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
1answer
20 views

Difficult inverse tangent identity

Prove that: $$\arctan\left(\frac{\sqrt{1 + x} - \sqrt{1-x}}{\sqrt{1 + x} + \sqrt{1-x}} \right) = \frac{\pi}{4} - \frac{1}{2}\arccos(x), -\frac{1}{\sqrt{2}} \le x \le 1$$ I'd multiply the ...
0
votes
3answers
55 views

Why does $\frac{49}{64}\cos^2 \theta + \cos^2 \theta$ equal $\frac{113}{64}\cos^2 \theta $?

I have an example: $$ \frac{49}{64}\cos^2 \theta + \cos^2 \theta = 1 $$ then what happens next: $$ \frac{113}{64}\cos^2 \theta = 1 $$ Where has the other cosine disappeared to? What operation ...
-3
votes
1answer
35 views

Verify these trig identities [on hold]

Can somebody verify these equations? $$\sin x\tan x+\cos x=\sec x$$ $$\tan x+\cot x=\sec x\csc x$$ $$\cos x\cot x+\sin x=\csc x$$
1
vote
2answers
54 views

Prove that $\sin(a)$ + $\cos(a)\leq\sqrt{2}$

$$\begin{align*} \sin (a) + \cos(a) &\leq \sqrt{2}\\ (\sin(a)+ \cos(a))^2 &\leq (\sqrt{2})^2\\ \sin^2(a) + 2\sin(a)\cos(a) + \cos^2(a) &\leq \text{2} \end{align*}$$ Am I doing it right? I ...
2
votes
5answers
91 views

Prove that $\sin (\theta) + \cos(\theta) \ge 1$

Let $\theta$ be an arbitrary acute angle. Prove that $\sin (\theta) + \cos(\theta) \ge 1$. $$\big(\sin (\theta) + \cos (\theta)\big)^2 = 1 + 2 \sin(\theta)\cos(\theta)\ge 0$$ so, ...
3
votes
2answers
69 views

$\sin x + c_1 = \cos x + c_2$

While working a physics problem I ran into a seemingly simple trig equation I couldn't solve. I'm curious if anyone knows a way to solve the equation: $\sin(x)+c_1 = \cos(x)+c_2$ (where $c_1$ and ...
1
vote
6answers
85 views

Find an acute angle $\gamma$ such that $\sin \gamma + \cos \gamma= \sqrt{2}$ [on hold]

Find an acute angle $\gamma$ such that $$ \sin \gamma + \cos \gamma = \sqrt{2} $$
0
votes
2answers
45 views

Fix the radius when drawing a circle.

I found this function and it draws an oval rather than a circle. What do I need to do to fix the calculations to make a circle? Thanks. ...
1
vote
1answer
52 views

Integral of $\frac{\sin^2(nx/2)}{\sin^2(x/2)}$ over $[-\pi,\pi]$.

I would like to show that $$\frac{1}{n\pi}\int_{-\pi}^\pi \frac{\sin^2(nx/2)}{2\sin^2(x/2)} dx = 1$$ My attempt is very similar to the accepted answer to this question. $$\int_{-\pi}^\pi ...
0
votes
3answers
41 views

If $\tan x=\sqrt{\frac{a}{b}}$ where a,b are positive real numbers and x is in 1st quadrant then find the value of $\sin x\sec^7x+\cos x\csc^7x$

The answer is $\frac{(a+b)^3(a^4+b^4)}{(ab)^{\frac{7}{2}}}$. I just want to now how to do it.
3
votes
3answers
47 views

How can I show this inequality: $-2 \le \cos \theta (\sin \theta +\sqrt{\sin ^2 \theta +3})\le 2$

Show that $$-2 \le \cos \theta ~ (\sin \theta +\sqrt{\sin ^2 \theta +3})\le 2$$ for all value of $\theta$. Trial: I know that $0\le \sin^2 \theta \le1 $. So, I have $\sqrt3 \le \sqrt{\sin ^2 ...
0
votes
3answers
57 views

why $ \sin \theta = \frac{7}{8} \cos \theta$?

I have an example: $$ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{7}{8} $$ and then this equation is true? why there is cos multiplied?: $$ \sin \theta = \frac{7}{8} \cos \theta$$
1
vote
0answers
37 views

Proving a trigonometric identity involving matrices

Matrix 1= $\begin{pmatrix} \frac{2}{3} \cos (\theta) & \frac{2}{3} \cos (\theta-\frac{2\pi}{3}) & \frac{2}{3} \cos (\theta+\frac{2\pi}{3}) \\[3px] \frac{-2}{3}\sin(\theta) & ...
1
vote
2answers
27 views

Trigonometric relationship in a triangle

If in a triangle $ABC$, $$3\sin^2B+4\sin A\sin B+16\sin^2A-8\sin B-20\sin A+9=0$$ find the angles of the triangle. I am unable to manipulate the given expression. Thanks.
1
vote
1answer
45 views

Suppose $x,y$ are fixed real numbers. Does there always exist a real number $z$ such that $\sin(x+z)$ and $\sin(y+z)$ are rational numbers?

Suppose $x,y$ are fixed real numbers. Does there always exist a real number $z$ such that $\sin(x+z)$ and $\sin(y+z)$ are rational numbers? I know that $\sin(x) \in \mathbb{Q}$ implies that $\sin(x) ...
-1
votes
2answers
51 views

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

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 ...
2
votes
0answers
28 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
83 views

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

$$ (|\cos A|)^2 \qquad\text{and}\qquad \cos^2 A $$ 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
16 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
39 views

Pythagorean theorem squared hypotenuse [on hold]

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 ...
23
votes
4answers
1k 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
27 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
90 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
26 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
85 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
40 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
124 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
83 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
19 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.
4
votes
3answers
79 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
42 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
68 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
43 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
15 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
25 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
43 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
42 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 ...