Numbers & Functions · 6 Trigonometric Functions

6.7 The third function, and the others

Sine and cosine are the two stars. The tangent is built from them: tan θ = sin θ cos θ.

Since sine is height and cosine is across-position, tangent is height divided by across — exactly the slope of the line from the center out to the walker. As the walker climbs toward straight-up, that line gets steeper and the slope shoots toward infinity.

slope = tanθ near a quarter-turn the line goes vertical slope blows up −π 2 0 π 2 π 3π 2 4 2 0 2 4 tanθ asymptotes where cosθ = 0 Figure 6.8. Tangent is the slope of the radius line — and it jams where cos θ = 0. As the walker nears a quarter-turn the line goes vertical and the slope blows up, giving the vertical asymptotes of tan θ.

This is the first real domain trouble of the chapter, and it’s the Chapter 3 jam: tangent is a fraction, so it jams wherever the bottom is zero — wherever cos θ = 0, at every odd multiple of π 2 . There the slope has gone vertical. The same rule governs all six trig functions: • sin θ, cos θ — no denominator, never jam. Domain: all angles. (This is why these two are the stars.)

• tan = sin cos and sec = 1 cos — jam where cos θ = 0.

• cot = cos sin and csc = 1 sin — jam where sin θ = 0.

Six functions, but really just two honest ones and four ratios — and every domain hole is nothing more than “you can’t divide by zero,” read off the circle.

6.8 The one identity to rule them all

Here is the single most important fact in all of trigonometry, and you already know why it’s true. Our walker is always at (cos θ, sin θ), and always exactly distance 1 from the center — that’s what being on the unit circle means. How do you measure that distance? The Pythagorean theorem.

1 cos θ sin θ θ Figure 6.9. The Pythagorean identity is just Pythagoras on the radius-1 triangle. The legs are cos θ and sin θ, the hypotenuse is the radius 1, so cos2 θ + sin2 θ = 1 for every angle, automatically.

The distance from the origin to (cos θ, sin θ) is √ cos2 θ + sin2 θ, and that distance is 1, so: cos2 θ + sin2 θ = 1.

This is the Pythagorean identity. It’s not a new fact — it’s Pythagoras, applied to a point on a circle. Everything in trigonometry leans on it. Trigonometric Functions

All 8 chapters of Numbers & Functions · Foundations · All courses