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

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-8
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1answer
31 views

Trigonometry Question $4/\sin 44 = 5/x$ [on hold]

What method would I use to get the answer to $\frac{4}{\sin(44)}=\frac{5}{x}$ and would it be 60.264337990587? This was the answer I have.
2
votes
5answers
38 views

How to evaluate $\cot(2\arctan(2))$?

How do you evaluate the above? I know that $\cot(2\tan^{-1}(2)) = \cot(2\cot^{-1}\left(\frac{1}{2}\right))$, but I'm lost as to how to simplify this further.
2
votes
3answers
38 views

Show that $\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x)$

The question asks to prove that - $$\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x) $$ I tried combining the first two or the last ...
1
vote
0answers
23 views

Simplifying cyclometric function

How does one simplify this function? $$ f(x) = \arccos(\frac{\pi}{2} - \sin(x)) $$ A plot in GeoGebra showed a graph that looked like semicircle, so can one expect something in this form: ...
0
votes
1answer
29 views

An equality with inverse trigonometric functions

I've stumbled on the equality $$ \tan ^{-1}\left(\frac{3}{4}\right) \left(\pi -\tan ^{-1}\left(4 \sqrt{3}\right)\right)=4 \tan ^{-1}\left(\frac{2}{\sqrt{3}}\right) \cot ^{-1}(3). $$ Out of ...
5
votes
2answers
70 views

Geometric intuition for derivatives of basic trig functions

I was inspired by this question to try and come up with geometric proofs for the derivatives of basic trig functions--basically, those that have simple representations on the unit circle ($\sin, \cos, ...
0
votes
6answers
62 views

De Moivre's Theorem (Trigonometry)

How to prove that $\cos^4 \theta+\sin ^4\theta=\frac{1}{4}(\cos4\theta+3)$ by using De Moivre's Theorem? I know that $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$, but how to apply this ...
1
vote
5answers
62 views

Question about a trigonometry proof?

I just want to ask how can you prove that 2α is twice the value of α in the following figure that depicts a proof of an arctangent identity (and likewise, for β as well).
0
votes
1answer
29 views

Subtraction of trigonometric functions

I was working on a problem booklet and came across the following equation. $$\sqrt2\sin(2x)-\cos(2x)=\sqrt3\sin(2x-a)$$ $a \in \mathbb{R}$ is a specific value that I'm supposed to find, but I don't ...
0
votes
3answers
62 views

De Moivre's Theorem (Trigo)

Prove the trigo identity by using method based on De Moivre's Theorem. $\sin^6\theta=\frac{1}{32}(10-15\cos2\theta+6\cos4\theta-\cos6\theta)$ My attempt, Using $z-\frac{1}{z}=2i\sin \theta$ ...
3
votes
4answers
103 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
vote
1answer
34 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
27 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 ...
4
votes
1answer
292 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
16 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
41 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
0answers
36 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
55 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
71 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
40 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
40 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
47 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
28 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
27 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
38 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
37 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
123 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
72 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
23 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
24 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 ...