1. SECTION 14-2 Trigonometric Functions of Angles Slide 14-2-1

2. TRIGONOMETRIC FUNCTIONS OF ANGLES Trigonometric Functions Undefined Function Values Slide 14-2-2

3. TRIGONOMETRIC FUNCTIONS Slide 14-2-3 To define the six basic trigonometric functions, start with an angle in standard position. Choose any point P having its coordinates (x, y) on the terminal side of angle (The point P must not be the vertex of the angle.) A perpendicular line from P to the x-axis at point Q determines a triangle having vertices O, P, and Q. r is the distance from the origin (0, 0) to the point P(x, y). See the next slide.

4. TRIGONOMETRIC FUNCTIONS Slide 14-2-4 P(x, y) y x x y r

5. TRIGONOMETRIC FUNCTIONS Slide 14-2-5 The six trigonometric functions of angle are called sine, cosine, tangent, cotangent, secant, and cosecant. In the definitions on the next slide, we use the customary abbreviations for the names of these functions.

6. TRIGONOMETRIC FUNCTIONS Slide 14-2-6 Let (x, y) be a point other than the origin on the terminal side of an angle in standard position. The distance from the point to the origin is The six trigonometric functions are:

7. EXAMPLE: FINDING FUNCTION VALUES OF AN ANGLE Slide 14-2-7 The terminal side of an angle in standard position goes through the point (3, 4). Find the values of the six trigonometric functions of angle (3, 4) 4 3 x y 5 Solution

8. EXAMPLE: FINDING FUNCTION VALUES OF AN ANGLE Slide 14-2-8 The terminal side of an angle in standard position goes through the point (–5, –12). Find the values of the six trigonometric functions of angle (–5, –12) –12 13 x y –5 Solution

9. UNDEFINED FUNCTION VALUES Slide 14-2-9 If the terminal side of an angle in standard position lies along the y-axis, any point on this terminal side has x-coordinate 0. Similarly, an angle with terminal side on the x-axis has y-coordinate 0 for any point on the terminal side. Because the values of x and y appear in the denominators of some of the trigonometric functions, some of the trigonometric function values of quadrantal angles will be undefined.

10. EXAMPLE: UNDEFINED FUNCTION VALUES Slide 14-2-10 The terminal side of an angle in standard position goes through the point (0, 2). Find the values of the six trigonometric functions of angle (0, 2) 2 x y Solution (undefined)

11. UNDEFINED FUNCTION VALUES Slide 14-2-11 If the terminal side of a quadrantal angle lies along the y-axis, the tangent and secant functions are undefined. If it lies along the x-axis, the cotangent and cosecant functions are undefined.

12. QUADRANTAL ANGLES Slide 14-2-12 0°0°010 Undefined 1 90°10 Undefined 0 1 180°0–10 Undefined –1 Undefined 270°–10 Undefined 0 –1 360°010 Undefined 1