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.
- أوكرانيا: الصف ٩
- روسيا: الصف ٩
- ألمانيا: الصف ٩
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الدليل
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 ≠ ٩٠°, tanα = yx.
So sin ٠° = ٠, cos ٠° = ١, sin ٩٠° = ١, cos ٩٠° = ٠, sin ١٨٠° = ٠, cos ١٨٠° = −١. For every angle ٠ ≤ sinα ≤ ١ and −١ ≤ cosα ≤ ١.
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(١٨٠° − α) = sinα, cos(١٨٠° − α) = −cosα, sin٢α + cos٢α = ١.
One grid square is ٠٫٢.