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
2answers
49 views

Need help solving a trigonometric equation

I am preparing for finals and there is one exercise in my book that i don`t know how to solve. $$\frac{\sin a}{\sin \frac{a}{b}}=b$$ I just need to solve this for b. I tried wolfram alpha but it does ...
4
votes
4answers
37 views

Converting $\cos\phi$ into $\frac{1−t^2}{1+t^2}$, given that $t = \tan\frac{\phi}{2}$

I have to figure out the working to convert $\cos\phi$ into $\dfrac{1−t^2}{1+t^2}$, given that $t = \tan\dfrac{\phi}{2}$. It would be amazing if someone could help I've been trying to do it for ...
-3
votes
0answers
21 views

Vectors applied on the arm [on hold]

I just received a maths question saying the body and its joints are subject to significant forces under load. Your challenge is to redesign a part of the body, and using vectors, explain how it could ...
1
vote
1answer
25 views

Height of lighthouse based on angle difference

I have a question in my maths book: A lookout in a lighthouse tower can see two ships approaching the coast. Their angles of depression are 25° and 30°. If the ships are 100 m apart, show that the ...
2
votes
0answers
27 views

Number of solutions of some trigonometric equations

Let $N > 1$ and let $S$ be a subset of the integers in the (real) interval $[1, N]$. Can we prove that there are only finitely many solutions $x \in [1,N] \setminus \mathbb{Z}$ to the equation $$ \...
3
votes
1answer
48 views

Does this inequality involving inverse tangent (arctan) hold?

I am wondering if the following statement is true for $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $x,y\in\mathbb{R}$: $$\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)\leq\...
0
votes
1answer
19 views

If $\cos\alpha = \frac{2\cos\beta - 1}{2-\cos\beta}$ , $(0<\alpha , \beta< \pi)$, then $\tan\frac{\alpha}{2}\cot\frac{\beta}{2}$ is equal to?

If $\cos\alpha = \frac{2\cos\beta - 1}{2-\cos\beta}$ , $(0<\alpha , \beta< \pi)$, then $\tan\frac{\alpha}{2}\cot\frac{\beta}{2}$ is equal to? MY ATTEMPT: I tried simplifying the equation to ...
2
votes
2answers
58 views

Simplify $\arccos\left(2\cos x\right)$.

Let $x\in[\pi/3,2\pi/3]$. We know that $\arccos (\cos x)=x$ but what we can say about $\arccos\left(2\cos x\right)$? Are there, for example, any "half-angle formula" also for inverse trigonometric ...
1
vote
4answers
36 views

$\cos2\theta +\cos\theta +k = 0 $ - set of all values of $k$ for which there is a solution

The set of all values of $k$ (real), such that the equation $\cos2\theta +\cos\theta +k = 0 $ admits a solution for $\theta$ is? MY ATTEMPT: I substituted $\cos2\theta$ with $2\cos^2\theta - 1 $. On ...
0
votes
3answers
49 views

If $x^2 - 2x\cos\alpha + 1 = 0$ and $y^2 - 2y\cos\beta + 1 = 0$, then $2\cos(\alpha + \beta)$ is equal to?

If $x^2 - 2x\cos\alpha + 1 = 0$ and $y^2 - 2y\cos\beta + 1 = 0$, then $2\cos(\alpha + \beta)$ is equal to? MY ATTEMPT: Using the fact that $\cos\alpha$ and $\cos\beta$ must be real, I know that $x$...
1
vote
0answers
39 views

Evaluation of trigonometric function without complex numbers

We are asked to prove $$\tan{\frac{3\pi}{11}} +4\sin{\frac{2\pi}{11}}=\sqrt{11}$$ So far the solution that I have come across all use complex numbers to get to the result I am searching for a ...
2
votes
2answers
38 views

$\cos28 + \sin28= k^3\cos17=$?

