Math 140
Class 13
Summary of Special Values
| t | sint | cost | tant |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π6 | 12 | √32 | √33 |
| π4 | √22 | √22 | 1 |
| π3 | √32 | 12 | √3 |
| π2 | 1 | 0 | undefined |
| quadrant | sint | cost | tant |
|---|---|---|---|
| I | + | + | + |
| II | + | − | − |
| III | − | − | + |
| IV | − | + | − |
Tangent and Cotangent
y=axx=1aya=tant1a=cott
tant=yx=sintcost
cott=xy=costsint
Secant and Cosecant
Summary
tant=sintcost
cott=costsint
sect=1cost
csct=1sint
cos2t+sin2t=1
Complex Unit Circle
cis(s+t)=cis(s)cis(t)
cis(s+t)=cos(s+t)+isin(s+t)
cis(s)cis(t)=(coss+isins)(cost+isint)=cosscost+icosssint+isinscost+i2sinssint=cosscost+icosssint+isinscost−sinssint=cosscost−sinssint+i(cosssint+sinscost)
cos(s+t)=cosscost−sinssintsin(s+t)=cosssint+sinscost
Addition formulas
cos(s+t)=cosscost−sinssintsin(s+t)=cosssint+sinscost
Subtraction Formulas
Double Angle Formulas
Example
cos(5π12)
Example
If s is in quadrant IV with coss=57, and π2≤t≤π with sint=√53, find cos(s+t).
Example
If tant=247 and t is in the third quadrant, find the values of the other 5 trigonometric functions.
Math 140 Class 13