Definition 3: Trigonometric Functions: The Unit Circle 3.4 JMerrill, 2009 Contributions from DDillon

Recall – Definitions of Trig Functions Definition 1 involved the ratios of 2 sides of a triangle (SOH CAH TOA) Definition 2 dealt with ratios using x- and y-coordinates and the distance from the origin to a point (using x’s, y’s, and r’s)

The Unit Circle This circle has radius of 1. It is centered at the origin. Endpoints are labeled as (1, 0) (0,1) (-1, 0) (0, -1) This is the standard that we use. All our function values are based on this standard.

Definition 3: The Unit Circle Let (x, y) be any point on the unit circle. If θ is the central angle that has the same measure as the arc length from the point (1,0) along the circumference to the point (x, y), then The coordinates of the points along the unit circle can be written (cosθ, sinθ).

Trig Function Values of Quadrantal Angles (0,1) (0, -1) (-1, 0) (1, 0) x’s are cosines y’s are sines 1.sin 180º = _____ 2. cos 90º = _____ 3. cot 270º = _____ 4. tan 90º = _____ 5. csc = _____ 6. Sec = _____ undef 1 1

45 º Recall: 45°-45°-90° Triangles 1 √2/2 sin 45° = √2/2 cos 45° = √2/2 tan 45° = 1 In any 45°-45°-90° triangle, the sides are in the ratio 1 :1 : √2. 45°

The Unit Circle

Recall: 30°-60°-90° Triangles In any 30°-60°-90° triangle, the sides are in the ratio 1 : 2 : √3 60 º 1 1/2 √3/2 30° sin 60° = √3/2 cos 60° = 1/2 tan 60° = √3 sin 30° = 1/2 cos 30° = √3/2 tan 30° = √3/3

The Unit Circle

Trig Function Values 1.sin 30º = _______4. cos π/4 = _______ 2.tan π/3 = _______5. sec π/6 = _______ 3. sin π = _______6. cot π/2 = _______ 1/2 √3 0 √2/2 2√3/3 0