Trigonometry and Vectors

Exponents and Logarithms handled size: how large, how fast growing. Size alone cannot describe a wind. A weather station reports 30 km/h from the northwest, and those two numbers, speed and heading, do a job one number never could. Direction is everywhere in engineering: a plane fights a crosswind, a robot arm pushes along an axis, current flows around a loop. Direction needs geometry, and the right triangle is the tool that turns direction into arithmetic.

Similar triangles fix the ratios

Two triangles are similar when they have the same angles, and similar triangles are scaled copies of each other. A triangle with sides 3, 4, and 5, and a triangle with sides 6, 8, and 10, hold the same three angles; the second is just the first photographed from twice as close. Every side of the big one is exactly twice the matching side of the small one.

The consequence that powers all of trigonometry is about ratios, not lengths. In the 3-4-5 triangle the ratio of the shortest side to the longest is 3/5 = 0.6, and in the 6-8-10 triangle it is 6/10 = 0.6, the same number. Every ratio of sides stays fixed when a triangle is scaled, because the scaling factor cancels between numerator and denominator. Ratios depend only on the angles. That single fact is the foundation: once an angle is fixed, all its side ratios are fixed with it, no matter how large or small the triangle is drawn.

This is also how heights get measured without climbing anything. A 3 meter pole stuck in the ground casts a 4 meter shadow at some moment; at that same moment a flagpole casts a 40 meter shadow. Sunlight strikes both at the same angle, so the two shadow-triangles are similar, and the flagpole's height is 40 times 3/4, which is 30 meters. Thales of Miletus is credited with exactly this trick on the Egyptian pyramids around 600 BC, and it needs nothing more than the fixed-ratio fact above.

Sine, cosine, and tangent

Name the parts of a right triangle relative to one chosen acute angle. The hypotenuse is the longest side, opposite the right angle. The opposite side is the one across from the chosen angle, and the adjacent side is the one next to it that is not the hypotenuse. Which side is "opposite" changes when the chosen angle changes, so the labels belong to the angle, not to the triangle.

Three ratios get names. The sine of the angle is opposite over hypotenuse. The cosine is adjacent over hypotenuse. The tangent is opposite over adjacent. For the 3-4-5 triangle with the angle chosen at the corner where the 3 side meets the hypotenuse, the opposite side is 4 and the adjacent side is 3, so sin of that angle is 4/5 = 0.8, cos is 3/5 = 0.6, and tan is 4/3, about 1.33. A few angles come out exact and are worth knowing cold: sin of 30 degrees is exactly 0.5, cos of 60 degrees is 0.5, and tan of 45 degrees is 1, because a 45 degree right triangle has two equal legs.

The tangent ratio connects straight back to the last lesson: for a line climbing at an angle, the tangent of that angle is the slope, rise over run once again. Trigonometry did not invent a new idea there; it named the slope of a hypotenuse.

Radians beat degrees

A degree is a historical accident: some ancient astronomer split the circle into 360 parts, and the convention stuck. A radian is a geometric definition: the angle that bends an arc of length equal to the radius around the edge of the circle. Since a full circle's edge has length 2 times pi times r, a full turn is 2 times pi radians, about 6.283. Half a turn is pi radians, so pi radians = 180 degrees, and a quarter turn is pi/2 radians, about 1.571. Converting is one multiplication: multiply degrees by pi/180 or radians by 180/pi.

Two things make radians the professional choice. First, arc length becomes trivial arithmetic: an angle of theta radians sweeps an arc of length r times theta, with no conversion constant in the way, which matters every time a wheel rolls or a motor turns. Second, and less obvious until the next lesson, the slope of the sine curve equals the cosine curve only when the angle is in radians; in degrees a nuisance factor of pi/180 attaches itself to every derivative. Working scientists write radians unless forced to do otherwise, and Python's math module agrees: math.sin takes radians, and math.radians converts when a degrees value arrives from elsewhere.

