Angles

*A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same direction.
When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle...
An obtuse angle is that which is greater than a right angle.
An acute angle is that which is less than a right angle.*

-Euclid, *The Elements: Book 1 Definitions IX-XII*, 300 BC
Angles are a rotational measurement between two lines, line segments, or rays. Angles can be measured in either degrees or radians and they can be converted using the relationship
$$ \pi\;\mathrm{radians} = 180^\circ $$

Engineering Context

Engineering design requires a firm grasp on angles and their relationships to one another. There are many examples of when angles are required in each of the disciplines and only a few will be given below.

MAE, BE, and CEE:

The horizontal component of a force is given by

$$ F_x = |F|\cos(\theta) $$

where θ is the angle between the force and the horizontal. Likewise, the vertical component of a force is given by

$$ F_y = |F|\sin(\theta) $$

ECE:

The impedance of an inductor and resistor in series can be written as a complex number as

$$ Z = R\hat{i} + X_L \hat{j}\;\Omega $$

with R the resistance of the resistor and X L the impedance of the inductor. This can be written in terms of a total impedance and phase in phasor notation as

Z = Z ∠ tan -1 X L / R Ω

where tan -1 X L / R is the phase angle.

A Deeper Dive

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