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

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4
votes
4answers
243 views

Solve $\frac{\cos x}{1+\sin x} + \frac{1+\sin x}{\cos x} = 2$

I am fairly good at solving trig equations yet this one equation has me stumped. I've been trying very hard but was unable to solve it. Can anyone help please? Thank you. $$\frac{\cos x}{1+\sin x} + ...
2
votes
2answers
706 views

amplitude of sine wave with multiple frequencies

I'm having some troubles determining the amplitude/magnitude of the following equation. $$ A\cos(2\omega t+\beta_1)+B\cos(3\omega t+\beta_2)+C\cos(5\omega t+\beta_3) $$ Since each part is at a ...
0
votes
1answer
836 views

Solve for $x$ in $[-\pi,\pi]$

Solve for $x$ in $[-\pi,\pi]$ $\cos x=\frac{-\sqrt{3}}{2}$ If I look up above Unit Circle,I can see that $\cos x=\frac{-\sqrt{3}}{2}$ is $\frac{5\pi}{3}$ but the right answer is ...
2
votes
1answer
78 views

Two questions on trigonommetric sums and integrals

Is it true that $\int_{0}^{\infty}\sin(mx)\sin(nx) \, dx = \delta (m-n) $ although using Euler formula I get a linear combination of $ \delta(m-n) $ and $ \delta (m+n)$? What is the sum ...
3
votes
2answers
130 views

Determine convergence of $\sum_{n=1}^{\infty} (\cos{\frac{2}{n}}-\cos{\frac{4}{n}})$

Determine convergence of $$\sum_{n=1}^{\infty} \left(\cos{\frac{2}{n}}-\cos{\frac{4}{n}}\right)$$ In the answer, it says $$\cos{\frac{2}{n}}-\cos{\frac{4}{n}} = 2\sin{\frac{3}{n}}\sin{\frac{1}{n}} ...
1
vote
4answers
410 views

Proving identities using Pythagorean, Reciprocal and Quotient

Back again, with one last identity that I cannot solve: $$\frac{\cos \theta}{\csc \theta - 2 \sin \theta} = \frac{\tan\theta}{1-\tan^2\theta}$$. The simplest I could get the left side to, if at all ...
1
vote
3answers
862 views

Proving an identity using reciprocal, quotient, or Pythagorean identities.

I've been trying to prove this for a while, to no avail. I am only allowed to use pythagorean, quotient, and reciprocal identities: $$\frac{\tan \theta}{1 + \cos \theta} = \sec \theta ...
5
votes
1answer
895 views

Trigonometric identities — working on both sides of the equation at once

When solving trigonometric identities, you aren't allowed to work on both sides of the equation at once. The reason for this is that if you do arrive at a valid conclusion, it doesn't provide the ...
3
votes
1answer
66 views

Can height of a curb be determined by the angles of scratches on the perimeter of a wheel that struck a curb?

At least to me, this turned into an interesting math question. All jokes aside. With any level of certainty, can the angles of scratches on the outermost edge of a wheel of known diameter be used to ...
6
votes
2answers
169 views

Trigonometry Problem

$$\tan\theta + \sec\theta =2\cos \theta,\quad 0\le \theta\le 2\pi$$Find all the possible solutions for the equations. Multiply both sides by $\sec\theta - \tan \theta$. $$\implies (\tan\theta + ...
2
votes
1answer
69 views

Equation with trigonometry

To solve $\sin^3x+\cos^3x=1$ So I just thought of a solution like: Let $\sin x=t$. Then we have: $$t^3+(1-t^2) \sqrt{1-t^2} =1 \\ (1-t^2) \sqrt{1-t^2} =1-t^3 \\ (1-t^2)^3=(1-t^3)^2 \\ ...
2
votes
1answer
1k views

Proof of Osborne's Rule

Osborne's rule is described here. Firstly, am I right that only signs of terms in the form $\sin^{4n+2} \theta$, $n \in \mathbb{Z^+}$ have their signs switched (i.e. terms like $\sin^4 \theta$ simply ...
9
votes
1answer
630 views

sum of series involving coth using complex analysis

I am self-studying complex analysis, so I am a rookie. I ran across an interesting series I am trying to evaluate using CA. Show that $$\sum_{n=1}^{\infty}\frac{\coth(\pi ...
2
votes
4answers
237 views

Show that $2\tan^{-1}(2) = \pi - \cos^{-1}(\frac{3}{5})$

Show that $2\tan^{-1}(2) = \pi - \cos^{-1}(\frac{3}{5})$ So, taking $\tan$ of both sides I get: LHS $=\frac{2\tan(\tan^{-1}(2))}{1 - \tan^2(\tan^{-1}(2))} = -\frac{4}{3}$ and RHS $= \tan(\pi - ...
6
votes
3answers
657 views

