to start this one, would it just be the regular Maclaurin series squared. For example: $ ...
0
votes
1answer
30 views

Solve using Law of Cosines or Law of Sines

I'm trying to solve these sets of problems please. Determine the number of triangles with the given parts and solve each triangle (if possible). $\alpha=39.6^\circ,c=18.4,a=3.7$ ...
5
votes
3answers
80 views

Proof that the Period of $\sin(x)$ is $2\pi$.

As I was walking through campus today, I had an interesting question pop into my head: How can we prove that the period of $\tan(x)$ is $\pi$ rather than $2\pi$? The answer to this was extremely ...
0
votes
2answers
26 views

How to find the phase of a wave

How do I calculate the phase $\phi$ of $\sin(4-3t)$ relative to $\sin(3t)$? Also what would the angular frequency $\omega$ be? With something like $\sin(2t + 2)$, I can see that the phase relative to ...
0
votes
1answer
34 views

Correct methodology for solving this trigonometric equation

I want the correct method for solving this trigonometric equation $$\sqrt{\cos(x)\sqrt{\cos(x)\sqrt{\cos(x)\dots}}} =1.$$ I have used my own technique: \begin{align*} ...
0
votes
1answer
21 views

What does it mean to have the same real part after taking the square root of two different complex number(Geometrical Interpretation)?

I have two different complex numbers. Say $a_1+b_1i$ and $a_2+b_2i$. I take the square root of these two complex numbers. Let $ \sqrt(a_1+b_1i) = c_1+d_1i $ and $ \sqrt(a_2+b_2i) = c_2+d_2i $. I can ...
0
votes
1answer
12 views

How to approach an inverse image problem with sinusoids?

I'm beginning some work on finding the inverse images of systems of equations that are primarily based on sinusoids. However, I'm a bit stuck in terms of the mathematical tools I have to work with. ...
0
votes
1answer
17 views

Integral of trigonometric function seems unit-dependent

for this problem I have to compute the fraction of a flux within a certain angle range. The flux distribution in $\phi$ is $I(\phi) = I_0 \cos^2(\phi)$, centred in 0 and spanning the flat angle. The ...
0
votes
2answers
32 views

Find distance from point to line using trigonometry

I have a scene with center $C$ and outer corners A and B. I have a camera with a fixed focal length that I want to place at a fixed height, at a distance $d$ such that the entire scene is in view, ...
0
votes
2answers
78 views

Find the number of solutions of $\sin x+2 \sin 2x- \sin 3x=3$

In $(0 \:\:\pi)$Find the number of solutions of $$\sin x+2 \sin 2x- \sin 3x=3$$ The equation can be written as $$\sin x+4 \sin x \cos x=3+\sin 3x$$ i.e. $$\sin x(1+4\cos x)=3+\sin 3x$$ i.e., ...
1
vote
2answers
53 views

Solving $\sin$($x$ - $\frac{π}{5}$) - $\cos$($\frac{π}{10}$) = 0.

Find all the exact angles between $0$ and $π$, which satisfy the equation $\sin(x - \frac{π}{5}) - \cos(\frac{π}{10}) = 0$. I have tried using the sum and difference formula for the $\sin$ function, ...
4
votes
1answer
62 views

Integral of $\frac{\sqrt {x}}{x^2+x}$

I'm trying to find $$\int_{1/3}^3 \frac{\sqrt{x}}{x^2+x} dx.$$ I used a $u$ substitution where $u = \sqrt{x}$ to get $$2 \int_\sqrt{1/3}^\sqrt{3} \frac{u}{u^4+u^2} du.$$ Substituting $u = \tan v$, I ...
1
vote
1answer
18 views

Converting theta into frequency

Sorry for a very elementry question. It's been a long time since I took trigonometry. In Daniel Shiffman's tutorial on drawing a sine wave in Processing, the "angular velocity" of the wave is given ...