If $\cos28^\circ +\sin28^\circ = k^3$ then $\cos17^\circ = $?. Find in terms of $k$. MY ATTEMPT: I tried finding $\cos28^\circ - \sin28^\circ$ in terms of $k$. Then I found out $\cos28^\circ$ with ...
0
votes
0answers
23 views

What is the significance of the multiplicative and additive identities? What can they tell us?

I have been aware of the these identities, the multiplicative and additive are widely known and introduced in most introductory mathematics classes.However, until I has begun to learn more about the ...
0
votes
0answers
23 views

Trigonometric Ratios with half angles [duplicate]

If $\tan\left(\frac{\pi}{4} + \frac{y}{2}\right) = \tan^3\left(\frac{\pi}{4} + \frac{x}{2}\right)$, prove that $\sin(y) = \sin(x)\left[\frac{3+\sin^2(x)}{1 + 3\sin^2(x)}\right]$ I come until the step ...
1
vote
0answers
15 views

Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
2
votes
3answers
85 views

Find the values of $x$ such that $2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ is independent of $x$.

Find the values of $x$ such that $$2\tan^{-1}x+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$$ is independent of $x$. Checking for $x\in [-1,1]$ In the taken domain $\sin^{-1}\left(\frac{2x}{1+x^2}\...
0
votes
1answer
34 views

Find sides of isosceles triangle inside a circle with line segment lengths as 5 and 4 as shown in the link. pls help!

Pls see the diagram below. I tried to use similar triangles and came to my wits end. Any help will be appreciated!
1
vote
1answer
26 views

$x, y , z$ are respectively the $sines$ and $p, q, r$ are respectively the $cosines$ of the …

$x, y , z$ are respectively the $sines$ and $p, q, r$ are respectively the $cosines$ of the angles $\alpha, \beta, \gamma$, which are in A.P. with common difference $\frac{2\pi}{3}$. 1. $yz + zx + xy ...
4
votes
1answer
91 views

The other $47$ roots of the minimal polynomial for $\cos 1 ^\circ$

The minimal polynomial for $x=\cos 1 ^\circ=\cos \frac{\pi}{180}$ is: $$281474976710656 x^{48}-3377699720527872 x^{46}+18999560927969280 x^{44}- \\ -66568831992070144 x^{42}+162828875980603392 x^{40}-...
0
votes
2answers
29 views

Find all values of parameter a, when sum of solutions of following equation is 100

Find all values of parameter $a$, when sum of solutions of following equation is $100$. $$ \sin(\sqrt{ax-x^2})=0 $$ I tried to get rid of that $sin$ and there was quadratic equation with two ...
1
vote
4answers
73 views

Solutions of $\sin^2\theta = \frac{x^2+y^2}{2xy} $ [on hold]

If $x$ and $y$ are real, then the equation $$\sin^2\theta = \frac{x^2+y^2}{2xy}$$ has a solution: for all $x$ and $y$ for no $x$ and $y$ only when $x \neq y \neq 0$ only when $x = y \neq 0$
0
votes
1answer
35 views

Prove that $-4\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq10$

Prove that $$\color\red{-4}\leq5\cos\theta+3\cos(\theta+\frac{\pi}{3})+3\leq\color\red{10}$$ My attempt:- I simplified the equation to $$\begin{align} &\;\;\phantom{=} 5\cos\theta+3\cos(\...
-1
votes
1answer
53 views

Find the value of x of $\frac{(3x^2-27)(8x^2)6}{4(9-3x)(x^2+3x)}=\frac{\tan (x+4)}{\log (x+\frac{1}{4})}$? [on hold]

$\frac{(3x^2-27)(8x^2)6}{4(9-3x)(x^2+3x)}=\frac{\tan (x+4)}{\log (x+\frac{1}{4})}$? How to find the value of $x$ I've been thinking for this question for quite a time. Hope can somebody solve it. ...
0
votes
0answers
33 views

How to solve this $80^\circ$-$80^\circ$-$20^\circ$ triangle ($60^\circ+20^\circ$ and $70^\circ+10^\circ$ variant)? [duplicate]

A friend of mine asked me for help with a math problem and I struggled with this for over an hour. I told him sorry, and I felt bad. It's been bugging me now for hours. I don't even so much care for ...
1
vote
2answers
49 views

Why should the solutions of $(\sin x)^2 = 0$ be rejected in the equation $((\sin x)^2)(\csc x + 1) = 0$?

