Trigonometric Identities
Sine and cosine on the unit circle
Draw a circle of radius 1 (the unit circle) centred
at the origin. A point P on the circle at angle
\theta (measured from the positive x-axis)
has coordinates
P = (\cos\theta,\ \sin\theta).
That point sits exactly distance 1 from the centre, so
Pythagoras
on its x and y coordinates gives our first
identity:
\sin^2\theta + \cos^2\theta = 1
Tangent is their ratio
Tangent isn't a new mystery — it's just sine divided by cosine:
\tan\theta = \frac{\sin\theta}{\cos\theta}
Together these two facts tie all three ratios into one family: know one of them and you can work
out the others.
- \sin^2\theta + \cos^2\theta = 1;
- \tan\theta = \dfrac{\sin\theta}{\cos\theta};
- so given any one ratio you can find the others.
See it on the unit circle
Step through the figure: a radius to P, then the right triangle whose
sides are \cos\theta and \sin\theta with
hypotenuse 1.