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

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2
votes
4answers
87 views

Why the anti derivative of $\sec(x) \cdot \tan(x)$ is $\sec(x)$?

I have discovered that $$\sec(x) = \frac{1}{\cos(x)}$$ but I do not understand why the indefinite integral of $\sec(x) \cdot \tan(x)$ is $\sec(x)$. I am watching the following videos: ...
-1
votes
1answer
26 views

angle sine and cosine identities problem 3

Write in terms no greater than one. $$\sin^3x$$ I originally thought the answer was $\sin x\sin x\sin x$, I was wrong. After using these sine and cosine identities, I came up with ...
0
votes
2answers
25 views

General solutions for trigonometry equations

I'm taught that how to find the general solution for example $\cos 5\theta=\frac{\sqrt{3}}{2}$. But the exercise given by the book is much more complex than the example. For example, $\sin^2 ...
3
votes
1answer
283 views

Prove this is an isosceles triangle

In a triangle ABC, $\sin B\cdot\sin C=\cos^2(\frac{A}{2})$ Prove that this is an isosceles triangle. Can anyone guide me to prove this? Thanks
0
votes
1answer
46 views

Help me prove $\cos A - \sin A = \sin (A \sqrt{2})$, given $\cos A + \sin A= \cos (A \sqrt{2})$. [on hold]

Prove that:$$\cos A-\sin A=\sin A \sqrt{2} \quad \rm{given} \quad \cos A+\sin A= \cos A \sqrt{2}.$$
0
votes
4answers
35 views

Find $\theta$ in $\frac{\sin(45º+\theta)}{850}$=$\frac{\sin 30º}{433}$

Find $\theta$ in the equation \begin{equation*} \frac{\sin (45º+\theta)}{850}=\frac{\sin 30º}{433}. \end{equation*} I know how to use the sum and difference but i still can't get the value of theta. ...
0
votes
1answer
15 views

Period of a solution in a trigonometric equation

This is more of a general question, which keeps confusing me when solving trigonometric equations. When is the period $k\pi$, and when is it $2k\pi$? For example, if I need to solve $\tan x=1$, is ...
-2
votes
1answer
40 views

Integration by parts prove integral of cos^n x dx [on hold]

I'm having a problem with one of my questions. How can I prove that $\begin{align}\int\cos^n x dx&=\sin x\cdot\cos^{n-1}x+(n-1)\int\sin^2x\cos^{n-2}x dx\end{align}$ ?
0
votes
1answer
33 views

Trigo Study plan

In what order of topics is probably the most effective in learning trigonometry for starters... where should I first start? and steps in between to De Moivre's Theorem (which is the last topic)... ...
1
vote
1answer
27 views

Explanation of two argument variant for arctan

Can someone please explain why $$\tan^{-1}\left(\frac{y}{x}\right)$$ has the additional conditions based on what the value of x and y are? I'm most specifically interested in the second equation: ...
1
vote
4answers
53 views

How do I solve the trigonometric equation $1 - \sin^2x - \cos(2x) = \frac{1}{2}$?

Solve for $x$ when $1-\sin^2x - \cos 2x = \dfrac{1}{2}$. I can' t change it into a form I can work with. It is rather complicated.
4
votes
5answers
69 views

find all possible solutions

The set of all $x$ in the interval $[0,\pi]$ for which $2\sin^2x-3\sin x+1 \geq 0$, is _________________. I have tried by factoring it first and then comparing it with the inequality. My final ...
0
votes
2answers
38 views

Trigonometry problem.

