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
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3answers
51 views

Simple trigonometry equation

The previous class we were doing trigonometry exercises. Before the class finished, our teacher wrote exercises on the table. I am stuck with the following one: $$ \cos(2x) + 1 + 3\sin x = 0 $$ I ...
0
votes
0answers
17 views

Forces of parallelograms

Two forces are applied to an object. The measure of the angle between the $30.2$ pound applied force and the $50.1$ pound resultant is $25$ degrees. a. Find the magnitude of the second applied force ...
1
vote
4answers
45 views

Evaluate $\sin\left(-\frac{\pi}{6} + \frac{1}{2}\arccos\left(\frac{1}{3}\right)\right)$.

My task is to evaluate $$\sin\left(-\frac{\pi}{6} + \frac{1}{2}\arccos\left(\frac{1}{3}\right)\right).$$ I think I've gotten most of the way there but I keep running into trouble... any suggestions?
1
vote
2answers
36 views

how to find trigonometric angle of any value?

how to find value for this- cot(1.3333) in degrees, without using a calculator? if it is possible please explain the process involved and how to find values of other similar questions.
-3
votes
0answers
31 views

Solve $\tan(w/2)$ , where $w = \arcsin(2x/(1+x^2)) + \arccos(1-y^2/(1+y^2))$ [on hold]

Solve $\tan(w/2)$ , where $w = \arcsin(2x/(1+x^2)) + \arccos(1-y^2/(1+y^2))$ Also, $|x|<1$ ; $y>0$ ; $xy>1$ The answer is given as $x+y/(1-xy)$
0
votes
1answer
651 views

Geometry Find the Radius of a circumcircle given the area of the triangle

Ok so here is what I know, the circumcircle of an equilateral triangle with an area of $4\sqrt{3}$ is drawn, calculate the radius lenght of the circumcircle. I also know that to find the radius I ...
7
votes
2answers
2k views

What exactly do the sin, cos, tan buttons do on a calculator?

I understand they mean sine, cosine, tangent, but what exactly is the calculator doing when I enter an angle and press those buttons? Edit: To help others better understand my question, my question ...
-4
votes
3answers
44 views

Are $\cos$ and $\sin$ linearly dependent in $[- π , π]$?

Are $\cos$ and $\sin$ linearly dependent on $[- π , π]$ If true, demonstrate; if False show a counterexample.
1
vote
4answers
47 views

Prove that $\sin 2\alpha=2 \sin \alpha \cos\alpha$

In this triangle $AD=AC=1$, $BC=a$, $BAC=2\alpha$ I thought $\sin 2\alpha=a$, but I don't know how to continue.
1
vote
3answers
48 views

Basic trigonometry intuition

I have already posted a question regarding the same function here However, now I simply can not grasp why the function has to have two solution sets:$$\cos y=\cos \Bigl(\frac{\pi ...
0
votes
1answer
13 views

The domain of logarithmic functions with sinx and cosx

I need to solve this equation: $\log_{\cos x} \sin x + \log_{\sin x} \cos x=2$ But in order to solve it, I first need to find the domain. What I did was this: $\cos x\neq1 \wedge \sin x\neq1 $ ...
1
vote
4answers
33 views

How to analytically evaluate $\cos(\pi/4+\text{atan}(2))$

This equals $\cos(\pi/4)\cos(\text{atan}(2))-\sin(\pi/4)\sin(\text{atan}(2))$. I'm just not sure how to evaluate $\cos(\text{atan}(2))$
0
votes
2answers
24 views

Will this equation with $\sin$ and $\arcsin$ cancel?

It can be said that $\arcsin(\sin(x))= x$ are inverses if $x \lt 2\pi$. Can it also be generalized so that $\arcsin(\sin(d\cdot x))= d\cdot x$ if $x \lt 2\pi\cdot d$ for a constant $d$?
6
votes
6answers
81 views

System of equations involving sin and cos

I'm trying to solve the following system: $$ \sin(x) + \cos(y) = 0.6\\ \cos(x) - \sin(y) = 0.2\\ $$ Solving for y in terms of x: $$ y=\arccos(0.6-\sin(x))=\arcsin(\cos(x) -0.2) $$ Therefore: $$ ...
2
votes
4answers
898 views

Prove that $\tan^{-1}\left(\frac{x+1}{1-x}\right)=\frac{\pi}{4}+\tan^{-1}(x)$

The question is: Prove that $\tan^{-1}\left(\frac{x+1}{1-x}\right)=\frac{\pi}{4}+\tan^{-1}(x)$. It's from A-level further mathematics.
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 ...
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 ...
2
votes
3answers
39 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 ...
2
votes
5answers
42 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.
5
votes
1answer
392 views

Continued fraction for $\tan(nx)$

I found this beautiful continued fraction expansion of $\tan(nx)$, $n$ being a positive integer, online but I don't remember the source now: $\displaystyle \tan(nx) = \cfrac{n\tan x}{1 -\cfrac{(n^{2} ...
5
votes
2answers
76 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, ...
-8
votes
1answer
32 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.
1
vote
5answers
63 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).
1
vote
0answers
26 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
3answers
66 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 $z-\frac{1}{z}=2i\sin \theta$ ...
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 ...
1
vote
1answer
61 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 ...
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
28 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
294 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
43
votes
10answers
10k views

Is π equal to 180°?

$$ \begin{array}{ccc} \sin{(\theta+180^{\circ})}=-\sin{\theta} & \cos{(\theta+180^{\circ})}=-\cos{\theta} & \tan{(\theta+180^{\circ})}=\tan{\theta} \\ \sin{(\theta+\pi)}=-\sin{\theta} & ...
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:$$\cos A-\sin A=\sin A \sqrt{2} \quad \rm{given} \quad \cos A+\sin A= \cos A \sqrt{2}.$$
1
vote
1answer
83 views

How do we define the branch cuts for $\sin^{-1}z = \frac{1}{i} \log(\sqrt{1-z^2} + iz)$ as $(-\infty,-1)$ and $(1,\infty)$?

As $\sin^{-1}z$ is a function of complex $\log$, it is multivalued. The branch cuts to make $\log$ single-valued are defined conventionally as $-\pi < Arg(z) \leq \pi$. Why wouldn't this carry over ...
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. ...
-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
1answer
17 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 ...
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
73 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
41 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 ...
1
vote
1answer
29 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 ...
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..
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
0answers
40 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 and positive) where $x$ is not of the form $90n^\circ$? (This question was also there on ...
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 ...