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Graphing the sine and cosine functions.
Graphing the sine and cosine functions. 1 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).
Which of the following functions are periodic? With what period?
The Amplitude of a periodic function f is given by M−m2 where M is the
maximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?
The Amplitude of a periodic function f is given by M−m2 where M is the
maximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?
The Amplitude of a periodic function f is given by M−m2 where M is the
maximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic
Periodic Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?
The Amplitude of a periodic function f is given by M−m2 where M is the
maximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic
Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?
The Amplitude of a periodic function f is given by M−m2 where M is the
maximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic
Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?
The Amplitude of a periodic function f is given by M−m2 where M is the
maximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not Periodic
Period = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?
The Amplitude of a periodic function f is given by M−m2 where M is the
maximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2
Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?
The Amplitude of a periodic function f is given by M−m2 where M is the
maximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2 Period=3
Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?
The Amplitude of a periodic function f is given by M−m2 where M is the
maximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?The Amplitude of a periodic function f is given by M−m
2 where M is themaximum value of f and m is the minimum value.
What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?The Amplitude of a periodic function f is given by M−m
2 where M is themaximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?The Amplitude of a periodic function f is given by M−m
2 where M is themaximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1
Amp=1−02 = 1
2 Amp=3−01 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?The Amplitude of a periodic function f is given by M−m
2 where M is themaximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12
Amp=3−01 = 1
Graphing the sine and cosine functions. 2 / 6
Plugging in any real number to the trig functions
A function f is called Periodic with Period p if f (x) = f (x + p).Which of the following functions are periodic? With what period?The Amplitude of a periodic function f is given by M−m
2 where M is themaximum value of f and m is the minimum value.What are the amplitudes of the periodic functions below
Periodic Periodic Periodic Not PeriodicPeriod = 2 Period=3 Period=1
Amp=1−(−1)2 = 1 Amp=1−0
2 = 12 Amp=3−0
1 = 1
Graphing the sine and cosine functions. 2 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic?
YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.
Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) =
0
sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? Yes
With what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.
Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) =
0
sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period?
2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.
Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) =
0
sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.
What is their amplitude? 1−(−1)2 = 1.
Let’s try to graph them.
Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) =
0
sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude?
1−(−1)2 = 1.
Let’s try to graph them.
Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) =
0
sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.
Let’s try to graph them.
Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) =
0
sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.
Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) =
0
sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) =
0
sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =
√2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) =
1/2
sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =
√3/2 ∼ .87
sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) =
1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) = 1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) = 1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) = 1
Use the fact that sin(π + θ) = − sin(θ)
Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) = 1
Use the fact that sin(π + θ) = − sin(θ)
Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) = 1
Use the fact that sin(π + θ) = − sin(θ)Fill it in
Use periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) = 1
Use the fact that sin(π + θ) = − sin(θ)Fill it in
Use periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) = 1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?Remember the identities sin(θ + 2π) = sin(θ)?Is sin(x) periodic? YesWith what period? 2π.What is their amplitude? 1−(−1)
2 = 1.Let’s try to graph them.Recall that we’ve computed some of the trig functions by hand:
sin(0) = sin(π) = sin(2π) = 0 sin(π/4) = sin(3π/4) =√
2/2 ∼ .71
sin(π/6) = sin(5π/6) = 1/2 sin(π/3) = sin(2π/6) =√
3/2 ∼ .87sin(π/2) = 1
Use the fact that sin(π + θ) = − sin(θ)Fill it inUse periodicity: draw the same thing every 2π.
Graphing the sine and cosine functions. 3 / 6
What about sin and cos?
There’s the graph of sin(x).
Remember that sin(π/2 + x) = cos(x)?How should we shift sin(x) to get the graph of cos(x)?Shift to the left by π/2
Comprehension exerciseWhere are the turning points of sin(x)?
π/2, 3π/2, −π/2, −3π/2, . . . .
Of cos(x)?
0, π, −π, 2π, −2π, . . . .
Graphing the sine and cosine functions. 4 / 6
What about sin and cos?
There’s the graph of sin(x).Remember that sin(π/2 + x) = cos(x)?
