# Derivatives of Trigonometric Functions

## Derivative of $\sin(x) = \cos(x)$ and other relationships between functions.

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X-Ray Vision
Teacher Contributed

Credit: COM SALUD
Source: https://www.flickr.com/photos/com_salud/13866052723

Radiation can cause mutations and cancer. When you get x-rays at the dentist, the technician covers you with a giant lead shield and then flees the room while the x-rays bombard your face. So, if x-rays are so dangerous, why do we also use them to diagnose and treat cancer? How can the same thing be deadly and life-saving?

#### Like Pool, with Electrons

Electromagnetic radiation behaves as both a wave and a particle. This means that it travels through space like a wave, but can also interact with other particles on the physical level. When an x-ray or a gamma ray comes in contact with an atom, it can hit an electron. If it has enough energy to knock the electron loose, it may obey the Newtonian laws for an elastic collision. The electron will bounce off in one direction while the ray moves off in another direction with less energy. The total momentum for the system will be conserved.

This reaction is called Compton Scattering. It can be described by the equation:

\begin{align*}\Delta \lambda = \frac{h}{m_e c}(1-\cos \theta)\end{align*}, where \begin{align*}\Delta \lambda\end{align*} is the change in wavelength, \begin{align*}h\end{align*} is Planck?s constant, \begin{align*}m_e\end{align*} is the mass of the electron at rest, \begin{align*}c\end{align*} is the speed of light, and \begin{align*}\theta\end{align*} is the angle of scattering. The change in wavelength depends on the angle of scattering, and that changes depending on the type of material that the x-ray has hit. In laboratory settings, scientists have used changes in the angle of scattering to distinguish healthy tissue from cancerous tissue.

When an x-ray or gamma ray hits an electron and triggers Compton scattering, the ray goes off in a new direction and hits other electrons. Meanwhile, the electron is on the loose and the original atom has become an ion. If this ion occurs in dividing DNA, it can change the genetic material of a cell and either trigger cell death or cause a mutation. When radiation causes cancer in a person, it?s because it changed atoms in dividing DNA, and created a mutation that led to tumor cells.

When radiation helps to cure cancer in a person, it?s because doctors have targeted a tumor cell with radioactive beams. The radiation causes Compton scattering that damages the rapidly dividing cell so badly that it dies.

Credit: Rhoda Baer for the National Cancer Institute
Source: https://visualsonline.cancer.gov/details.cfm?imageid=4477

While radiation is more likely to cause permanent damage in rapidly dividing cancer cells, it can also damage healthy cells in the body. The cells most susceptible to radiation damage include bone marrow, lymph nodes, the GI tract and the hair follicles. That?s why radiation treatments often leave cancer patients with poor immune systems, constant nausea, and hair loss.

Researchers are constantly seeking ways to improve radiation treatments so that they do more damage to cancer cells and less damage to healthy tissue. In 2011, a pair of high school students came up with a plan to use Compton Scattering to make cancer treatments safer and less expensive.

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