The unit circle

Draw a circle of radius 1 centered where the axes cross, start measuring the angle at due east, and sweep counterclockwise. The point where the angle's ray meets the circle has coordinates (cos of the angle, sin of the angle): cosine is the east-west coordinate and sine is the north-south coordinate. Every value of sine and cosine comes from this one picture. At 0 degrees the point sits at (1, 0), so cos is 1 and sin is 0. At 90 degrees it sits at (0, 1). Sweep past 90 degrees and the east-west coordinate goes negative, because the point has crossed into the western half of the circle; a negative cosine is not an error, it is the arithmetic reporting direction. This one diagram is why sine and cosine are called circular functions, and it is the picture to summon whenever a sign comes out unexpected.

A vector carries size and direction

A vector is a quantity that carries a magnitude and a direction together. A wind of 30 km/h from the northwest, a push of 50 newtons on a crate, a walk of 5 kilometers due east: all vectors. A scalar is a bare quantity with no direction: mass, temperature, time, price. Adding scalars is ordinary arithmetic; adding vectors needs care, because 30 km/h of wind plus 400 km/h of plane does not produce 430 km/h unless both point the same way.

On paper a vector is an arrow whose length shows the magnitude and whose orientation shows the direction. The picture is honest but hard to compute with. The fix is to store the arrow as two numbers, its reach along the east-west axis and its reach along the north-south axis, and those numbers are called components.

Components turn arrows into numbers

Components fall out of the triangle ratios directly. For a vector of magnitude M pointing at an angle measured up from due east, the eastward component is M times cos of the angle and the northward component is M times sin of the angle. The hypotenuse is the vector itself, the adjacent side is its eastward reach, and the opposite side is its northward reach, so cosine and sine are just doing their original jobs.

Take the 30 km/h wind at 30 degrees north of east. The eastward component is 30 times cos 30, about 26.0 km/h. The northward component is 30 times sin 30, exactly 15.0 km/h, because sin 30 is exactly 0.5. Every part of that arrow now sits in ordinary numbers, and numbers add. Adding two vectors means adding their east components together and their north components together, two separate sums of ordinary arithmetic.

The conversion also runs backward. Given components, the magnitude comes back out of the Pythagorean theorem, sqrt(x^2 + y^2), and the angle comes from the tangent: the tangent of the angle above due east is the north component divided by the east component, which is just the slope of the vector's line. A vector of 3 east and 4 north has magnitude 5 and an angle whose tan is 4/3: the 3-4-5 triangle wearing a compass rose. Every arrow is a triangle, and every triangle is a pair of numbers, so nothing in this section ever needs a drawing to be computed.

A plane heads due east at 400 km/h while that 30 km/h wind pushes it. The east components add to 400 + 26.0 = 426.0, the north components to 0 + 15.0 = 15.0. The plane's actual path, called the resultant, has magnitude found by the Pythagorean theorem, sqrt(426.0^2 + 15.0^2), about 426.2 km/h, and a heading of a bit over 2 degrees north of east. The wind nudged a 400 km/h plane by only 0.2 km/h of speed but visibly bent its heading, which is exactly the kind of insight components make obvious and arrow-drawing hides.

import math

def add(a, b):
    return [a[0] + b[0], a[1] + b[1]]

def components(speed, angle_deg):
    r = math.radians(angle_deg)
    return [speed * math.cos(r), speed * math.sin(r)]

plane = [400.0, 0.0]                  # due east, km/h
wind = components(30.0, 30)           # 30 km/h, 30 degrees north of east
total = add(plane, wind)
speed = math.hypot(total[0], total[1])
heading = math.degrees(math.atan2(total[1], total[0]))
print(round(speed, 1), round(heading, 1))

The print shows 426.2 and 2.0: the ground speed and the heading drift in degrees north of east. The function components is worth keeping; the same ten lines convert any magnitude-and-angle pair into machine-addable numbers, and atan2 reports the resultant's angle from its components. No protractor appears anywhere in the calculation, and that is the point: components reduce geometry to arithmetic, and arithmetic does not need a steady hand.

Components give every quantity a size and a direction, both in plain numbers that add. What the arithmetic still cannot answer is how any of it changes from one moment to the next: speeds grow, headings swing, temperatures climb. Rates and Derivatives builds the tool for change itself, and it starts by shrinking a time interval down to nothing.