How to prove $\cosh(x) \ge 1$ without the $\cosh^2x-\sinh^2x=1$ identity

I think it's pretty easy question, maybe even dumb one, but still I can't find a nice way to solve it. How do you prove that $\forall x . \cosh(x) \ge 1$ , without using the identity: ...
0
votes
1answer
153 views

Trigonometric factorization method

I'm looking at some textbook question and there is a part where it writes this: $$[ 16\cos^4\theta-10\cos^2\theta+1]$$ as this: $$ \left( 2\cos^2 \theta-1 \right) \left( 8\cos^2 \theta-1 \right)$$ ...
1
vote
1answer
249 views

Integrate $(\cos x) ^ 4$

Integrate $(\cos x)^4$. I see solutions using power reduction everywhere. I vaguely remember doing it based on some manipulation of trig identities $(\cos x)^2 = 1 - (\sin x)^2$ and $u$-substitution ...
0
votes
1answer
113 views

How to sketch: inverse $\csc(x) \in (0,\frac{\pi}{2}]$

How to sketch: $\csc^{(-1)}(x) \in (0,\frac{\pi}{2}]$ I can sketch the graph of $f(x) = \csc(x)$ fine, and I can see that an inverse does exist on the given interval as a bijection occurs over ...
2
votes
3answers
528 views

Please help with word problem involving Trigonometry

I'm kind of clueless, apart from its a min/max thing. The question is as follows: The water levels in a dock follow (approximately) a 12-hour cycle, and are modeled by the equation $D = A+ ...
10
votes
5answers
6k views

Best self study math books?

I graduated highschool a while ago, hardly remember anything and have no idea where to begin relearning. I am looking for math text/book recommendations from basic algebra to precalculus. cheers
0
votes
1answer
5k views

Coterminal Angles?

I understood coterminal angles as angles that have the same terminal angle value. By this logic, why aren't 135 and 315 coterminal? They both have a terminal angle of 45. Is my interpretation of ...
0
votes
1answer
1k views

Eccentricity of an ellipse

How is $\frac{PF}{PD} = e = \frac{C}{A}$ ? where e is eccentricity. What is the answer NOT using analytic geometry? (Using trigonometry) P stands for any point on the ellipse. $F$ stands for one of ...
1
vote
3answers
531 views

Showing that $\cos\left(\frac{\pi}{5}\right)=\frac{1}{2}\phi?$

What is the usual way of proving things like $$\cos\left(\frac{\pi}{5}\right)=\frac{1}{2}\phi?$$ I know that there is an identity which claims the above, but how was it derived? Are other identities ...
2
votes
1answer
276 views

Convergence and closed form of this infinite series?

If we have a circle of radius $r$ with an $n$-gon inscribed within this circle (i.e. with the same circumradius), we can find the difference of the areas using: $$A_n =\overbrace{\pi r^2}^\text{Area ...
1
vote
2answers
528 views

Simplify $\arcsin(\sin(x))$ when $\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}$

Simplify $\arcsin(\sin(x))$ when $\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}$ I realize that $\arcsin(\theta)$ is restriced to $\frac{-\pi}{2} \leq \theta \leq \frac{\pi}{2}$ in order to be one to ...
1
vote
3answers
2k views

Using the unit circle to prove the double angle formulas for sine and cosine?

How do you use the unit circle to prove the double angle formulas for sine and cosine?
4
votes
4answers
603 views

What is the most elegant and simple proof for the law of cosines?

Given 2 sides, and an angle between those two sides, what is the simplest proof you can come up with to find the measure of the 3rd side?
1
vote
3answers
252 views

Which of the following expressions completes the identity 1 + sec^2 x?

I simplified $1 + \sec^2 x$ to $1 + 1/\cos^2 x$ then $2/\cos^2 x$ but I am stuck here.
1
vote
1answer
97 views

How do you write the cosecant function of a negative number as the trigonometric function of a positive number?

More specifically, how would you write $\csc(-2\pi/5)$ as the trigonometric function of a positive number?
-1
votes
1answer
198 views

Why do opposite angles have equal cosines?

Why is $x$ in Figure 1 $x$ and not $-x$? This has caused me to not understand why $\cos(-\theta) = x$ and $\cos(\theta) = x$.
3
votes
4answers
2k views

Trigonometric ratios for angles greater than $90^\circ$?

The trigonometric ratios of an angle greater than $90^\circ$ are equal to the supplementary angle's ratios. I'm just clarifying this, but the ratios don't actually exist for angles greater than ...
0
votes
5answers
362 views

Solve that $3 \sec θ = 4 \cos θ$.

