1.4 Definition of the Trigonometric FunctionsOBJ: Find the values of the six trigonometric functions of an angle in standard position

DEF: Standard Position

Angle with its vertex

at the origin,

initial side on the x-axis,

and terminal side

containing a point P

with coordinates (x,y)

P (x,y) y

x

5

5

-5-5

Draw in standard position.

Choose any point P

with coordinates (x, y)

on the terminal side of

(P cannot be the

vertex of ) P (x,y)

y

x

5

5

-5-5

y

x

5

5

-5-5

Draw the to the x-axis at Q, forming ∆OPQ. PO = r > 0.

x2 + y2 = r2

P

Q O

DEF: Trigonometric Functions

X2 + Y2 = R2

P

(Opp) Y R (Hyp)

(Adj) X

SOHCAHTOASYRCXRTYX

Sin = Y Opp Csc = R Hyp R Hyp Y

Opp

Cos = X Adj Sec = R Hyp

R Hyp X Adj

Tan = Y Opp Cot = X Adj X Adj Y Opp

y

x

5

5

-5-5

All Students Take Calculus

S A

T C

Sin All

Tan Cos

y

x

5

5

-5-5

y

x

5

5

-5-5

EX: 1 The terminal side of an angle in standard position goes through the point ( 8 , 15 ). Find the values of the six trigonometric functions of P 35 (8, 15)

Sin = Y 15 Csc = R 17 R 17 Y 15

Cos = X 8 Sec = R 17 R 17 X 8 15

Tan = Y 15 Cot = X 8 X 8 Y 15

8

y

x

5

5

-5-5

y

x

5

5

-5-5

EX 2 The terminal side of an angle in standard position goes through the point (-3, - 4 ). Find the values of the six trigonometric functions of .P 36

Sin = Y = -4 Csc = R = 5 R 5 Y -4 Cos = X = -3 Sec = R = 5

R 5 X -3Tan = Y = -4=4 Cot = X = -3 3 -3 X -3 3 Y -4 4 -4

(-3,-4)

y

x

5

5

-5-5

EX 4 Find the values of the six trigonometric functions for an angle of 90 .P 39Sin = Y = 1 Csc = R = 1

R 1 Y 1 (0, 1)

Cos = X = 0 Sec = R = 1 = Ø

R 1 X 0

Tan = Y = 1 = Ø Cot = X = 0 X 0 Y 1

y

x

5

5

-5-5

DEF: Quadrantal angles

TABLE OF VALUES OF THE SIX TRIGONOMETRIC FUNCTIONS OF QUAD. S

0

90

180

270

360

450

DEF: Coterminal Angles

Angles with the same initial side and

the same terminal side

EX: Find the angles of smallest possible measure coterminal

with the following angles: 900

y

x

5

5

-5-5

EX: Find the angles of smallest possible measure coterminal

with the following angles: 900 900 – 720 180

y

x

5

5

-5-5

DEF: Quadrantal angles

TABLE OF VALUES OF THE SIX TRIGONOMETRIC FUNCTIONS OF QUAD. S

Sin Cos Tan Cot Sec Csc

0 0 1 0 Ø 1 Ø

90 1 0 Ø 0 Ø 1

180 0 -1 0 Ø -1 Ø

270 -1 0 Ø 0 Ø -1

360 0 1 0 Ø 1 Ø

450 1 0 Ø 0 Ø 1