1 – tan 2 u
1
1 – tan 2 u 2 3 cos u sin u sin = csc = cos = sec = tan = cot = Definition of the Six Trigonometric Functions Right triangle definitions, where 0 < < /2 Hypotenuse Opposit e Adjacent sin = csc = cos = sec = tan = cot = opp. hyp. hyp. opp. adj. hyp. hyp. adj. opp. adj. adj. opp. Circular function definitions, where is any angle. r y x (x, y) y x r = x 2 + y 2 y r r y x r r x y x x y Reciprocal Identities sin u = cos u = tan u = csc u = sec u = cot u = 1 1 1 1 1 1 csc u sec u cot u sin u cos u tan u Quotient Identities tan u = cot u = sin u cos u Pythagorean Identities sin 2 u + cos 2 u = 1 1 + tan 2 u = sec 2 u 1 + cot 2 u = csc 2 u Cofunction Identities sin( ) = cos u cos( ) = sin u tan( ) = cot u cot( ) = tan u sec( ) = csc u csc( ) = sec u – u 2 – u 2 – u 2 – u 2 – u 2 – u 2 Even/Odd Identities sin(– u) = – sin u cot(– u) = – cot u cos(– u) = cos u sec(– u) = sec u tan (– u) = – tan u csc(– u) = – csc u Sum and Difference Formulas sin(u + v) = sin u cos v + cos u sin v sin(u – v) = sin u cos v – cos u sin v cos(u + v) = cos u cos v – sin u sin v cos(u – v) = cos u cos v + sin u sin v tan(u + v) = tan(u – v) = tan u + tan v tan u – tan v 1 – tan u tan v 1 + tan u tan v 6 4 3 2 3 4 6 6 4 3 3 4 6 2 30 o 45 o 90 o 60 o 120 o 135 o 150 o 180 o 210 o 225 o 240 o 270 o 300 o 315 o 330 o 0 o 360 o 2 0 (1,0 ) (– 1,0) (0,1 ) (0, – 1) 2 3 2 2 1 2 x y ( , ) 2 3 1 2 ( , ) 2 2 2 2 (– , – ) ( , ) 1 2 (– , ) 2 3 1 2 (– , ) 2 2 2 2 (– , ) 1 2 2 3 1 2 2 3 (– , – ) ( , – ) 2 2 2 2 (– , – ) 2 3 1 2 ( , – ) 1 2 2 3 ( , – ) 2 2 (x,y) (cos u, sin u) Double Angle Formulas sin 2u = 2 sin u cos u cos 2u = cos 2 u – sin 2 u = 2 cos 2 u – 1 = 1 – 2 sin 2 u tan 2u = 2 tan u Power-Reducing Formulas sin 2 u = cos 2 u = 1 – cos 2u 1 + cos 2u 2 2 tan 2 u = 1 – cos 2u 1 + cos 2u Sum-to-Product Formulas sin u + sin v = 2 sin ( ) cos ( ) u + v u – v 2 2 sin u – sin v = 2 cos ( ) sin ( ) u + v u – v 2 2 cos u + cos v = 2 cos ( ) cos ( ) u + v u – v 2 2 cos u – cos v = – 2 sin ( ) sin ( ) u + v u – v 2 2 Product-to-Sum Formulas sin u sin v = ½ [cos( u – v ) – cos( u + v )] cos u cos v = ½ [cos( u – v ) + cos( u + v )] sin u cos v = ½ [sin( u + v ) + sin( u – v )] cos u sin v = ½ [sin( u + v ) – sin( u – v )] Unit Circle RVCC – ASC : RME/AMS 03-31-2011 Half-Angle Formulas sin( ) = u + 1 – cos u 2 – 2 cos( ) = u + 1 + cos u 2 – 2 tan ( ) = = u 1 – cos u sin u 2 sin u 1 + cos u The signs of sin(u/2) and cos(u/2) depend on the quadrant in which u/2 lies
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Definition of the Six Trigonometric Functions. Unit Circle. q. 90 o. 120 o. 60 o. 135 o. 2 p. 11 p. 7 p. 5 p. 3 p. 5 p. 7 p. 5 p. 3 p. 4 p. 45 o. 3. 3. 2. 4. 6. 6. 4. 3. 4. 6. 150 o. 30 o. p. 180 o. 0. 0 o. y. x. r. y. r. x. 2 p. 360 o. y. r. x. x. y. - PowerPoint PPT Presentation
Transcript of 1 – tan 2 u
1 – tan2 u
2 3
cos u sin u
sin =csc =
cos =sec
=
tan = cot =
Definition of the Six Trigonometric FunctionsRight triangle definitions, where 0 < < /2
Hypotenuse
Opp
osite
Adjacent
sin =csc =
cos =sec
=
tan = cot =
opp. hyp.hyp. opp.
