What is $ \sin(x)+\sin(x−π)+\sin(x+π) $?

As shown in some other answers, this is very simple if you know that: $$\sin(x-\pi)=-\sin x \quad\mbox{and}\quad \sin(x+\pi)=-\sin x$$ If you don't know these formulas or you have a hard time understanding why they are true, you should spend some time to carefully study the unit circle and how symmetry there leads to these simple relations.

The image below should help you understand why $\sin(x+\pi)=-\sin x$.

enter image description here

Then note that by "adding a full cirle", the same holds for the angle $x-\pi$: $$\sin(x-\pi)=\sin(x-\pi\color{blue}{+2\pi})=\sin(x+\pi)=-\sin x$$


$$\sin(x)+\color{green}{\sin(x-\pi)}+\color{red}{\sin(x+\pi)}$$ $$=\sin(x)+\color{green}{\sin(x)\cos(-\pi)+\cos(x)\sin(-\pi)}+\color{red}{\sin(x)\cos(\pi)+\cos(x)\sin(\pi)}$$ $$=\sin(x)\color{green}{-\sin(x)}\color{red}{-\sin(x)}=-\sin(x)$$

using the formula for $\sin(x+\theta)$ and the facts that $\cos(\pm\pi)=-1$ and $\sin(\pm\pi)=0$

Tags:

Trigonometry

Related

When is composition of meromorphic functions meromorphic Kummer sequence etale topology Why doesn't Gödel's incompleteness theorem apply to false statements? Every connected orientable smooth manifold has exactly two orientations, Lee Proposition 15.9 How to prove that function $f(p)= \frac{1}{(p^2 - 1)} \sum_{q=3}^p \frac{q^2-3}{q}[q\text{ is prime}]$ is less than 0.3 for all $p$? $a,b \in \mathbb Z$ we have that $a+ b\phi \in \mathbb Z[\phi]^* \iff a^2 +ab-b^2 = \pm 1$ What is the tangent at a sharp point on a curve? Zero divisors of $\mathbb{Z}_{7}[x] / (x^4+x^3-3)$ and inverse element of $\overline{x+1}$ Integral of $\sin^5x\cdot\cos^{14}x$ Largest multiple of $7$ lower than some $78$-digit number? Combinatorial argument for solution of recursion behaving similarly as Pascals triangle? Find the minimum value of $4(a^3 + b^3 + c^3) + 15abc$ with $a + b + c = 2$.

Recent Posts