Articles of trigonometry

Longest pipe that fits around a corner.

This question already has an answer here: Intuitive explanation for formula of maximum length of a pipe moving around a corner? 3 answers

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+…..+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using the product-to-sum formula for $\cos(x)+\cos(y)$?

Finding a closed form for $\cos{x}+\cos{3x}+\cos{5x}+\cdots+\cos{(2n-1)x}$

This question already has an answer here: $\sum \cos$ when angles are in arithmetic progression [duplicate] 1 answer

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the Roll and it is \begin{pmatrix} cos(Pitch) sin(Yaw)\\ cos(Yaw) cos(Pitch)\\ sin(Pitch) \end{pmatrix}. How should it […]

Fixed point iteration convergence of $\sin(x)$ in Java

This question already has an answer here: Convergence of the fixed point iteration for sin(x) 1 answer

Evaluating $\frac{\sum_{k=0}^6 \csc^2(a+\frac{k\pi}{7})}{7\csc^2(7a)}$

The question is to evaluate $$\frac{\sum_{k=0}^{6}\csc^2(a+\frac{k\pi}{7})}{7\csc^2(7a)}$$ where $a=\pi/8$ without looking at the trigonometric table. I tried to transform the $\csc^2$ term to $\cot^2$ term and use addition formula but it made the problem too cumbersome.I also tried to manipulate the numerator in the form so that it telescopes but couldnot succeed.I am not in need […]

solve a trigonometric equation $\sqrt{3} \sin(x)-\cos(x)=\sqrt{2}$

$$\sqrt{3}\sin{x} – \cos{x} = \sqrt{2} $$ I think to do : $$\frac{(\sqrt{3}\sin{x} – \cos{x} = \sqrt{2})}{\sqrt{2}}$$ but i dont get anything. Or to divied by $\sqrt{3}$ : $$\frac{(\sqrt{3}\sin{x} – \cos{x} = \sqrt{2})}{\sqrt{3}}$$

Computation of $\int _{-\pi} ^\pi \frac {e^{in\theta} – e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$

I need to compute $$\int \limits _{-\pi} ^\pi \frac {e^{in\theta} – e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$$ Does anyone see any good strategy? Thanks.

If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$

If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$ My Attempt: $$\sin x + \sin^2 x=1$$ $$\sin x = 1-\sin^2 x$$ $$\sin x = \cos^2 x$$ Now, $$\cos^8 x + 2\cos^6 x + \cos^4 x$$ $$=\sin^4 x + 2\sin^3 x +\sin^2 x$$ $$=\sin^4 x […]

Evaluating a double integral involving exponential of trigonometric functions

I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ I will be satisfied with an answer involving special functions (e.g. modified Bessel function). I tried expanding the product terms using the product-to-sum identities, as well applied other trigonometric identities, to no avail. Can anyone help? I suspect that the solution will involve […]