adj. hyp.hyp. adj.
opp. adj.adj. opp.
Circular function definitions, where is any angle.
r
y
x
(x,y)y
x
r = x2 + y2
yr
ry
xr
rx
yx
xy
Reciprocal Identitiessin u = cos u = tan u =
csc u = sec u = cot u =
1 1 1
1 1 1
csc u sec u cot u
sin u cos u tan u Quotient Identitiestan u = cot u =
sin u cos u
Pythagorean Identitiessin2 u + cos2 u = 11 + tan2 u = sec2 u 1 + cot2 u = csc2 u
Cofunction Identities
sin( ) = cos u cos( ) = sin u tan( ) = cot u
cot( ) = tan u sec( ) = csc u csc( ) = sec u
– u2
– u2
– u2
– u2
– u2
– u2
Even/Odd Identities
sin(– u) = – sin u cot(– u) = – cot u
cos(– u) = cos u sec(– u) = sec u
tan (– u) = – tan u csc(– u) = – csc u
Sum and Difference Formulas
sin(u + v) = sin u cos v + cos u sin v sin(u – v) = sin u cos v – cos u sin v cos(u + v) = cos u cos v – sin u sin v cos(u – v) = cos u cos v + sin u sin v
tan(u + v) =
tan(u – v) =
tan u + tan v
tan u – tan v
1 – tan u tan v
1 + tan u tan v
6
4
3
2
34
6
6
4 3
3
4
6
2
30o
45o
90o
60o120o
135o
150o
180o
210o
225o
240o
270o300o
315o
330o
0o
360o
2
0 (1,0)(–1,0)
(0,1)
(0, –1)
2 3
2 2
12
x
y
( , )2 3 1
2
( , )2 2
2 2
(– , – )
( , )12(– , )2
3 12
(– , )2 2
2 2
(– , )122
3
122
3
(– , – )
( , – )
2 2
2 2
(– , – )2 31
2 ( , – )12 2
3
( , – )2 2
(x,y) (cos u, sin u)
Double Angle Formulassin 2u = 2 sin u cos u cos 2u = cos2 u – sin2 u = 2 cos2 u – 1 = 1 – 2 sin2 u
tan 2u = 2 tan u
Power-Reducing Formulas
sin2 u = cos2 u = 1 – cos 2u 1 + cos 2u
2 2
tan2 u = 1 – cos 2u1 + cos 2u
Sum-to-Product Formulas
sin u + sin v = 2 sin ( ) cos ( )u + v u – v 2 2
sin u – sin v = 2 cos ( ) sin ( )u + v u – v 2 2
cos u + cos v = 2 cos ( ) cos ( ) u + v u – v 2 2
cos u – cos v = – 2 sin ( ) sin ( ) u + v u – v 2 2
Product-to-Sum Formulassin u sin v = ½ [cos( u – v ) – cos( u + v )] cos u cos v = ½ [cos( u – v ) + cos( u + v )] sin u cos v = ½ [sin( u + v ) + sin( u – v )] cos u sin v = ½ [sin( u + v ) – sin( u – v )]
Unit Circle
RVCC – ASC : RME/AMS 03-31-2011
Half-Angle Formulas
sin( ) =u + 1 – cos u 2 – 2
cos( ) =u + 1 + cos u 2 – 2
tan ( ) = =u 1 – cos u sin u 2 sin u 1 + cos u
The signs of sin(u/2) and cos(u/2) depend on the quadrant in which u/2 lies
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