Let’s extend our knowledge of trigonometric functions… …in Section 4.3a

Trigonometric Functions of Any Angle (not just positive or acute angles) A more “dynamic” view of angles: Initial Side – beginning position of a ray (part of the angle) Vertex – the endpoint of this ray Terminal Side – final position of a ray, after being rotated about the vertex Measure of an Angle – a number that describes the amount of rotation from the initial side to the terminal side of the angle Positive angles are generated by counterclockwise rotations and negative angles are generated by clockwise rotations.

Trigonometric Functions of Any Angle (not just positive or acute angles) terminal side terminal side initial side initial side A positive angle in standard position A negative angle in standard position

Trigonometric Functions of Any Angle (not just positive or acute angles) Coterminal Angles – angles that have the same initial side and the same terminal side, but have different measures. Positive and negative coterminal angles Two positive coterminal angles

Guided Practice Find and draw a positive angle and a negative angle that are coterminal with the given angle. Note: there are infinitely many possible solutions… Here are two: (a)

Guided Practice Find and draw a positive angle and a negative angle that are coterminal with the given angle. (b) The diagram? (c) The diagram? I believe that the diagram should always come first!

Definition: Trigonometric Functions of any Angle Let be any angle in standard position and let P(x, y) be any point on the terminal side of the angle (except the origin). Let r denote the distance from P to the origin ( ). Then P(x,y) y r x

Guided Practice Let be the acute angle in standard position whose terminal side contains the point (5, 3). Find the six trig functions of . The diagram? P(5,3) 3 5

Guided Practice Let be any angle in standard position whose terminal side contains the point (–5, 3). Find the six trig functions of . The diagram? P(–5,3) 3 –5

Guided Practice Find the six trigonometric functions of . First, let’s draw a diagram, complete with a reference triangle: P(1,–1)

Whiteboard Practice Find each of the following without a calculator by using ratios in a reference triangle: (a) (b) (c)