The Unit Circle and Trig Ratios

0:00The unit circle has radius 1 and is centred at the origin.

0:05An angle in standard position starts on the positive x-axis, the initial arm, and turns counterclockwise to the terminal arm.

0:14The terminal arm meets the circle at a point P, with coordinates x and y.

0:20Drop a line from P to the x-axis. The right triangle has adjacent side x, opposite side y, and hypotenuse r equals 1.

0:31So cosine theta is x over r, and sine theta is y over r. Since r is 1, the point P is cosine theta, sine theta.

0:42Let theta be 30 degrees. Cosine 30 degrees is root 3 over 2, about 0.866, and sine 30 degrees is one half.

0:53Check with Pythagoras: x squared plus y squared must equal 1. Root 3 over 2 squared is three quarters, one half squared is one quarter, and they add to 1.

1:06This holds for every angle: cosine squared theta plus sine squared theta equals 1.

1:13Tangent theta is y over x, when x is not 0. For 30 degrees, one half divided by root 3 over 2 is 1 over root 3, about 0.577.

1:27In the same way, 45 degrees is at root 2 over 2, root 2 over 2, 90 degrees is at 0, 1, and 180 degrees is at negative 1, 0.

1:40Reflect the 30 degree point across the y-axis to get 150 degrees. Only x changes sign: negative root 3 over 2, one half.

1:51A common mistake is writing sine theta, cosine theta. Cosine comes first, because it is the x-coordinate.

2:00On the unit circle, the point is cosine theta, sine theta, and tangent theta is y over x.