I learned geometry many decades ago.
I learned the formulas for the trigonometric functions.
They never made sense to me.
I knew what a tangent was. Why was it also sin/cos?
What was a secant, anyway?
My first realization was π. I thought I knew π. I knew the formulas for the circumference and the area of a circle. I didn't know that the formula for circumference was actually the definition of π.
First came the desire to know the circumference of a circle given the diameter. Second came the realization that it was always the ratio 3.14159265... Third came the name for the ratio, π.
My next realizations were for the trig functions. Where did they originate?
They originated with triangles and the pythagorean theorem. If two sides of the triangle are known, then the third side is known.
Some basic triangles were needed that could be scaled to other triangles. The basic framework was the unit circle. A circle whose center was the origin of the x-y coordinate system with radius 1. A radius of 1 is used to create simple formulas.
A tangent is a line that intersects the circle in exactly one point.
A secant is a line that intersects the circle in two points. Only the part of the secants that are in the first quadrant will be drawn here.
Draw a secant with angle θ from the origin to a point outside the circle, in the first quadrant.
Drop a vertical line to the x-axis from the point where the secant intersects the circle.
Draw another secant from the origin to a point outside the circle with an angle 90-θ.
The angle will be called θ. The complimentary angle is 90-θ.
Create a vertical tangent line at x=1.
Trigonometric functions.
The point where the secant crosses the circle defines sin θ and cos θ.
All we really need is sin, since cos θ = sin (90 - θ)
The point where the secant crosses the tangent line is the definition of tan.
By using similar triangles, the ratio of the sin θ / cos θ is the same as the ratio of tan θ / 1.
tan θ = sin θ / cos θ
The intersection of the complimentary secant and the circle is not in the diagram, but it would have coordinates (sin θ, cos θ) and its intersection with the tangent line would be (1, cot θ).
cot θ = cos θ / sin θ
The length of the secant line from the origin to the tangent line is the definition of sec.
Using the pythagorean theorem on the triangle in the circle gives the identity that sin² θ + cos² θ = 1.
Using the distance formula yields the formula for sec in terms of cos.
First, calculate the square of the sec.
sec² θ = (tan θ - 0)² + (1 - 0)² = tan² θ + 1 = sin² θ / cos ² θ + 1 = (sin² θ + cos² θ) / cos ² θ = 1 / cos ² θ
Then take the square root.
sec θ = 1 / cos θ
A similar calculation with the complement yields
csc θ = 1 / sin θ