If $ \sin\theta = n\sin(\theta + 2\alpha)$, then $\tan(\theta + \alpha) $ is equal to? I tried evaluating $n$, however I got no conclusive answer. I tried expanding $\sin(\theta + 2\alpha)$, but to ...
0
votes
1answer
38 views

Trigonometric identity [on hold]

I have troubles solving the following problem: Assume that $\alpha, \beta$ and $\gamma$ are the three angles in triangle. Show that: $$\cot \biggl( \frac{\alpha}{2}\biggl) + \cot \biggl( ...
2
votes
2answers
44 views

Establishing a trigonometric identity for $n\in\mathbb{N}$

The original problem was showing that this infinite sum converges to $\tan\theta$: $$\sum_{n=1}^\infty \frac{\tan\dfrac{\theta}{2^n}}{\cos\dfrac{\theta}{2^{n-1}}}$$ One hint was given: the series ...
0
votes
4answers
44 views

How to convert $\cos4\theta$ into $\cos3\theta$

How do i show that: $\cos 4θ = − \cos 3θ$ for each of the values θ = $\frac{\pi}7, \frac{3{\pi}}7, \frac{5{\pi}}7, \pi.$ How is $\cos4\theta$ related to $\cos3\theta$? Can someone please explain..
1
vote
1answer
26 views

Intersections of Trig Functions with different periods

There are 2 trig functions on the same set of axis. $f(x)=600\sin(\frac{2\pi}{3}(x-0.25))+1000$ and $f(x)=600\sin(\frac{2\pi}{7}(x))+500$ How do I go about finding the points of intersections of ...
2
votes
0answers
26 views

Finding an angle which satisfies two equations

I'd like to prove the following: Given any two real numbers $a$ and $b$, not both zero, there exists $c \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ such that $\sin c = \frac{a}{\sqrt{a^2 + b^2}}$ and $\cos c ...
0
votes
1answer
26 views

Solving for all equations of x trigonometry

Solve for all the values of $x$. $$\tan^2 x=\tan x $$ I don't know how to do this. I've tried similar examples but have failed to get this one.
2
votes
4answers
62 views

Trigonometric equation $\sec(3\theta/2) = -2$ - brain dead

Find $\theta$ with $\sec(3\theta/2)=-2$ on the interval $[0, 2\pi]$. I started off with $\cos(3 \theta/2)=-1/2$, thus $3\theta/2 = 2\pi/3$, but I don't know what to do afterwards, the answer should be ...
3
votes
1answer
48 views

weird trig problem $\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \leq 2\pi$

$\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \lt 2\pi$ I started off with $[(\sin(\theta)/\cos(\theta)] \times (1/\sin(\theta) )= - \sqrt 2$, then after simplification i got ...
1
vote
1answer
27 views

How can I find these two limits?

How can I find these two limits? I've no idea how to improve or continue now. Can someone give me a hint? 1)$$\lim_{x \to 0^+}\left(\frac{\cos^{\pi}(25x)} {\tan^3(x)}\right)=\lim_{x \to 0^+} ...
1
vote
1answer
19 views

Triangle in circumference of circle

Points $A$, $B$, and $C$ are on the circumference of a circle with radius 2 such that $\angle BAC = 45^\circ$ and $\angle ACB = 60^\circ$. Find the area of $\triangle ABC$. How would I start this ...
3
votes
2answers
37 views

Limit of cos function in a sequence

In my assignment I have to calculate to following limit. I wanted to know if my solution is correct. Your help is appreciated: $$\lim_{n \to \infty}n\cos\frac{\pi n} {n+1} $$ Here's my solution: ...
1
vote
0answers
33 views

Complex numbers and simple argument question

Yesterday, i encountered a question: $z=a+bi$ $Arg(z-\overline z + 4) = {4\pi \over 3}$ $b=?$ I solved the question using basic method: $$\overline z = a-bi$$ $$ w = z - \overline z + ...
1
vote
7answers
122 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
48 views

Is there another way to solve this Trigo in series? [duplicate]

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
2answers
38 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
33 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
70 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
32 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
41 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
60 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
28 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
22 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
2answers
63 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
55 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
115 views

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

Assume that $A,B,C$ are positive real numbers and $A,B,C \in (0,\frac{\pi}{2}]$ and we have $$\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C = 1 $$ prove or disprove that $$A+B+C=\pi$$
0
votes
2answers
35 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
1answer
19 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
51 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: $π/12±2πn,11π/12±2πn,−π/12±2πn,13π/12±2πn$
2
votes
3answers
88 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
107 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 ...
-2
votes
1answer
65 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
22 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
80 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 ...