How should we shift sin(x) to get the graph of cos(x)?Shift to the left by π/2
Comprehension exerciseWhere are the turning points of sin(x)?
π/2, 3π/2, −π/2, −3π/2, . . . .
Of cos(x)?
0, π, −π, 2π, −2π, . . . .
Graphing the sine and cosine functions. 4 / 6
What about sin and cos?
There’s the graph of sin(x).Remember that sin(π/2 + x) = cos(x)?How should we shift sin(x) to get the graph of cos(x)?
Shift to the left by π/2
Comprehension exerciseWhere are the turning points of sin(x)?
π/2, 3π/2, −π/2, −3π/2, . . . .
Of cos(x)?
0, π, −π, 2π, −2π, . . . .
Graphing the sine and cosine functions. 4 / 6
What about sin and cos?
There’s the graph of sin(x).Remember that sin(π/2 + x) = cos(x)?How should we shift sin(x) to get the graph of cos(x)?Shift to the left by π/2
Comprehension exerciseWhere are the turning points of sin(x)?
π/2, 3π/2, −π/2, −3π/2, . . . .
Of cos(x)?
0, π, −π, 2π, −2π, . . . .
Graphing the sine and cosine functions. 4 / 6
What about sin and cos?
There’s the graph of sin(x).Remember that sin(π/2 + x) = cos(x)?How should we shift sin(x) to get the graph of cos(x)?Shift to the left by π/2
Comprehension exerciseWhere are the turning points of sin(x)?
π/2, 3π/2, −π/2, −3π/2, . . . .
Of cos(x)?
0, π, −π, 2π, −2π, . . . .
Graphing the sine and cosine functions. 4 / 6
What about sin and cos?
There’s the graph of sin(x).Remember that sin(π/2 + x) = cos(x)?How should we shift sin(x) to get the graph of cos(x)?Shift to the left by π/2
Comprehension exerciseWhere are the turning points of sin(x)?
π/2, 3π/2, −π/2, −3π/2, . . . .
Of cos(x)?
0, π, −π, 2π, −2π, . . . .
Graphing the sine and cosine functions. 4 / 6
What about sin and cos?
There’s the graph of sin(x).Remember that sin(π/2 + x) = cos(x)?How should we shift sin(x) to get the graph of cos(x)?Shift to the left by π/2
Comprehension exerciseWhere are the turning points of sin(x)?π/2, 3π/2, −π/2, −3π/2, . . . .Of cos(x)?
0, π, −π, 2π, −2π, . . . .
Graphing the sine and cosine functions. 4 / 6
What about sin and cos?
There’s the graph of sin(x).Remember that sin(π/2 + x) = cos(x)?How should we shift sin(x) to get the graph of cos(x)?Shift to the left by π/2
Comprehension exerciseWhere are the turning points of sin(x)?π/2, 3π/2, −π/2, −3π/2, . . . .Of cos(x)?0, π, −π, 2π, −2π, . . . .
Graphing the sine and cosine functions. 4 / 6
Using the graph
Here is the graph of the first quarter-period of cos(x) with grid lines drawnevery tenth.
What is cos(.5)? Give you answer correct to the nearest tenth.What x in [0, π/2] solves cos(x) = .8?What are all the solutions to cos(x) = .8? (Use that cos(x) is symmetricand periodic.)
Graphing the sine and cosine functions. 5 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?
Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.
So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) =
[−1, 1]rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]
rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) =
[0, π]dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]
dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]dom(sin−1) =
[−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]
rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]rng(sin−1) =
[−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.
Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x).
Graphing the sine and cosine functions. 6 / 6
Inverse trig functionsHere are the graphs of cos(x) over [0, π] and sin(x) over [−π/2, π/2].
Do they pass the Horizontal line test?Yes.So they each have inverse functions.cos−1(x) and sin−1(x).
dom(cos−1) = [−1, 1]rng(cos−1) = [0, π]dom(sin−1) = [−1, 1]rng(sin−1) = [−π/2, π/2]
Using a computer to do algebra:You can use this solve equations involving trig functions.Example: Solve 3 cos(x) + 2 = 0.Exit quiz: find all the solutions to sin(x) = 1/3. Remember thatsin(x) = sin(π − x). Graphing the sine and cosine functions. 6 / 6