Lesson 4.2

A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be (1,0) (0,1) (0,-1) (-1,0)

Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value? (1,0) (0,1) (0,-1) (-1,0) x = 1/2

(1,0) (0,1) (0,-1) (-1,0)  Now, draw a right triangle from the origin to the point when x = ½. Use this triangle to find the values of sin, cos, and tan. Sine is the y value of the unit circle point. Cosine is the x value. Tangent is y/x

(1,0) (0,1) (0,-1) (-1,0)  Find the sin, cos, and tan of the angle

How many degrees are in each division? 45° 90° 0°0° 135° 180° 225° 270° 315°

How about radians? 45° 90° 0°0° 135° 180° 225° 270° 315°

How many degrees are in each division? 30° 90° 0°0° 120° 180° 210° 270° 330° 60° 150° 240° 300°

How about radians? 30° 90° 0°0° 120° 180° 210° 270° 330° 60° 150° 240° 300°

Let’s think about the function f(  ) = sin  Domain: All real numbers. Range: -1  sin   1 (1, 0) (0, 1) (-1, 0) (0, -1)

Let’s think about the function f(  ) = cos  Domain: All real numbers Range: -1  cos   1 (1, 0) (0, 1) (-1, 0) (0, -1)

Let’s think about the function f(  ) = tan  Domain: Range:All real numbers x ≠ n(  /2), where n is an odd integer

Trig functions:

Look at the unit circle and determine sin 420°. So sin 420° = sin 60°. Sine is periodic with a period of 360° or 2 .

Reciprocal functions have the same period. PERIODIC PROPERTIES sin(  + 2  ) = sin  cosec(  + 2  ) = cosec  cos(  + 2  ) = cos  sec(  + 2  ) = sec  tan(  +  ) = tan  cot(  +  ) = cot  This would have the same value as 1 (you can count around on unit circle or subtract the period twice.)

Positive vs. Negative angles. Remember negative angle means to go clockwise

Cosine is an even function.

Sine is an odd function.

EVEN-ODD PROPERTIES sin(-  ) = - sin  (odd) cosec(-  ) = - cosec  (odd) cos(-  ) = cos  (even) sec(-  ) = sec  (even) tan(-  ) = - tan  (odd) cot(-  ) = - cot  (odd) Problem Set 4.2