Advanced trigonometric applications use sine, cosine, and tangent to model real-world quantities
that involve periodic motion, rotation, direction, or components.
1. Harmonic motion
A common sinusoidal model is:
\[
y=A\sin(\omega t+\phi)+D
\]
Here, \(A\) is the amplitude, \(\omega\) is the angular frequency, \(\phi\) is the phase,
\(D\) is the vertical shift, and \(t\) is time.
The period is:
\[
T=\frac{2\pi}{\omega}
\]
2. Rotating point
A point rotating on a circle of radius \(r\) can be modeled by:
\[
x=h+r\cos\theta
\]
\[
y=k+r\sin\theta
\]
If the angle changes with time, then:
\[
\theta=\omega t+\phi
\]
3. Vector components
A vector with magnitude \(R\) and standard angle \(\theta\) can be split into:
\[
x=R\cos\theta
\]
\[
y=R\sin\theta
\]
To add vectors, add their \(x\)-components and \(y\)-components separately.
4. Navigation bearings
In navigation, a bearing is measured clockwise from north. If a displacement has length \(d\)
and bearing \(B\), then:
\[
E=d\sin B
\]
\[
N=d\cos B
\]
\(E\) is the east component and \(N\) is the north component.
5. Formula summary
| Application |
Model |
Meaning |
| Harmonic motion |
\(y=A\sin(\omega t+\phi)+D\) |
Models periodic displacement |
| Rotating point |
\(x=h+r\cos\theta,\ y=k+r\sin\theta\) |
Models circular motion |
| Vector components |
\(x=R\cos\theta,\ y=R\sin\theta\) |
Converts magnitude and direction into components |
| Navigation bearing |
\(E=d\sin B,\ N=d\cos B\) |
Converts a bearing into east and north components |
6. Example: harmonic motion
Suppose:
\[
y=4\sin\left(\frac{\pi}{3}t+\frac{\pi}{6}\right)+1
\]
At \(t=2\):
\[
y=4\sin\left(\frac{2\pi}{3}+\frac{\pi}{6}\right)+1
\]
\[
y=4\sin\left(\frac{5\pi}{6}\right)+1
\]
\[
y=4\cdot\frac12+1=3
\]
7. Common mistakes
- For standard vectors, use \(x=R\cos\theta\) and \(y=R\sin\theta\).
- For navigation bearings, use \(E=d\sin B\) and \(N=d\cos B\), because the angle is measured from north.
- Keep radians and degrees consistent.
- Angular frequency \(\omega\) is usually measured in radians per time unit.
- In harmonic motion, the period is \(T=\frac{2\pi}{\omega}\), not \(\frac{\omega}{2\pi}\).
- When adding vectors, add components first, then find magnitude and direction.