Q: Determine the number of solutions for $((\sin x)^2)(\csc x + 1) = 0$ over the interval $0 \leq x < 2\pi$ with the correct reasoning. Correct answer: There is one solution because the solutions ...
2
votes
1answer
39 views

Prove that $\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$ using given condition [duplicate]

If $$\tan(\frac{\pi}{4}+\frac{\theta}{2})=\tan^3(\frac{\pi}{4}+\frac{\alpha}{2})$$, then prove that $$\sin \theta=\frac{3 \sin \alpha+\sin^3 \alpha}{1+3\sin^2 \alpha}$$ I tried using the fact that $\...
5
votes
2answers
101 views

Roots of $y=x^3+x^2-6x-7$

I'm wondering if there is a mathematical way of finding the roots of $y=x^3+x^2-6x-7$? Supposedly, the roots are $2\cos\left(\frac {4\pi}{19}\right)+2\cos\left(\frac {6\pi}{19}\right)+2\cos\left(\...
-3
votes
0answers
23 views
-1
votes
0answers
53 views

How to derive two angles and a length from this diagram [on hold]

I'm familiar with sohcahtoa, the sin and cosine rule, I just can't seem to apply them here. I know angles alpha and ow, I know lengths z and al. I need to know length ? and angle ?? and angle Q
2
votes
2answers
37 views

Inverse trigonometric function identity doubt: $\tan^{-1}x+\tan^{-1}y =-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)$, when $x<0$, $y<0$, and $xy>1$

According to my book $$\tan^{-1}x+\tan^{-1}y =-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)$$ when $x<0$, $y<0$, and $xy>1$. I can't understand one thing out here that when the above ...
0
votes
3answers
40 views

$\tan x= \cot (x+\phi)$ for some $\phi$

Suppose we graphed the equation $ y = \tan x $. Is it possible to describe this graph with an equation of the form $ y = \cot (x + \phi) $, for some number $ \phi $? Why or why not?
5
votes
4answers
110 views

Tips for integrating $\int \frac{dx}{1+\cos(x)}$

I tried the following $$ \int \frac{dx}{1+\cos(x)}=\int \frac{1-\cos(x)}{1-\cos^2(x)}\,dx= \int \frac{1-\cos(x)}{\sin^2(x)}\,dx\\ =\int \frac{1}{\sin^2(x)}\,dx-\int \frac{\cos(x)}{\sin^2(x)}\,dx=\int ...
3
votes
2answers
39 views

Is it possible to convert the polar equation $\ r = k \cos (\theta n) + 2$ into cartesian form?

Is it possible to convert the polaer equation $$\ r = k \cos (\theta n) + 2$$ into cartesian form? Here, $k$ is some constant and $n$ is any positive whole number greater than $2$. The ...
1
vote
1answer
24 views

Inverse trigonometry simple doubt.

I recently started learning the inverse trigonometric function and got stuck at one point. In the question involving the expression of tangents, It was given that $\frac{x}{y}>1$ The authors ...
0
votes
4answers
46 views

Small problem about domain of a function .

I want to know that whether $f:\mathbb{R}^2/\lbrace(0,0)\rbrace \to \mathbb{R}$ defined by $f(x,y) = \arctan(\frac{x}{y})$ is a function or not? I think this is very silly problem but i think it is ...
0
votes
1answer
81 views

Why are trigonometric functions defined so abruptly?

