1. 4.4 Trigonometric Functions of Any Angle
Objectives
Use the definitions of trigonometric functions of any angle
Use the signs of the trigonometric functions
Find reference angles
Use reference angles to evaluate trigonometric functions

2. Applications
Cycles govern many aspects of life – heartbeats, sleep patterns, seasons, and tides all follow regular, predictable cycles. Trigonometric functions are used to model phenomena that occur in cycles.

3. Example
Let P = (1, -3) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ.

4. Examples
Evaluate, if possible, the six trigonometric functions of the four quadrantal angles.

5. Signs of Trigonometric Functions
If θ is not a quadrantal angle, the sign of a trigonometric function depends on the quadrant in which θ lies. y
Quadrant II
Quadrant I x
Quadrant III
Quadrant IV

6. Signs of Trigonometric Functions
If θ is not a quadrantal angle, the sign of a trigonometric function depends on the quadrant in which θ lies. y
Quadrant II
Sin and csc
positive
Quadrant I
All
positive x
Quadrant III
Tan and cot positive
Quadrant IV
Cos and sec positive

7. Example
If sin θ < 0 and cos θ < 0, name the quadrant in which angle θ lies.

8. Example
If tan θ < 0 and cos θ > 0, name the quadrant in which angle θ lies.

9. Example
Given that tan θ = and cos θ > 0, find cos θ and csc θ.

10. Example
Given that tan θ = and cos θ < 0, find sin θ and sec θ.

11. SAT PREP QUESTION OF THE DAY
An exhibitor is selling decorative wreaths at an arts and crafts show. The net profit P, in dollars, from the sales of the wreaths is given by P(n) = 0.75n – 50, where n is the number of wreaths sold. How many wreaths must the exhibitor sell in order to earn a net profit of $100? 25 75
150
175
200

12. SAT PREP QUESTION OF THE DAY
An exhibitor is selling decorative wreaths at an arts and crafts show. The net profit P, in dollars, from the sales of the wreaths is given by P(n) = 0.75n – 50, where n is the number of wreaths sold. How many wreaths must the exhibitor sell in order to earn a net profit of $100? 25 75
150
175
200

13. Assignment
Page 499, #s 1-33 odd

14. Reference Angles
Let θ be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle θ formed by the terminal side of θ and the x-axis.

15. Reference Angles
When considering a reference angle, the angle MUST be:
Acute
Positive
With respect to the x-axis !!

16. Examples
Find the reference angle for each of the following angles: a) 210o b) c) -240o d) 3.6

17. Examples
Find the reference angle for each of the following angles: a) 665o b) c)

18. Using Reference Angles
Use the following procedure:
Sketch the angle
Find the associated reference angle (remember it is with respect to the x-axis)
Find the trig function of the reference angle
Use the quadrant in which the original angle lies to determine the appropriate sign.

19. Examples
Use reference angles to find the exact value of the following trigonometric functions: a) sin 300o b) c)

20. Examples
Use reference angles to find the exact value of the following trigonometric functions: a) b)

21. Examples
Use reference angles to find the exact value of the following trigonometric functions: a) b)

22. SAT PREP QUESTION OF THE DAY
If x2 + y2 = 73 and xy = 24, what is the value of (x + y)2 ? 73 97
100
121
144

23. SAT PREP QUESTION OF THE DAY
If x2 + y2 = 73 and xy = 24, what is the value of (x + y)2 ? 73 97
100
121
144

24. Assignment
Page 499, #s odd

25. Insert video clip “Hand Trick” http://www. youtube. com/watch

26. Review for Test 4A ( )
Page 498; Page 500, #s 1-30; Pages Page 563, #s 1-67; Page 567, #s 1-13, 20-22, CD; “Writing in Math” at the end of each section

27. SAT PREP QUESTION OF THE DAY
If 30 percent of 40 percent of a positive number is equal to 20 percent of w percent of the same number, what is the value of w? 80 60 50 15 10

28. SAT PREP QUESTION OF THE DAY
If 30 percent of 40 percent of a positive number is equal to 20 percent of w percent of the same number, what is the value of w? 80 60 50 15 10

29. Insert video clip “Trig Modeling”
Introduction – not necessary to view.