5.3-part 1 The Circular Functions The trigonometric functions, when defined for all real values, are referred to as the circular functions. To define the circular functions for any real number s, use the unit circle, the circle with center at the origin and radius one unit. Start at the point (1,0) and measure an arc of length s, counterclockwise if s > 0 and clockwise if s < 0.

5.3 The Circular Functions

5.3 The Circular Functions Evaluating a Circular Function Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians. This applies to both methods of finding exact values (such as reference angle analysis) and calculator approximations. Calculators must be in radian mode when finding circular function values.

5.3 Evaluating Circular Functions Example Evaluate and . Solution An angle of radians intersects the unit circle at the point (0, -1)

5.3 Special Angles and The Circular Functions The special angles and their corresponding points on the unit circle are summarized in the figure.

5.3 Evaluating Circular Functions Example (a) Find the exact values of and . (b) Find the exact value of .

5.3 Evaluating Circular Functions Solution (a) From the figure The angle -5p/3 radians is coterminal with an angle of p/3 radians. From the figure

5.3 Trigonometric Functions and the Unit Circle The figure relates trigonometric functions, triangles and the unit circle. Horizontal segments to the left of the origin and vertical segments below the x-axis represent negative values.

5.3 Applications of Circular Functions The phase F of the moon is given by where t is called the phase angle. F(t) gives the fraction of the moon’s face illuminated by the sun.

5.3 Modeling the Phases of the Moon Example Evaluate and interpret. (a) (b) (c) (d) Solution (a) new moon

5.3 Modeling the Phases of the Moon Solution (b) first quarter (c) full moon (d) last quarter