2.5 Properties of the Trig Functions

Warm-up (IN) – none!

Learning Objective: To find the exact values of the Trig functions using the unit circle, to know the domain and range of the trig functions, and to use the Periodic and Even-Odd Properties to find the Exact values of the Trig functions.

If t is a real # and P is any point on the unit circle...

( , )P a b sin t b

cos t a

tan t b

a

csc t 1

b

sec t 1

a

cot t a

b

t

0a

0b

0a

0b

NOTES

Ex 1 – Find all 6 trig functions if P = is the point on the unit circle that corresponds to t.

1 3,

2 2

sin t 3

2

cos t 1

2

tan t 3

csc t 2 3

3

sec t 2

cot t 3

3

If P is any point on circle and is on the terminal side of …

2 2 2x y r

sin t b

r

cos t a

r

tan t b

a

csc t r

b

sec t r

a

cot t a

b

2 2r x y

Ex 2 – Find all 6 trig functions if P = 2, 4

sin t 4

2 5

cos t 2

2 5

tan t 4

2

csc t 5

2

sec t 5

cot t 1

2

222 4r

20r

2 5r 2 5

5

5

5

2

Periodic Properties

periodic – means repeating

sin, cos, csc and sec are the same every360º or 2π radians

tan and cot are the same every180º or π radians

Ex 3 – Find the exact value:

So… sin 2 sink

cos 2 cosk

tan tank

k meaning any multiple of 2π

a. sin 420

5b. tan

4

4tan

4 4

tan4

1

11c. cos

4

3 8cos

4 4

3

cos4

2

2

)60360sin( 60sin 2

3

D:

R:

Domain and Range of 6 trig functions

sin cos tan csc sec cot

( , )P a bgiven

b a b

a1

b1

a

a

b

Except odd multiples of

2

Except odd multiples of

Except odd multiples of

2

Except odd multiples of

1 1y 1 1y

Except odd multiples of

2

1

and

1

y

y

1

and

1

y

y

Except odd multiples of

Even-Odd Functions

sin -sin

cos cos

tan - tan

csc -csc

sec sec

cot cot

Odd

Even

Odd

Ex 4 – Find the exact value:

a. sin 45 sin 45 2

2

b. cos cos 13

c. cot2

3

cot2

0

37d. tan

4

37

tan4

1

HW – pg. 172 #13-17,21-24,29-33,37-41, 47-51,55-57,61-64, 67-72

Out – What does periodic mean?

Summary - Today was …

Don’t forget

about POW!!