I will make use of a diagram to ask the question I am asking. First of all, consider this diagram: This will be enough for me to ask a question. Now, let $AC=AB=1$, which means both hypotenuse are ...
8
votes
1answer
95 views

Fake proof that $\frac{e^x-1}{e^x+1}=e^x$, via integrating $\operatorname{sech} x$ in two ways

We start with the integral: $$\int \text{sech}(x)dx$$ Method 1 \begin{align} \int \text{sech}(x)dx & = \int\frac{2}{e^x+e^{-x}}dx \\ &= \int\frac{2e^x}{e^{2x}+1}dx \end{align} Using the ...
1
vote
2answers
34 views

If $a\sin x + b\cos(x+\theta) +b\cos(x-\theta) = d$, then what is the minimum value of $|\cos\theta|$?

If $$a \sin x + b \cos(x+\theta)+ b \cos(x-\theta) = d$$ then what is the minimum value of $|\cos\theta|$? The answer is given: $\dfrac{\sqrt{d^2 - a^2}}{2|b|}$ I tried simplifying the ...
3
votes
1answer
65 views

Find real parts of the complex roots of this $9^{th}$ order polynomial in explicit form

I have a following polynomial. (See WolframAlpha ): $$x^9-6x^8+14x^7-16x^6+36x^5-56x^4+ 24x^3-320x+\frac{640}{9}=0 \tag{1}$$ Wolfram says that $(1)$ has three real roots and three pairs of complex ...
0
votes
5answers
76 views

How do you solve trig integrals using recursions?

My calculus professor gave us the problem $\int \sin^2(x)\cos^2(x)dx$ and told us to solve it via recursion but I can't seem to find how to do it in my textbook.
0
votes
1answer
32 views

Generalize $r\cos(\theta n)$ Into Polynomials in Terms of x

I understand that it is possible to generalize $\cos(\theta n)$ via Chebyshev polynomials of the first kind, and I was also wondering if it is possible to generalize $r\cos(\theta n)$ in a similar ...
3
votes
0answers
69 views

Sine identity involving (3/p) for prime p greater than 3.

I am working through Ireland and Rosen's "Classical Introduction to Modern Number Theory" and am very stuck on this problem (#34 in Chp 5, 2nd edition): Note that $(a/b)$ is the Legendre symbol (or ...
1
vote
1answer
34 views

How to calculate $\lim_{x\to4}(x-4)\cdot\cot(x-4)$

It's been a while since I've done calc, so I'm trying to review by reading "Calculus Demystified" by Steven G. Krantz. Question 1c at the end of chapter 2 has me stumped: $$\lim_{x\to4}(x-4)\cdot\cot(...
3
votes
1answer
75 views

I want to show that, ${\Phi\tan{9^\circ}-\phi\tan{27^\circ}\over \sin^2{9^\circ}-\sin^2{27^\circ}}=4$

$\phi$: golden ratio, $\Phi={1\over \phi}$ I want to show that, $${\Phi\tan{9^\circ}-\phi\tan{27^\circ}\over \sin^2{9^\circ}-\sin^2{27^\circ}}=4$$ Using $\sin^2{x}={1\over 2}(1-\cos{2x})$ $${\Phi\...
0
votes
1answer
37 views

Eliminate $x$ from this system: $a \sec{ x} +b\tan{ x} +c = 0 $ and $ p \sec {x} +q\tan {x} + r = 0$

Eliminate x from this system of equations: $$\begin{align} a \sec{ x} +b\tan{ x} + c &= 0\\ p \sec {x} +q\tan {x} + r &= 0 \end{align}$$ I tried expressing the value of the trigonometric ...
1
vote
7answers
177 views

Solve definite integral: $\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$

I need to solve: $$\int_{-1}^{1}\arctan(\sqrt{x+2})\ dx$$ Here is my steps, first of all consider just the indefinite integral: $$\int \arctan(\sqrt{x+2})dx = \int \arctan(\sqrt{x+2}) \cdot 1\ dx$$ ...