Trigonometry

Topics

• Trigonometry

  • Beyond triangles

  • Circles

• Sinusoidal functions

• Trig estimations

New models

• Triangle model

• Sinusoidal model

Trigonometry or triangle model

• Trigonometry, as it applies to triangles, is the use of proportions

  with names like sine and tangent.

• The lengths of any two sides of a right triangle of a similar shape

  are proportional.

• The constant of proportionality is the sin, cos, or tangent

• We define "similar shape" by specifying the angles in the triangle

Triangle model

• What can we model with a triangle?

• The height of a distant object

• An arc length for a small angle

Sinusoidal model

• We can generalize the triangle and use the unit circle

Sinusoidal model

• What can we model with a sinusoid?

• Planetary motion

• Synthesizers add sine waves together to make sounds

• Sound is made up of sine waves

• Light is made up of sine waves

• Populations have actions similar to sine waves

Sine and cosine animation

• Unit Circle, Lucas Barbosa

Different measurements of angle

• Rotations

• Degrees

• Radians

Rotations

• Divers use full rotations as a unit of angle

• Isn't easy to express small angles

Degrees

• Divide the circle into 360 parts

Radians

• Divide the arc length of a wedge by the radius

• Radians have no dimension

• The most beautiful and natural way to describe angles

Description of the radian

• Radian, Lucas Barbosa

Degree and radian conversion factors

• How many degrees in a circle?

• How many radians in a circle?

• Form a conversion factor

Mathematically expressing a sine wave

$\sin\theta$

$\sin \omega t$

We can think of this as another unit conversion. In this case instead of converting degrees to radians, we are converting meters or seconds or days to radians. We can think of the greek letter $\omega$ as a conversion factor.

Dimensions

• The quantity in the sine or cosine must be dimensionless

• To get a sinusoid with time or space as the argument we use a unit conversion

• $\omega$, $k$ are popular

• $\omega$ has dimensions of radians per unit time

• $k$ has dimensions of radians per distance

Calculation examples

• Excel

• Calca

• Julia, Jupyter

Calculator exercises

• Sine and cosine functions

  • Find the sine of 30 degrees

  • Find the cosine of 60 degrees

• Switching between degrees and radians

  • Find the cosine of 1 radian

  • Find the sine of 1/2 a radian

Pi in unexpected places

$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$

How many radians in a thumb?

• Written

• For small angles, we can estimate the arc length as the vertical

  length

• This means in radians for small angles $\sin\theta \sim \theta$

Activities

• Measure angle subtended by your thumb

• Measure height of bacon and eggs

• Measure height of Darwin