Definition of the sine and cosine of an angle

(1) If an angle has measure between 0 and 90 degrees, the sine and cosine can be defined using a right triangle, as in a beginning trigonometry course. Let ABC be a triangle with the measure of the angle at vertex A equal to α (the Greek letter alpha), and the measure of the angle at vertex C equal to 90 degrees. Then the sine of the angle α is defined by

sin α = BC/AB = opposite/hypotenuse,

and the cosine of the angle α is defined by

cos α = AC/AB = adjacent/hypotenuse.

Note that the sine and cosine of an angle depend only on the measure of the angle, not on the particular right triangle used. For if another right triangle A'B'C' has the same angle α at vertex A' and a right angle at vertex C', then the triangles ABC and A'B'C' are similar, by the angle-angle similarity theorem, so B'C'/A'B' = BC/AB, and A'C'/A'B' = AC/AB.

For this definition we are using absolute angle measure, which is the same as GSP degrees.

(2) For arbitrary angles the sine and cosine can be defined using Cartesian coordinates and the unit circle, as in a precalculus course. Suppose we want to define the sine and cosine of angles with initial side the ray R with endpoint O. Let L be the line containing R. (L is like the x-axis.) Let S be the ray with endpoint O such that the angle from R to S is +90 degrees. Let M be the line containing S. (M is like the y-axis.) Let C be the circle with center O and radius 1.

Now let T be a ray with endpoint O so that the measure of the angle from R to T is α. Let P be the intersection of T and C. Let PA be the perpendicular segment from P to L and let PB be the perpendicular segment from P to M. Then the absolute value |cos α| is the length |x| = PB = AO, and |sin α| is the length |y| = PA = BO. The plus or minus signs of sin α and cos α are determined by the quadrant in which the point P lies. If P lies in the first quadrant (on the same side of M as R and on the same side of L as S) then sin α and cos α are positive, i.e. (x,y) = (cos α, sin α) has signs (+,+). If P lies in the second quadrant, (cos α, sin α) has signs (-,+). If P lies in the third quadrant, (cos a, sin a) has signs (-,-). If P lies in the fourth quadrant, (cos α, sin α) has signs (+,-).

Note that here we can use either angle measure mod 360 (which we have denoted @ in this course) or we can use proper angle measure (which we have denoted @'). Proper angle measure is the same as GSP directed degrees.

Some consequences of the definition of sine and cosine:

If α + β = 0 then cos α = cos β, and sin α = - sin β. In other words,

cos(-α) = cos α,

sin(-α) = - sin α.

If α + β = 90 (α and β are complementary), then cos α = sin β. In other words,

cos(90 - α) = sin α,

sin(90 - α) = cos α.

If α + β = 180 (α and β are supplementary), then cos α = - cos β, and sin α = sin β. In other words,

cos(180 - α) = - cos α,

sin(180 - α) = sin α.

These formulas work for angle measure mod 360 and for proper angle measure.

C. McCrory 10/17/07