Trig Functions of Real Numbers

Characteristics of the six trig graphs (5.3)(2)

POD

If sin θ = 4/5, and θ is in quadrant II, findcos θ

sin (π-θ)

sin (-θ)

What can we say about the sine of any obtuse angles?

How about the sine of opposite angles?

Review from last time

Using the unit circle and the graphs on the handout or calculator, compare

cos (30°) cos (-30°)

sin (π/4) sin (-π/4)

tan (π/6) tan (-π/6)

What might that tell us about the nature of these functions?

Consider a reciprocal function

What do you think the graph of y = csc θ would look like? Let’s build it off of the sine graph.

Consider a reciprocal function

Start with the sine graph.

Plot reciprocal y-values for x-values.

Where do we not get y-values?

Consider a reciprocal functionSee how the ranges of

the reciprocal functions are related?

If we remove the sine graph, we have this. Where are the vertical asymptotes?

What are the domain and range?

Is it even, odd, neither?

Consider a reciprocal function

y = csc (x)

an odd function

asymptotes at x = ±πnwhere sin(θ) = 0

,11,:

:

R

nD

Consider another reciprocal function

How would the graph of y = sec θ compare with this?

Consider another reciprocal function

How would the graph of y = sec θ compare with this?

Where are the vertical asymptotes?

What are the domain and range?

Even, odd, or neither?

Consider another reciprocal function

y = sec θ

an even function

vertical asymptotes at x = π/2±πn,where cos(θ) = 0

,11,:2

:

R

nD

Consider the third reciprocal function

y = cot θ

Where are the vertical asymptotes? Why?

What are the domain and range?

Consider the third reciprocal function

y = cot θ

vertical asymptotes at x = ±πn, where sin(θ) = 0or where tan (θ) = 0

:

:

R

nD

Summary chart– do we need to do this?

Fill in the chart below for the characteristics of the trig functions.

Function domain range even/odd symmetric element

Summary chart—let’s do this.

Fill in the chart below for the characteristics of three primary trig functions.

Function period amplitude asymptotes

The full chart for all six trig functions is on p. 401.

The Chart

Formulas for negative angles

Since sine and tangent are odd functions,

sin(-x) = -sin(x)tan(-x) = -tan(x)csc(-x) = -csc(x)cot(-x) = -cot(x)

In other words, change the sign of the angle, change the sign of the trig value. You can see this especially clearly on the graph.

Formulas for negative angles

Since cosine is an even function

cos(x) = cos(-x)sec(x) = sec(-x)

In other words, change the sign of the angle, the trig value stays the same. You can see this on the graph.

Practice an identity

Use the negative angle formulas to verify the identity.

xxxx sec)cos()tan()sin(

Practice an identity

Use the negative angle formulas to verify the identity.

xx

xx

xx

x

x

x

xxxx

xxxx

xxxx

secsec

seccos

1

seccos

cos

cos

sin

seccostansin

seccos)tan)(sin(

sec)cos()tan()sin(

22

Finally…

… an interesting graph. On calculators, graph f(x) = sin(x)/x on the interval . What does the graph do as and ?

, 0x 0x

Finally…

Although we know there is a hole at x = 0, it appears that as x approaches 0 from either direction.

1)( xf

Finally…

An interesting result from this interesting graph is that, if x is in radians and close to 0, then

which means thatfor very small angles. Test if for x = .03, .02, .01.

1sin

x

x

xx sin