Solve the equation $$3 \sec θ = 4 \cos θ \text{ where ; }0 \leq \theta \leq 90$$ Help find and explain theta. Thanks.
0
votes
1answer
2k views

height and distance

A pole fixed on the ground is leaning away from the vertical. When the Sun is directly overhead, the length of its shadow is 7.5m. An observer standing 30m away from the base of the pole in the ...
2
votes
3answers
188 views

Dividing by possibly zero value when proving an identity?

In my trigonometry course, we're currently proving identities. I'm wonder if I can divide by something that could be zero while proving it. For example, $\sin{x}$, is it still a valid proof if I ...
1
vote
3answers
135 views

$\int_0^{2\pi}\sin\frac{x}{2}\cos^2\frac{x}{2}\,dx$

$$\int_0^{2\pi}\sin\frac{x}{2}\cos^2\frac{x}{2}\,dx$$ Tried substitution ($u = \cos\frac{x}{2}$), but I get $-\frac{\cos^3\frac{x}{2}}{3}$ ($-\frac{2}{3}$) instead of the correct answer, which is ...
3
votes
2answers
134 views

Sum the series: $ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \cdots \ \text{ad inf}$

How do I sum the following series? $$ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \sin{3\alpha} + \cdots \ \text{ad ...
1
vote
5answers
3k views

Evaluate the $\sin$, $\cos$ and $\tan$ without using calculator?

Evaluate the $\sin$, $\cos$ and $\tan$ without using calculator? $150$ degree the right answer are $\frac{1}{2}$, $-\frac{\sqrt{3}}{2}$and $-\frac{1}{\sqrt{3}} $ $-315$ degree the ...
3
votes
1answer
1k views

Calculate Length in Perspective View

I have a 4-sided plane in a perspective view. Each side is equal in length to the side across from it. Given the length of two sides and the fore-shortened length of one side, how can we solve for the ...
1
vote
4answers
3k views

Finding the perimeter for a triangle $ABC$ given a side, angle, and the area

Given a triangle $ABC$, $\angle C$ is $65$ degrees, side $C$ is 10. The area of the triangle is $20.$ What is the perimeter?
4
votes
4answers
2k views

How does $\cos(\pi ) = -1$?

I know this is a very elementary question, but how does $\cos(\pi) = -1$? I thought the cosine function required a minimum of 2 numbers, the adjacent side and hypotenuse of a triangle?
2
votes
1answer
383 views

Showing that $\cos n\pi\theta$ is periodic unless $\theta$ is an even integer.

I am trying to show that $\cos n\pi\theta$ is periodic unless $\theta$ is an even integer. I wish to provide a proof based on an example of the $\sin n\pi\theta$ case of the first result, I would ...
40
votes
4answers
2k views

Do “Parabolic Trigonometric Functions” exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) ...
6
votes
2answers
924 views

How was Euler able to create an infinite product for sinc by using its roots?

In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
8
votes
2answers
912 views

Elegant Trigonometric Sums

While studying characters of a finite field and the Polya-Vinogradov inequality, I've found some nice identities (verified by simulations) that I'm not sure how to prove. They seem to be related to ...
1
vote
3answers
417 views

How to express $\cosh(4x)$ as a polynomial in $\cosh(x)$

My friend needs help with this question and I don't know it.
3
votes
2answers
96 views

question about binomial expansion's coefficients

I am trying to show that if $$\left( 1+x\right) ^{n}=p_{0}+p_{1}x+p_{2}x^{2}+\ldots $$ and n being a positive integer, then $$p_{0}-p_{2}+p_{4}+\ldots = 2^{\frac {n} {2} }\cos \dfrac {n\pi } {4}$$ and ...
0
votes
1answer
146 views

Determination of center and radius of circles

I have a problem that I have to solve mathematically. I need the solution1 for drawing circles in a GUI: I have a rect with width s, the height is not important. Inside this rect I draw a circle at ...
4
votes
3answers
2k views

Prove $\sin{2\theta} = \frac{2 \tan\theta}{1+\tan^{2}\theta}$

How to prove: $$\sin{2\theta} = \frac{2 \tan\theta}{1+\tan^{2}\theta}$$ Help please. Don't know where to start.
5
votes
2answers
198 views

How to transform the factored form of $\sin(x)$?

We know $\sin(x)=0$ has solutions $0,\pm\pi,\pm2\pi,\pm3\pi,\dots$. So $\sin(x)$, if interpreted as a polynomial, could be written as: $a_0x^0+a_1x^1+a_2x^2+\cdots$ and we know this polynomial too: ...
1
vote
2answers
72 views

How do I solve this equation?

$\displaystyle \large \cos 2x + 1 - \sin 2x=\frac{2 \cos 2x \cos x}{\cos x + \sin x}$ I've been trying for a long time but I can't get it.