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