1. The Trigonometric Functions ---Mandy

2. 6.1 Trigonometric Functions of Acute Angles

3. The Trigonometric Ratios An acute angle is an angle with measure greater than 0° and less than 90°.

4. Letters that are used to denote an angle : α(alpha), β(beta),γ(gamma), θ(theta), and φ(phi).

5. Trigonometric Function Values of an Acute Angle θ sin θ= length of side opposite θ÷ length of hypotenuse cos θ= length of side adjacent θ ÷ length of hypotenuse tan θ = side opposite θ ÷ side adjacent θ csc θ = hypotenuse ÷side opposite θ sec θ = hypotenuse ÷ side adjacent to θ cot θ =side adjacent to θ ÷ side opposite θ

6. 6.2 Applications of Right Triangles

7. Example : Cloud Height To measure cloud height at night, a vertical beam of light is directed on a spot on the cloud. From a point 135 ft away from the light source, the angle of elevation to the spot is found to be 67.35°. Find the height of the cloud. From the figure, we have tan 67.35° = h / 135 ft h = 135 ft · tan 67.35° ≈ 324 ft

8. 6.3 Trigonometric Function of Any Angle

9. Trigonometric Functions of Any Angle θ Suppose that P(x, y) is any point other than the vertex on the terminal side of any angle θ in standard position, and r is the radius, or distance from the origin to P(x, y). Then the trigonometric functions are defined as follows:

10. sin θ = y-coordinate / radius = y / r cos θ = x-coordinate / radius = x / r tan θ = y-coordinate / x-coordinate = y / r csc θ = radius / y-coordinate = r / y sec θ = radius / x-coordinate = r / x cot θ = x-coordinate / y-coordinate = x / y

11. 6.4 Radians, Arc Length, and Angular Speed

12. Linear Speed in Terms of Angular Speed The linear speed v of a point a distance from the center of rotation is given by v = rω, where ω is the angular speed in radians per unit of time.

13. 6.5 Circular Functions: Graphs and Properties

14. Basic Circular Functions For a real number s that determines a point (x,y) on the unit circle: sin s = second coordinate = y cos s = first coordinate = x tan s = second coordinate / first coordinate = y/x csc s = 1 / second coordinate = 1/y sec s = 1 / first coordinate = 1/x cot s = first coordinate / second coordinate = x/y

15. Domain and Range of Sine and Cosine Functions The domain of the sine function and the cosine function is (-∞, ∞). The range of the sine function and the cosine function is [-1, 1].

16. 6.6 Graphs of Transformed Sine and Cosine Functions

17. Graphing a Transformation of Sine and Cosine

18. Amplitude The amplitude of the graphs of y =Asin(Bx- C)+D and y = Acos(Bx-C)+D is |A|. Period The period of the graphs of y =Asin(Bx-C)+D and y = Acos(Bx-C)+D is |2π/B| Phase Shift The phase shift of the graphs is the quantity C/B y = A sin(Bx-C)+D = A sin [B (x - C/B) ] + D and y = A cos(Bx-C)+D = A cos [B (x - C/B) ] + D

19. Thank You