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Trig Functions of Real Numbers

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Trig Functions of Real Numbers

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

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

cos θ

sin (π-θ)

sin (-θ)

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

How about the sine of opposite angles?

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?

What do you think the graph of

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

Start with the sine graph.

Plot reciprocal y-values for x-values.

Where do we not get y-values?

See 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?

y = csc (x)

an odd function

asymptotes at x = ±πn

where sin(θ) = 0

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

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?

y = sec θ

an even function

vertical asymptotes at

x = π/2±πn,

where cos(θ) = 0

y = cot θ

Where are the vertical asymptotes? Why?

What are the domain and range?

y = cot θ

vertical asymptotes at

x = ±πn,

where sin(θ) = 0

or where tan (θ) = 0

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

Functiondomainrangeeven/oddsymmetric element

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

Functionperiodamplitudeasymptotes

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

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.

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.

Use the negative angle formulas to verify the identity.

Use the negative angle formulas to verify the identity.

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

Although we know there is a hole at

x = 0, it appears that as x approaches 0 from either direction.

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

which means that

for very small angles. Test if for

x = .03, .02, .01.