Chapter 8: Trigonometric Functions And Applications

2. Chapter 8: Trigonometric Functions And Applications 8.1 Angles, Arcs, and Their Measures 8.2 The Unit Circle and Its Functions 8.3 Graphs of the Sine and Cosine Functions 8.4 Graphs of the Other Circular Functions 8.5 Functions of Angles and Fundamental Identities 8.6 Evaluating Trigonometric Functions 8.7 Applications of Right Triangles 8.8 Harmonic Motion

3. 8.2 The Unit Circle and Its Functions • Interpretations of trigonometric functions based on the graph of the unitcirclex² + y² = 1. • We refer to them as circular functions.

4. 8.2 Trigonometric Functions Circular (Trigonometric) Functions If (x, y) is the point on the unit circle that corresponds to the real number s, then

5. 8.2 Relationship Between Sine and Cosine • An important relationship between sin s and cos s for any real number s.

6. 8.2 Domains • Since the point (x,y) = (cos s,sin s) represents a point on the unit circle, • For any value of s, sin s and cos s exist, so the domain is the set of all real numbers. • Note that tan s = This is undefined when x = 0. This occurs when and so on. So the domain of tan s is the set of all real numbers s satisfying

7. 8.2 Domains • The secant function also has an x in the denominator, so the domain is the same as the tangent function. • Cotangent and cosecant functions are undefined when y = 0, so the domain for cot s and csc s is the set of all real numbers of s satisfying s n, n any integer. The domains of the six circular functions are as follows. Assume that n is any integer and s is a real number. Sine and Cosine Functions: (–,) Tangent and Secant Functions: Cotangent and Cosecant Functions: {s | s n}

8. 8.2 Identities • The following are true for the real number s, provided the denominator is not zero. • If cos s 0, 1 + tan² s = sec² s. • If sin s 0, 1 + cot² s = csc² s. • These statements are called identities. • An identity is a statement that is true for all values in the domain of the variable. More detail in §8.5.

9. 8.2 Finding Function Values Using the Unit Circle Example For s = , determine the point on the unit circle to which it corresponds, and its trigonometric function values. Solution The point (–1,0) corresponds to s = .

10. 8.2 Exact Function Values for Quadrantal Angles

11. 8.2 Using a Calculator to Find Function Values Example Use a calculator in radian mode to find the six trigonometric function values for . Solution Set the calculator in radian mode. Caution The calculator functions cos-1, sin-1, and tan-1 do not represent the reciprocals of cosine, sine, and tangent. These are inverse trigonometric functions and will be introduced later.

12. 8.2 Function Values at Multiples of 2 • Because the circumference of a circle is 2, any integer multiple of 2 added to s corresponds to the same point as s on the unit circle. • For any integer n, cos(s + 2n) = cos s and sin(s + 2n) = sin s. • For example:

13. 8.2 Signs of Values of Trigonometric Functions Example Use a calculator to find approximations for cos 4.5 and sin 4.5. Then determine the quadrant in which 4.5 lies. Solution cos 4.5  –.2107957994 and sin 4.5  –.9775301177. Since cos 4.5 < 0 and sin 4.5 < 0, the point corresponding to 4.5 lies in quadrant III.

14. 8.2 Exact Function Values for /4 • The line y = x bisects the quadrant I portion of the unit circle giving exact values for

15. 8.2 Exact Function Values for /6 and /3 • Observe the chord QR in the following figure. Q(x,y) corresponds to and Q(x,–y) corresponds to The length of arc QR is

16. 8.2 Exact Function Values for /6 and /3 • Therefore, the real number  /6 corresponds to the point • These results lead to the exact values of the trigonometric function values for

17. 8.2 The Unit Circle

18. 8.2 Finding Circular Function Values with the Unit Circle Example Use the unit circle to find each value. Solution • The point corresponds to on the unit circle. So the sin = ½. • The point on the unit circle for s = is the same as So the tan