1. WARM UP
Find the value of the angle θ in degrees:

2. What you’ll learn about
Trigonometric Functions of Any Angle
Acute angles
Obtuse angles
Positive angles
Negative angles
… and why
Trigonometry is a mathematical tool that allows us to solve real-world problems involving right triangle relationships…we can now move beyond acute angles, to consider any angle

3. Vocabulary The terms we will use today are: Standard position Vertex
Initial side
Terminal side
Positive angle
Negative angle
Coterminal angles
Reference triangle

4. Initial Side, Terminal Side
Vertex

5. Positive Angle, Negative Angle

6. Coterminal Angles
Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. These angles are called coterminal angles.
In other words, what happens when the positive angle runs into the negative angle?

7. Coterminal Angles

8. Coterminal Angles Angles of 90˚, 450˚, and 270˚ are all coterminal
Angles of π radians, 3π radians, and 9π radians are all coterminal
Angles are coterminal whenever they differ by an integer multiple of 360 degrees or by an integer multiple of 2π radians

9. Example: Finding Coterminal Angles
Find a positive and a negative angle that are coterminal with 30˚
Add 360˚
30˚ + 360˚ = 390˚
Subtract 360˚
30˚– 360˚ = –330˚

10. Example: Finding Coterminal Angles
Find a positive and a negative angle that are coterminal with 30˚: 390˚ and –330˚

11. Classwork: Finding Coterminal Angles
Find a positive and a negative angle that are coterminal with –150˚
Sketch the angles
Find a positive and a negative angle that are coterminal with 2π/3 radians

12. Investigating First Quadrant Trigonometry
Let P(x, y) be any point in the first quadrant (QI), and let r be the distance from P to the origin

13. Investigating First Quadrant Trigonometry
What is sin θ in terms of x, y and/or r?
What is cos θ in terms of x, y and/or r?
What is tan θ in terms of x, y and/or r?

14. Investigating First Quadrant Trigonometry
Let θ be the acute angle in standard position whose terminal side contains the point (3, 5). Find the six trig ratios of θ.

15. Investigating First Quadrant Trigonometry

16. Example: Trigonometric Functions of any Angle
Find the six trig functions of 315˚
Reference triangle for 315˚

17. Example: Trigonometric Functions of any Angle
Find the six trig functions of 315˚
Draw an angle of 315˚ in standard position
Pick a point P on the terminal side and connect it to the x-axis with a perpendicular segment
The reference triangle formed is a special triangle
Choose the horizontal and vertical sides of the reference triangle to be of length 1
P (x, y) has coordinates (1, –1)

18. Example: Trigonometric Functions of any Angle
Find the six trig functions of 315˚

19. Trigonometric Functions of any Angle

20. Evaluating Trig Functions of an Angle θ
Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant.
Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ.
Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator.
Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant.
Use the coordinates of point P and the definitions to determine the six trig functions.

21. HOMEWORK Page 381 # 1 to 12 EXIT TICKET
Define one of the following in your own words:
Vertex
Initial side
Terminal side
Positive angle
Negative angle
Coterminal angles