SECTION 1.2 Triangle Geometry 23
1.2.6 Brief interlude: laws of sines and cosines
In a right triangle 4 ABC, whereĈ
is a right angle, we have the familiar
trigonometric ratios: settingθ=
Â, we have
sinθ=a
c
, cosθ=b
c;
the remaining trigonometric ratios (tanθ, cscθ, secθ, cotθ) are all
expressable in terms of sinθand cosθin the familiar way. Of crucial
importance here is the fact that by similar triangles, these
ratios depend only on θ an not on the particular choices of
side lengths.^5
We can extend the definitions of
the trigonometric functions to ar-
bitrary angles using coordinates in
the plane. Thus, ifθ is any given
angle relative to the positivex-axis
(whose measure can be anywhere
between−∞and∞degrees, and if
(x,y) is any point on the terminal
ray, then we set
sinθ=
y
√
x^2 +y^2, cosθ=x
√
x^2 +y^2.
Notice that on the basis of the above definition, it is obvious that
sin(180−θ) = sinθand that cos(180−θ) =−cosθ. Equally important
(and obvious!) is thePythagorean identity: sin^2 θ+ cos^2 θ= 1.
(^5) A fancier way of expressing this is to say that by similar triangles, the trigonometric functions
arewell defined.