6.3 trig functions of angles

6.3 TRIG FUNCTIONS

OF ANGLES

So P = (x, y) is a point on the terminal side of theta. Using the Pythagorean theorem, r =

O

P

Q

opposite

adjacent

hypo

tenu

se

O

P (x, y)

Q

r

x

y

2 2x y

So:

sin cos tan =

csc sec = cot =

0 x 0 y 0

y x yr r x

r r xy x y

y

ExLet (-8, -6) be a point on the terminal side of theta. Find all six trig functions of theta.

Quadrantal Angels – angels that are coterminal with the coordinate axis. When the terminal side of an angle theta, that is in

standard position , lies on the coordinate axes. (The angles for which either the x- or y-coordinate of a point on the terminal side of the angle is 0.)

Since division by 0 is undefined, certain trig functions are not defined for certain angles. EX is tan 90º degrees = y/x because x = 0.

Evaluating Trig Functions at Any Angle ‘r’ is always positive because it is the

distance from the origin to the point P(x, y). The position of the terminal side will determine whether the trig function is positive or negative.

Q I: all positive Q2: (-x, y): sin, csc are + Q3 (-x, -y): tan, cot are + Q4 (-x, -y): cos, sec are +

EX sin (-180º)

3tan sec 42

cos 270 csc2

cot( 90 )

We see that the trig functions for angles that aren’t acute have the same value except possibly for the sign, as the corresponding trig functions of an acute angle. Let theta be an angle in standard position. The reference angle (theta bar) associated with theta is the acute angle formed by the terminal side of theta and the x-axis.

ExFind the reference angle for:

30023

176

137

ExFind the following: Cos 135º =

tan 390º

cos 60º

tan 240º

Evaluating Trig Functions for any angle:1. Find the reference angle (theta bar)

associated with the angle theta.2. Determine the sign of the trig function of

theta by noting the quadrant in which theta lies.

3. The value of the trig function of theta is the same, except possibly for the sign, as the value of the trig function theta bar.

Evaluate the Trig Functions using the Reference Angle.

15sin120 csc4

5cos120 tan3

7 5tan sin6 6

sec( 135 )

Fundamental Identities

2 2

2 2

2 2

R eciprocal Identities:1 1 1csc sec = cot =

sin cos tansin costan cotcos sin

Pythagorean Identities:

sin cos 1

tan 1 sec

cot 1 csc

Ex express sin in terms of cos

Exexpress tan in terms of sin

Ex

1tan and is in the fourth quadrant.2

Find cos

If

Ex

2If tan and is in the third quadrant3

Find cos

Ex If sec =6 and is in the fourth quadrant,find the other 5 trig functions of .

Area of Triangles: With sides of length a & b and with included angle theta.

Find the area of triangle ABC:

1 sin2

A ab

120º

AB

C

18cm

3cm