Topic 11 Periodic and Exponential Functions I
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Topic 11Topic 11
Periodic and Exponential Periodic and Exponential Functions IFunctions I
Recall
•A S T C
sinsinθθ= = = + = = = +
coscosθθ= = = += = = +
tantanθθ= = = = = = ++
X
Y
-1 1
-1
1
0
opp
hyp
+
+
adj
hyp
+
+
opp
adj
+
+
θ
sinsinθθ= = = + = = = +
coscosθθ= = = -= = = -
tantanθθ= = = - = = = - X
Y
-1 1
-1
1
0
opp
hyp
+
+
adj
hyp
-
+
opp
adj
+
-
θ
sinsinθθ= = = - = = = -
coscosθθ= = = -= = = -
tantanθθ= = = = = = ++
X
Y
-1 1
-1
1
0
opp
hyp
-
+
adj
hyp
-
+
opp
adj
-
-
θ
sinsinθθ= = = - = = = -
coscosθθ= = = += = = +
tantanθθ= = = - = = = - X
Y
-1 1
-1
1
0
opp
hyp
-
+
adj
hyp
+
+
opp
adj
-
+
θ
X
Y
-1 1
-1
1
0
ALL +
COS +TAN +
SIN +
A S T Cll entralotations
X
Y
-1 1
-1
1
0
ALL +
COS +TAN +
SIN +
A S T Cll lasseseachaints
X
Y
-1 1
-1
1
0
ALL +
COS +TAN +
SIN +
A S T Cll onstantlyalkheilas
Blah blah blah !!!!
Yappity yappity…
Sine of any angle
Cosine of any angle
Tangent of any angle
Recall
•A S T C
•tan = sin / cos
•sin (90-) = cos
•cos (90-) = sin
•sin2 + cos2 =1
•sin, cos and tan of 30o , 60o and 45o
sin 30o = ½
cos 30o =
tan 30o =
23
31
sin 60o =
cos 60o = ½
tan 60o =
23
13
30
60
sin 45o =
cos 45o =
tan 45o = 1
21
21
22
22
45
45
θsinθcos
θtan1
θcot
θcos1
θsec
θsin1
θcosec
θcosθsin
θtan
cosθθcos
sinθθsin
So we can express every trig ratio in terms of sin and cos
Model: Evaluate (a) cosec Model: Evaluate (a) cosec /4/4 (b) cot 5 (b) cot 5/3/3
31
32
21
23
21
3
3
35
21
sin
cos
cot)(
2
1
)(
b
sin
1coseca
4π4
π
Model: If sec Model: If sec = 5/4, find cot = 5/4, find cot if if is acuteis acute
34
43
cot
tan
cos
)(
54
45seca
3
45
Model: Prove(a) cosec2x = sec x cosec x cot x
(b) 1/(cosec x + 1) + 1/(cosec x -1) = 2tan x sec x
LHE
xcosec
xxecxsecRHEa
x
xx
xxcos1
2
sin1
sincos
sin1
2
cotcos)(
Model: Prove(a) cosec2x = sec x cosec x cot x
(b) 1/(cosec x + 1) + 1/(cosec x -1) = 2tan x sec x
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