Download inverse trig functions!

Warm up
 Find the values of the following trig functions:
1. tan 𝑥
x
−𝜋
2
−𝜋
4
0
𝜋
4
𝜋
2
2. cot 𝑥
𝒕𝒂𝒏 𝒙
x
0
𝜋
4
𝜋
2
3𝜋
4
𝜋
𝒄𝒐𝒕 𝒙
Graphing Other Trigonometric
Function
Properties of tangent function
Domain:
Range:
y-intercept:
x-intercepts:
Continuity:
Symmetry:
Extrema:
End Behavior:
Period of the tangent function
 The period of a tangent
function is the distance
between any two consecutive
vertical asymptotes.
 For 𝑦 = 𝑎 tan(𝑏𝑥 + 𝑐) where
𝑏 ≠ 0, 𝑝𝑒𝑟𝑖𝑜𝑑 =
𝜋
𝑏
You can find two consecutive vertical asymptotes for any
tangent function of the form 𝑦 = 𝑎 tan 𝑏𝑥 + 𝑐 + 𝑑 by solving
𝜋
𝜋
the equations 𝑏𝑥 + 𝑐 = − and 𝑏𝑥 + 𝑐 =
2
2
Period of the cotangent function
 The period of a cotangent
function is the distance
between any two consecutive
vertical asymptotes.
 For 𝑦 = 𝑎 cot(𝑏𝑥 + 𝑐) where
𝑏 ≠ 0, 𝑝𝑒𝑟𝑖𝑜𝑑 =
𝜋
𝑏
You can find two consecutive vertical asymptotes for any
tangent function of the form 𝑦 = 𝑎 tan 𝑏𝑥 + 𝑐 + 𝑑 by solving
the equations 𝑏𝑥 + 𝑐 = 0 and 𝑏𝑥 + 𝑐 = 𝜋
x
−π/2
−𝜋/4
0
𝜋/4
𝜋/2
3𝜋/4
𝜋
5𝜋/4
3𝜋/2
2𝜋
𝒔𝒊𝒏𝒙
𝒄𝒔𝒄𝒙
x
−π/2
−𝜋/4
0
𝜋/4
𝜋/2
3𝜋/4
𝜋
5𝜋/4
3𝜋/2
2𝜋
𝒄𝒐𝒔𝒙
𝒔𝒆𝒄𝒙
More Props
 Like the sinusoidal functions, the period of the cosecant
and secant functions is
2𝜋
.
𝑏
 To sketch the graph of a cosecant or secant function, locate
the asymptotes of the function and find the corresponding
relative maximum and minimum values.
Helpful Reciprocities
 What are some relationships between the graphs of
sin 𝑥 and csc 𝑥 or cos 𝑥 and sec 𝑥?
Tired of Graphing? ?
NEXT UP… INVERSE TRIG FUNCTIONS!
1.
2.
WHAT IS AN INVERSE
FUNCTION?
WHAT DOES IT DO?
Evaluate the following inverse trig functions:
1. arcsin 1
4.
2.
−1
−1
sin
2
1
−1
cos (− )
2
5. arccos
3. arctan
3
3
− 2
2
Does Sine have an inverse function?
 In order to have an inverse function, the function
must be one to one and pass the ______________
 Does sine pass the HLT?
 Restrict the domain:
Inverse Sine
 sin−1 𝑥 can be interpreted as the angle between
−𝜋
2
𝑎𝑛𝑑
𝜋
2
with the exact sine value of x.
Inverse Cosine
 Over what domain will cosine be one to one?
Inverse Tangent
 Over what domain will tangent be one to one?
Summary of inverse trig functions
Practice
Find the exact value of each expression, if it exists
Sketch the graph of inverse trig functions
1. Sketch the graph of 𝑦 = cos −1 2𝑥
y
0
𝜋
4
𝜋
6
𝜋
2
5𝜋
6
3𝜋
4
𝜋
𝟏
𝒙 = 𝒄𝒐𝒔𝒚
𝟐
Compositions
 If x is in the domain of 𝑓(𝑥) and 𝑓 −1 (𝑥), then
𝑓 𝑓 −1 (𝑥) = 𝑥 and 𝑓 −1 𝑓(𝑥) = 𝑥
Because the domains of the trigonometric functions
are restricted to obtain the inverse trig functions, the
properties do not apply for all values of x.
For example,
sin 𝑠𝑖𝑛−1 𝑥 = 𝑥 is true when?
sin−1 𝑠𝑖𝑛𝑥 = 𝑥 is true when?
Domain restrictions
Find the exact value of each expression, if it
exists.
a) sin
−1
−1
sin
4
𝜋
b) arctan(tan )
2
c) arcsin
7𝜋
sin
4
d) tan
𝜋
−1
tan
3
cos −1
3𝜋
cos
4
f) arcsin
2𝜋
sin
3
e)
Evaluate compositions of different inverse trig
functions
 Find the exact value of:
1. cos tan−1
2. cos
3. sin
−1
−3
4
𝜋
sin
3
5
arctan
12
Evaluate compositions of trig functions
 Write tan (arcsin 𝑎 ) as an algebraic expression of 𝑎 that
does not involve trigonometric functions.
 Let 𝑢 = arcsin 𝑎
 so sin 𝑢 = 𝑎
 What is the domain of inverse sine?
 Where must u lie?
 Write each as an algebraic expression of a that does
not involve trigonometric functions.
1. sin(arccos 𝑥)
2. cot sin−1 𝑥