Articles of trigonometry

finding $\int {\tan^{3/2} 3x\sec 3x\,dx}$

How do I integrate this expression? $$\int \tan^{3/2} 3x\sec 3x dx$$

Triangular sides

In a triangle the least angle is $45º$ and the tangents of the angle are in arithmetic progression. If its area is $27\text{cm}^2$, find the length of the sides. I tried to solve the problem in this way Let the smallest angle in the triangle $ABC$ (say) be $A=45º$, where $\tan A$, $\tan B $ […]

Formula for solving for Cx and Cy…

I’m trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: $C(x, y)$ Assumptions: $Ax = 0, Ay = 0, Bx = AB, By = 0$ Also feel free to […]

Prove that $\frac{1}{90} \sum_{n=1}^{90} 2n \cdot \sin((2n)^\circ) = \cot (1^{\circ})$

Show that $$\frac{(2\sin(2^\circ)) + (4\sin(4^\circ))+ (6\sin(6^\circ)) + \ldots +(180\sin(180^\circ))}{90} = \cot(1^\circ).$$ I used a lot of steps, and typing it all down on here would take me an hour, but here are my last few steps up to the point where I got stuck: $$180 (\sin(2^\circ) + \sin(4^\circ) + \sin(6^\circ) +…..+ \sin(88^\circ)) + 90$$ Using […]

Proving a function is continuous and periodic

Suppose we are given a function $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$ and $$g\left ( k\pi \right )=\lim _{x\rightarrow k\pi}g\left ( x \right )$$ I found that $\lim _{x\rightarrow k\pi}g\left ( x \right )= \sum_{n=1}^{\infty}\frac{1}{10^{n}}$ for both odd and even $k$. However, […]

How to calculate $\left( 1+\tan 5^\circ\right) \left( 1+\tan 10^\circ\right)\left( 1+\tan 15^\circ\right)\cdots\left( 1+\tan 40^\circ\right)$

I curious practical solution.(Step by step) $\left( 1+\tan 5^\circ\right) \left( 1+\tan 10^\circ\right)\left( 1+\tan 15^\circ\right)\cdots\left( 1+\tan 40^\circ\right)$ Answer is $16$.

$W_n=\int_0^{\pi/2}\sin^n(x)\,dx$ Find a relation between $W_{n+2}$ and $W_n$

Set $$W_n=\int_0^{\pi/2}\sin^n(x)\,dx.$$ Compute $W_0$ and $W_1$. Find a relation between $W_n$ and $W_{n+2}$ and deduce a formula for $W_n$. What I have so far is: $$W_{2k}=\frac{1}{2^k}\left( \sum_{\substack{0\leq j \leq k \\ j \text{ even }}} \binom{k}{j} W_j \right)\\[30pt]W_{2k+1}= \sum_{j=1}^k \binom{k}{j}\frac{(-1)^j}{2j+1}.$$ How I got this is: $$\int_0^{\pi/2}\sin^{2k}x\,dx=\int_0^{\pi/2}\frac{1}{2^k}\left(1-\cos2x\right)^k = \frac{1}{2^{k+1}}\int_0^{\pi}(1-\cos x)^k$$ the odd powers evaluate to zero, […]

Calculate coordinate of any point on triangle in 3D plane

I am really stuck and can’t find right way to write a formula(s) that will calculate Z coordinate of point on triangle plane in 3D plane. I know all coordinates of triangle points ( Ax, Ay, Az, Bx, By, Bz, Cx, Cy, Cz ), and I know x and y of point, but I need […]

Calculating $\sum_{k=0}^{n}\sin(k\theta)$

This question already has an answer here: How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? 5 answers

inverse Laplace transfor by using maple or matlab

I want to use inverse Laplace transform to F function by using maple or matlab. However I cannot get any answer. I know the answer from table but I want to use one of softwares. from table: $$\mathscr{L}^{-1}({1\over \sqrt{p}}~ e^{-\sqrt{p~a}}~\cos(\sqrt{p~a}))={1\over \sqrt{\pi~t}}~\cos \left({a \over 2t} \right)$$ Maple: with(inttrans); F := (p^(-1/2))*exp(-(p*a)^(1/2))*cos((p*a)^(1/2)); f := invlaplace(F, p, t); […]