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

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-1
votes
2answers
47 views

how to understand if the number is already squared or it should be squared in future?

for example: $$ \cos^2 \theta + \sin^2 \theta = 1 $$ $$ \cos^2 \theta = 1 - \sin^2 \theta $$ $$ \cos \theta = \sqrt{1 - \ \sin^2 \theta} $$ I mean in this case how can I know if the sin is already ...
0
votes
0answers
18 views

Periodic Functions and Trigonometry

Find a positive and a negative co-terminal angle for an angle that measures 1485 degrees. (I would like to know how to solve it and the formula I'm supposed to use for problems like these, if there is ...
3
votes
0answers
82 views

what is the difference between $(\cos A)^2$ and $\cos^2 A$

$$ (|\cos A|)^2 \qquad\text{and}\qquad \cos^2 A $$ For example if $\cos A = 0.5$, and $0.5 \times 0.5 = 0.25$, are here some difference in the notations or are they equal?
-1
votes
0answers
15 views

If P(4, -3) is a point on angle A, find the exact value of tan(2s) [on hold]

*I need to brush up on this. Could someone explain?
-2
votes
0answers
37 views

Pythagorean theorem squared hypotenuse

by default the formula is: $$ a^2 + b^2 = c^2 $$ but let's see the real numbers: $$ 1^2 + 2^2 = 5^2 $$ when I see the number $$ 5^2 $$ I was thinking that it should be 25 but looks like in ...
9
votes
3answers
239 views

Axiomatic definition of sin and cos?

I look for possiblity to define sin/cos through algebraic relations without involving power series, integrals, differential equation and geometric intuition. Is it possible to define sin and cos ...
2
votes
0answers
19 views

How to find coefficients in a power of a trig series?

Suppose $m$ is a positive integer, and we know that the following identity holds for all $0<x<2\pi$: ...
3
votes
3answers
85 views

why $\tan x = \frac{\sin x}{\cos x}$? and not $\tan x$ = opposite/adjacent?

we know that $\tan x =\left(\frac{\text{opposite}}{\text{adjacent}}\right)$, but sometimes I see that $\tan x = (\frac{\sin x}{\cos x})$, is that the same thing or why it is different sometimes? ...
0
votes
1answer
22 views

Obtaining $\sum_{n=1}^{\infty} a^n \cos{(n\theta)} = \frac{a \cos{\theta}-a^2}{1-2a\cos{\theta}+a^2}$

This is a homework problem. From Fourier Series and Boundary Value problems, Brown/Churchill 8th ed. I should begin with $2\cos{A}\cos{B}=\cos{(A+B)}+\cos{(A-B)}$, substitute with $A=n\theta$ and ...
-1
votes
2answers
57 views

Limits and Trigonometry

Consider an function $f$ , defined as : $$f^k (\theta) =\sum_{r=1}^n \left( \frac{\tan \left( \frac {\theta}{2^r} \right) }{2^r} \right)^k +\frac 1 3 \sum _{r=1}^n \left( \frac { \tan \left( ...
1
vote
1answer
37 views

For what values of $x\in \Bbb N$ is $\tan^{-1}\left(\frac{360}{x}\right)$ rational?

For what values of $x\in \Bbb N$ is $$\tan^{-1}\left(\frac{360}{x}\right)$$ rational? Just wondering if there is any method to accomplish this.
5
votes
2answers
89 views

for which values of $\theta$ does this equation $x^{\cos\theta} +y^{\sin\theta }=1$ have solutions in integers .?

for which values of $\theta$ does this equation $$x^{\cos\theta} +y^{\sin\theta}=1$$ have solutions in integers ? Note : $x, y$ integers, $\theta$ is real number. Thank you for your help
3
votes
3answers
80 views

Solve $\cos3x=\cos4x$

I want to solve the equation $\cos3x=\cos4x$. The given solutions are $x= 0$, $2\pi/7$, $4\pi/7$ and $6\pi/7$. My first approach was to write the whole thing in terms of $\cos x$ this gave, ...
1
vote
0answers
17 views

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$, Calculate $[CM,CB,BF]$.

$ABCD$ has area $9$. $M$ is in the middle of $AB$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that $\mathbb{V}^3$ is oriented by a positive basis. ...
0
votes
2answers
31 views

Trigonometry Identity proving

If $\sin(x-y) =\cos y$ prove that $\tan y = \frac{1+ \sin y}{\cos y} $. Is there an error with the question? I don't seem to be able to get the answer. Should it be $\tan x$ instead of $\tan y = ...
0
votes
0answers
4 views

Determining pitch and roll angles from the coordinates of a vector

I want to know, given the measurement of an accelerometer at rest (so not really an acceleration but a force per unit of mass) the inclination of this accelerometer, along the X and Y axis. So, In ...
-1
votes
1answer
16 views

Area of a triangle with one given measurement [on hold]

The hypotenuse of an isosceles right triangle is 4 centimeters long. How long are its sides? Round your answers to two decimal places.
2
votes
3answers
68 views

Could someone please explain double-angle identities?

I don't understand how to do maths, mostly because I don't understand why formulae work they way they do, or the reasoning behind equations, etc. I tried to explain the $\sin(2\theta)$ double-angle ...
4
votes
2answers
42 views

Triangle with Ratio of Sides Equal to Ratio of Angles

In an equilateral triangle, the side lengths are in ratio 1:1:1, as are the angle measures. Are there also non-equilateral triangles in which the ratio of the side lengths is the same as the ratio of ...
1
vote
1answer
39 views

Why is the chain rule applied to derivatives of trigonometric functions?

I'm having trouble to understand why is the Chain rule applied to trigonometric functions, like: $$\frac{d}{dx}\cos 2x=[2x]'*[\cos 2x]'=-2 \sin 2x$$ Why isn't it like in other variable derivatives? ...
4
votes
5answers
77 views

Given $\csc\theta=-\frac53$ and $\pi<\theta<\frac32\pi$, evaluate sine ,cosine, and tangent of $2\theta$

If $\csc\theta=\frac{-5}{3}$, what is the exact value of $\tan(2\theta)$, $\sin(2\theta)$, and $\cos(2\theta)$ on the interval of $\left(\pi, \frac{3\pi}{2}\right)$? I think I'm getting the fraction ...
-3
votes
0answers
21 views

Forces of parallelograms [on hold]

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 ...
3
votes
3answers
66 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 ...
-4
votes
3answers
46 views

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

Are $\cos$ and $\sin$ linearly dependent on $[- π , π]$ If true, demonstrate; if False show a counterexample.
0
votes
2answers
42 views

how to find trigonometric angle of any value? [on hold]

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.
1
vote
4answers
75 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.
0
votes
1answer
14 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
3answers
60 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 ...
1
vote
4answers
51 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?
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 $d$?
1
vote
4answers
35 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 $\cos(\text{atan}(2))$
-3
votes
0answers
32 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 $x+y/(1-xy)$
-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.
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.
2
votes
3answers
40 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
27 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
30 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
79 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
64 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
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 $z-\frac{1}{z}=2i\sin \theta$ ...
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
35 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
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
18 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 $\begin{align}\int\cos^n x dx&=\sin x\cdot\cos^{n-1}x+(n-1)\int\sin^2x\cos^{n-2}x dx\end{align}$ ?