Upload
kasie
View
72
Download
0
Embed Size (px)
DESCRIPTION
The Trigonometric Functions. What 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:. The 6 Trigonometric Functions of angle are:. The Trigonometric Functions. - PowerPoint PPT Presentation
Citation preview
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:
sin yr
cos
tan ,
sin 0
0
0
csc ,
sec ,
cot ,
r yyr xxx yy
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 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, 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 Angles (terminal side lies along an axis)
Trig values of quadrantal angles:
0° 90° 180° 270° 360°0 1 0 –1 01 0 –1 0 10 undefined 0 undefined 0
undefined 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
xx
1cossec
xx
1tancot
xx
sintancosxxx
coscotsinxxx
Quotient Identities
Trigonometric Identities
2 2 1sin cosx x 2 21 cot cscx x
2 21tan secx x
Pythagorean IdentitiesThe fundamental Pythagorean
identity:
Divide the first by sin2x : Divide the first by cos2x :