4.6 Other Inverse Trig Functions

To get the graphs of the other inverse trig functions we make similar efforts we did to get inverse sine & cosine.

We will also do the same type of computational problems!y = Tan–1 x and y = Cot –1 x

Restricting domains: want (+) and (–) values AND no asymptotes in between

y = tan x y = cot x

So y = Tan–1 x Domain = RRange =

III

III IV

(+)(–)

(+) (–)

close

und

und,

2 2

III

III IV

(+)(–)

(+) (–)

undund

So y = Cot–1 xDomain = RRange = (0, π)

close

Let’s trace with Sharpie on our WS of graphs to discover what the inverse trig functions look like

- Trace the axes & tick marks- Write a (+) where x & y are positive- Trace asymptotes that are “pinning” in our values- Trace the graph between the asymptotes- Flip paper “over line y = x”- Label on new graph

y = Tan–1 x

2y

y = Cot–1 x

2y

y = π

y = 0

++

+

+

y = Csc–1 x and y = Sec –1 x

Same domain as reciprocal function

y = csc x y = sec x

So y = Csc–1 x

Domain =

Range =

III

III IV

(+)(+)

(–) (–)

closeund

,2 2

III

III IV

(+)(–)

(–) (+)

und

So y = Sec–1 x

Domain =

Range = [0, π] &

close

1x

& y ≠ 0

1x

2y

Time for more tracing & flipping

- Trace the axes & tick marks- Write a (+) where x & y are positive- Now think about domains to make it a function- y = sec x asymptote at

- y = csc x asymptote at x = 0

- Reflect over y = x- Label on new graph

y = Sec–1 x

2x

y = Csc–1 x

2y

(–1, π)

y = 0

,2 2

trace values between [0, π]

trace values between

1,2

1,2

(1, 0)

Calculator & Reference Triangle work* Remember what type of answer we are going for!

Ex 1) Evaluate to 4 decimal places.

a) Tan–1 1.54 0.9949

in range ? Yes!

b) y = Arccot (–5.1)

–0.1936 in range [0, π]? No So, –0.1936 + π = 2.9480

,2 2

1 1Tan

5.1

c) y = Arccsc (–3.86)

–0.2621

in range ?

Yes!

,2 2

1 1Sin

3.86

Ex 2) Evaluate to nearest tenth of a degree.

a) Arcsec (–1.433)

134.3°in range [0, 180°]? Yes!

b) y = Cot–1 4.317

13.0° in range (0, 180°)? Yes!

1 1Tan

4.317

1 1Cos

1.433

Ex 3) Determine the exact value.

a)

Arcsec sec4

Arcsec 24

b)

2ratio

θ2

2

1 5cot Cos

12

5 5 119cot

119119

x

y

angle = θ

c) 1 2sin Cot

2

2 2 6 6sin =

6 36

2

6

θ

(Draw those pictures!!)

θ

12

5

119

angle = θ

2

Ex 4) Rewrite y = sin (Cos–1 t) as an algebraic expression.

22

sin

11

1

y

tt

angle = θ

θ

1

t

21 t

Homework

#407 Pg 226 #1–15 odd, 16–24 all, 32, 35–39