# Trigonometry/Applications and Models

## Simple harmonic motion

Simple harmonic motion (SHM) is the motion of an object which can be modeled by the following function:

$x=A\sin \left(\omega t+\phi \right)$

or

$x=c_{1}\cos \left(\omega t\right)+c_{2}\sin \left(\omega t\right)$
where c1 = A sin φ and c2 = A cos φ.

In the above functions, A is the amplitude of the motion, ω is the angular velocity, and φ is the phase.

The velocity of an object in SHM is

$v=A\omega \cos \left(\omega t+\phi \right)$

The acceleration is

$a=-A\omega ^{2}\sin \left(\omega t+\phi \right)=-\omega ^{2}x$

An alternative definition of harmonic motion is motion such that

$\displaystyle a=-\omega ^{2}x$

### Springs and Hooke's Law

An application of this is the motion of a weight hanging on a spring. The motion of a spring can be modeled approximately by Hooke's law:

F = -kx

where F is the force the spring exerts, x is the extension in meters of the spring, and k is a constant characterizing the spring's 'stiffness' hence the name 'stiffness constant'.

#### Calculus-based derivation

From Newton's laws we know that F = ma where m is the mass of the weight, and a is its acceleration. Substituting this into Hooke's Law, we get

ma = -kx

Dividing through by m:

$a=-{\frac {k}{m}}x$

The calculus definition of acceleration gives us

$x''=-{\frac {k}{m}}x$
$x''+{\frac {k}{m}}x=0$

Thus we have a second-order differential equation. Solving it gives us

$x=c_{1}\cos \left({\sqrt {\frac {k}{m}}}t\right)+c_{2}\sin \left({\sqrt {\frac {k}{m}}}t\right)$  (2)

with an independent variable t for time.

We can change this equation into a simpler form. By lettting c1 and c2 be the legs of a right triangle, with angle φ adjacent to c2, we get

$\sin \phi ={\frac {c_{1}}{\sqrt {c_{1}^{2}+c_{2}^{2}}}}$
$\cos \phi ={\frac {c_{2}}{\sqrt {c_{1}^{2}+c_{2}^{2}}}}$

and

$c_{1}={\sqrt {c_{1}^{2}+c_{2}^{2}}}\sin \phi$
$c_{2}={\sqrt {c_{1}^{2}+c_{2}^{2}}}\cos \phi$

Substituting into (2), we get

$x={\sqrt {c_{1}^{2}+c_{2}^{2}}}\sin \phi \cos \left({\sqrt {\frac {k}{m}}}t\right)+{\sqrt {c_{1}^{2}+c_{2}^{2}}}\cos \phi \sin \left({\sqrt {\frac {k}{m}}}t\right)$

Using a trigonometric identity, we get:

$x={\sqrt {c_{1}^{2}+c_{2}^{2}}}\left[\sin \left(\phi +{\sqrt {\frac {k}{m}}}t\right)+\sin \left(\phi -{\sqrt {\frac {k}{m}}}t\right)\right]+{\sqrt {c_{1}^{2}+c_{2}^{2}}}\left[\sin \left({\sqrt {\frac {k}{m}}}t+\phi \right)+\sin \left({\sqrt {\frac {k}{m}}}t-\phi \right)\right]$
$x={\sqrt {c_{1}^{2}+c_{2}^{2}}}\sin \left({\sqrt {\frac {k}{m}}}t+\phi \right)$  (3)

Let $A={\sqrt {c_{1}^{2}+c_{2}^{2}}}$  and $\omega ^{2}={\frac {k}{m}}$ . Substituting this into (3) gives

$x=A\sin \left(\omega t+\phi \right)$

Next Page: Using Fundamental Identities
Previous Page: Inverse Trigonometric Functions

Home: Trigonometry