Motivation for the definition of sine

what's the motivation for letting sin(x) be defined for angles greater than 90°? how is it defined without a right triangle?

In elementary geometry, sine is introduced via a right-triangle: for an acute angle θ,

$$ \sin θ = \frac{\text{opposite side}}{\text{hypotenuse}}\,.
$$

But once you want to talk about angles beyond 90° (or even negative angles, or angles > 360°, or non-geometric “angles” like radians), that triangle picture no longer suffices. Here’s the motivation and the standard ways we extend sin to all real (and even complex) arguments:

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1. Motivation for extending beyond 90°

1. Continuity & periodicity If you try to piece together right-triangle definitions only for 0°–90°, you’d have gaps and “jumps” when you pass 90°, 180°, etc. But in nature (waves, rotations, oscillations) the sine-wave is smooth and repeats forever. Extending sin θ to every real θ makes it a continuous, infinitely differentiable, 2π-periodic function.

2. Applications in physics & engineering Rotations in the plane (e.g. turning a vector by an arbitrary angle), harmonic motion, signal processing, Fourier analysis—all require sine (and cosine) at angles of any size, not just acute ones.

3. Solving differential equations The basic ODE

   $$ \frac{d^2y}{dx^2} + y = 0 $$

   has solutions y = sin x and y = cos x defined for all real x. Restricting to 0–90° would deprive us of half of the solutions!

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2. The unit‐circle definition

We place ourselves in the Cartesian plane and draw the unit circle (radius 1 centered at the origin). Then for any real angle x (measured, say, in radians, counter-clockwise from the positive-x axis):

• Travel around the circle by arc‐length x.

• Let (X,Y) be the coordinates of the point you land on.

• Define

  $$ \cos(x) = X,\quad \sin(x) = Y. $$

This single picture handles:

• x between 0 and π/2 (0°–90°) → first quadrant
• x between π/2 and π (90°–180°) → second quadrant (sin is still positive, cos flips negative)
• x between π and 3π/2 → third quadrant (both sin, cos negative)
• and so on, for all real x, positive or negative, even beyond 2π.

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3. The power‐series (analytic) definition

Analytically, we can define sine by its infinite Taylor series about 0:

$$ \boxed{\displaystyle \sin x = x \;-\;\frac{x^3}{3!}\;+\;\frac{x^5}{5!}\;-\;\frac{x^7}{7!} \;+\;\cdots} $$

This converges for every real (and complex!) x. You no longer need a triangle or circle—just algebra and limits. From this you can prove all the familiar properties:

• Oddness: sin(−x)=−sin x
• Periodicity: sin(x+2π)=sin x
• Derivatives: d/dx sin x=cos x, etc.

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4. The differential‐equation definition

Yet another equivalent is to say “sin x is the unique solution of”

$$ \frac{d^2y}{dx^2} + y = 0,\quad y(0)=0,\;y'(0)=1. $$

This defines it for all x, independent of any triangle picture.

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In practice

• For intuition: think “y‐coordinate on the unit circle.”
• For computation: use the series or numerical algorithms derived from it.
• For theory: you can lean on the ODE or on complex‐analytic continuation to define sin z for complex z.

All of these views agree and ensure that sin x makes perfect sense (and behaves beautifully) for angles of any measure—acute, obtuse, reflex, negative, fractional, or purely imaginary.

The output contained a hallucinated imgur image. I only saw it after exporting it for this post. I have commented it out, but you can see it in the HTML of this page.