WARM UP

Find the value of the angle θ in degrees:

1. 65

2. 6

43.

3

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

Vocabulary

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

Initial Side, Terminal Side

Vertex

Positive Angle, Negative Angle

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?

Coterminal Angles

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

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˚

Example: Finding Coterminal Angles• Find a positive and a negative angle that are

coterminal with 30˚: 390˚ and –330˚

Classwork: Finding Coterminal Anglesa) Find a positive and a negative angle

that are coterminal with –150˚Sketch the angles

b) Find a positive and a negative angle that are coterminal with 2π/3 radiansSketch the angles

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

• 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?

Investigating First Quadrant Trigonometry

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.

Slide 4- 15

The distance from (3,5) to the origin is 34.

5 34sin 0.857 csc 1.166

534

3 34cos 0.514 sec 1.944

3345 3

tan cot3 5

Investigating First Quadrant Trigonometry

Example: Trigonometric Functions of any Angle

Find the six trig functions of 315˚• Reference triangle for 315˚

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 45-45-90 special

triangle• Choose the horizontal and vertical sides of the

reference triangle to be of length 1• P (x, y) has coordinates (1, –1)

Example: Trigonometric Functions of any Angle

Find the six trig functions of 315˚

Example: Trigonometric Functions of any Angle

Trigonometric Functions of any Angle

2 2

Let be any angle in standard position and let ( , ) be any point on the

terminal side of the angle (except the origin). Let denote the distance from

( , ) to the origin, i.e., let . Then

si

P x y

r

P x y r x y

n csc ( 0)

cos sec ( 0)

tan ( 0) cot ( 0)

y ry

r y

x rx

r xy x

x yx y

Evaluating Trig Functions of an Angle θ

1. Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant.

2. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ.

3. 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.

4. 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.

5. Use the coordinates of point P and the definitions to determine the six trig functions.

HOMEWORKPage 381 # 1 to 12

EXIT TICKETDefine one of the following in your own words:•Vertex•Initial side•Terminal side•Positive angle•Negative angle•Coterminal angles