# Questions tagged [trigonometry]

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

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### Is there a trigonometric function to calculate average temperature of a region based on longitude?

It seems to me that average temperature of a region is based on the distance and angle to the sun. (Excluding the Earth's tilt for a moment). If this is the case, there should be a way to convert ...
0answers
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### Evaluating $\sum_{n=0}^{\lfloor n/2\rfloor} f(\frac{2 p \pi n}{N+1})g(\frac{2 q \pi n}{N+1})$ where $f$ and $g$ are either $\sin$ or $\cos$

I want to know the value of the followings. \begin{aligned}\sum_{n=0}^{\lfloor N/2\rfloor } \cos \left(\frac{2 p \pi n}{N+1}\right) \cos \left(\frac{2 q \pi n}{N+1}\right) \\[8pt] \sum_{n=0}^{\lfloor ...
0answers
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0answers
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### Find earth radius [on hold]

Given two equations: 1) A=inverse sine(21/R). 2) 0.0008=R-R*cos(A) . A Laser beam points horizontally at height of 1.5 meter above sea level to distance of 21 km. The drop on the the other end is ...
1answer
49 views

### What connection do the hyperbolic trig functions have to the actual trig functions?

As far as my understanding goes, trigonometry is the math of right triangles. Sine is the opposite side over the hypotenuse, cosine is the adjacent side over the hypotenuse, etc. The unit circle ...
1answer
44 views

### Given the equations $y=3\sin x+2$ and $y=x+c$, which statements are true?

$y=3\sin x+2$ $y=x+c$ where c is a constant Which of the following statements is/are true? For some value of c: there is exactly one solution with $0\leq x\leq \pi$ and there is at ...
1answer
59 views

### Proof of $\sin(x) > \frac{2}{\pi} \cdot x$ [duplicate]

I want to prove that $\left(\forall x \in \left( 0; \frac{\pi}{2} \right)\right) \left[ \sin(x) > \frac{2}{\pi} \cdot x \right]$. This is quite easy to see when drawing the functions, but I wonder ...
3answers
70 views

### For non-negative $a$ and $b$ with $a+b \leq c$ for a small constant $c$, what is the minimum of $\cos a + \cos b$?

Let $a,b \geq 0$ with $a+b \leq c$ for a small constant $c$ between $0$ and $1$. What is the minimum of $\cos(a) + \cos(b)$? I conjecture it is $\cos(0)+\cos(c) = 1 + \cos(c)$ but I have no ...
2answers
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0answers
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### Cotangent property

If I have $\cot(k(x-a+π+π/k))$ Then, is $\cot(kx-ka+kπ+π)=\cot(kx-ka+kπ)=\cot(k(x-a+π))=\cot(k(x-a))$ correct if I use $\cot(a+π)=\cot(a)$ property?
3answers
62 views

### Trigonometric equations: cotangent

If I have $cot(x-a)=cot(x-b)$ Where x is in radians and equal on both the sides and not equal to $0$ or $π$ Also for a and b, they are not equal to $0$ or $π$ Does the above equality mean $a=b$? ...
2answers
41 views

### Given that $\sin θ = 5/6$, what is the exact value of $\sin(2θ)$? ($0^\circ < θ < 90^\circ$) [on hold]

Can anyone please explain this to me? Thank you!
2answers
30 views

### Find distance AB given bearings and angles of elevation

I'm having issues with the following problem, which I think are due to how I am modelling the problem. Seems like I have to think in 2D and 3D which is where I get confused. Could someone show me how ...
2answers
57 views

### Show that the general value of $\theta$ satisfying $\sin\theta=\sin\alpha$ and $\cos\theta = \cos\alpha$ is given by $\theta = 2n\pi + \alpha$ [duplicate]

The general value of $\theta$ simultaneously satisfying equations $$\sin\theta = \sin\alpha \quad\text{and}\quad \cos\theta = \cos\alpha$$ is given by $\theta = 2n\pi + \alpha$, where $n\in\mathbb{Z}$ ...
2answers
142 views

### value of $(\cos\frac{2\pi}{7})^{{1}/{3}}+ (\cos\frac{4\pi}{7})^{{1}/{3}} + (\cos\frac{8\pi}{7})^{{1}/{3}}$

This question was on my list. I was trying to apply the $n$-th roots of unity, but other ideas are welcome. I also tried Newton's sums, but it's not working. I searched around here and I didn't find ...
1answer
76 views

### Finding global extrema of $a\left(\frac{1}{2}-b\frac{\sin(cx)}{x} \right) - b(1-\cos(cx))$, for $x\geq 0$

I have the following equation: \begin{equation} f(x) = a\left(\frac{1}{2}-b\frac{\sin(cx)}{x} \right) - b(1-\cos(cx)), \quad x\geq0, \end{equation} where a, b and c are strictly positive constants....
3answers
60 views

### If $u=\sqrt {a\cos^{2}x+b\sin^{2}x} + \sqrt {b\cos^{2}x+a\sin^{2}x}$ , find the maximum and minimum value of $u^2$.

If $u=\sqrt {a\cos^{2}x+b\sin^{2}x} + \sqrt {b\cos^{2}x+a\sin^{2}x}$ , find the maximum and minimum value of $u^2$. This problem was bothering me for a while. The minimum value of $u$ seemed ...
4answers
172 views

1answer
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### Why do the sines of the numerators of $\pi$’s continued fraction convergents approach zero?

I was messing with the sine function and tried getting values close to zero with integer inputs. I found a peculiar pattern. If you take pi’s continued fraction and write them out as one whole ...
1answer
50 views

### Twice angle conditions with a point in a triangle

Let $X$ be a point lying in the interior of the acute triangle $ABC$ such that $\angle BAX = 2\angle XBA$ and $\angle XAC = 2\angle ACX$. Denote by $M$ the midpoint of the arc $BC$ of the ...
2answers
40 views

### Is there an easy way to quickly prove (or memorize) inverse trig formulas such as $\arcsin(a) = \arctan(\frac{a}{\sqrt{1-a^2}})$?

Is there an easy way to quickly prove these formulas? If not, is there any easy mnemonic way to memorize them fast? \begin{align} \arcsin(a) &= \arctan\left(\frac{a}{\sqrt{1-a^2}}\right) \...
1answer
22 views

### How to find the angle of a vector based on origin? [closed]

How can I find the angle between a vector and unit vector [1, 0], but I don't want the shortest angle, but always the right side angle like this: ...
2answers
59 views

### How to form functions similar to $\frac 12 - \frac12 \cos \pi x$

I wanted to know how to form curves similar to that of $y = \frac12 - \frac12\cos(\pi x)$. What I need is a function that forms a smooth curve been $x = 0$ to $x = 1$. As you see with the \$y = \frac12 ...