The Trigonometric FunctionsWhat about angles greater than 90°? 180°?The trigonometric functions are defined in terms of a

point on a terminal side r is found by using the Pythagorean Theorem:

22 yxr

The 6 Trigonometric Functions of angle are:

siny

r

cos

tan ,

sin 0

0

0

csc ,

sec ,

cot ,

ry

y

rx

xxy

y

0x

The Trigonometric FunctionsThe trigonometric values do not depend

on the selected point – the ratios will be the same:

First Quadrant:

sin = +

cos = +

tan = +

csc = +

sec = +

cot = +

Second Quadrant:

sin = +

cos = -

tan = -

csc = +

sec = -

cot = -

Third Quadrant:

sin = -

cos = -

tan = +

csc = -

sec = -

cot = +

y

x

Fourth Quadrant:

sin = -

cos = +

tan = -

csc = -

sec = +

cot = -

y

x

All Star Trig Class Use the phrase “All Star Trig Class” to

remember the signs of the trig functions in different quadrants:

AllStar

Trig Class

All functions are positive

Sine is positive

Tan is positive Cos is positive

The value of any trig function of an angle is equal to the value of the corresponding trigonometric function of its reference anglereference angle, except possibly for the sign. The sign depends on the quadrant that is in.

So, now we know the signs of the trig functions, but what about their values?...

Reference AnglesThe reference angle, reference angle, α, is the angle between the

terminal side and the nearest x-axis:

All Star Trig Class Use the phrase “All Star Trig Class” to

remember the signs of the trig functions in different quadrants:

AllStar

Trig Class

All functions are positive

Sine is positive

Tan is positive Cos is positive

Quadrantal AnglesQuadrantal Angles (terminal side lies along an axis)

Trig values of quadrantal angles:

0° 90° 180° 270° 360°

0 1 0 –1 0

1 0 –1 0 1

0 undefined 0 undefined 0undefined 0 undefined 0 undefined

1 undefined –1 undefined 1undefined 1 undefined –1 undefined

sin

tan

cos

cot

sec

csc

Trigonometric Identities

Reciprocal Identities

1sin

csc x

x

1cos

secx

x

1tan

cotx

x

sintan

cos

xx

x

coscot

sin

xx

x

Quotient Identities

Trigonometric Identities

2 2 1sin cosx x 2 21 cot cscx x

2 21tan secx x

Pythagorean Identities

The fundamental Pythagorean identity:

Divide the first by sin2x : Divide the first by cos2x :