Sine, cosine and tangent of an angle from 0° to 180°
Define sine and cosine by the coordinates of a point on the unit semicircle, find the values for obtuse angles and use sin²α + cos²α = 1.
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Guide
In Ukrainian Grade 9 geometry (§1) the sine and cosine of any angle from 0° to 180° are defined on the unit semicircle: the upper half of the circle with centre at the origin and radius 1.
Let M with coordinates (x,y) lie on the unit semicircle and α be the angle between the ray OM and the positive direction of the x-axis. Then cosα = x, sinα = y and, for α ≠ 90°, tanα = yx.
So sin 0° = 0, cos 0° = 1, sin 90° = 1, cos 90° = 0, sin 180° = 0, cos 180° = −1. For every angle 0 ≤ sinα ≤ 1 and −1 ≤ cosα ≤ 1.
For an obtuse angle the x-coordinate is negative, so the cosine of an obtuse angle is negative and the sine is positive. Also sin(180° − α) = sinα, cos(180° − α) = −cosα, sin2α + cos2α = 1.
One grid square is 0.2.