Tangent Function
f (x) := tan x
Domain:
So the domain of f (x) := tan x is all real numbers except x =
What is happening at the ”ends” of this domain?
Consider the value x =
π
:
2
π
+ kπ, k an integer.
2
Similar logic can be used to show that the behavior around each of the other ”ends” of the domain
3π
is the same. For example, for x =
:
2
lim − tan x = ∞ and
x→ 3π
2
lim + tan x = −∞
x→ 3π
2
Intercepts:
Putting all of this together gives the tangent graph:
y
4
2
-2 Π - 32Π
-Π
- Π2
Π
2
Π
3Π
2
-2
-4
Figure 1: y = tan(x)
x
2Π
Inverse Trig Functions
All of the trig functions are periodic and thus are not one-to-one. To define the inverse trig functions, we must restrict the domain to an interval that is one-to-one.
1) Inverse Sine Function
Notations: f (x) = sin−1 (x) or f (x) = arcsin(x)
Consider the graph of y = sin x:
2
y
1
-2 Π - 32Π
-Π
- Π2
Π
2
Π
3Π
2
x
2Π
-1
-2
Figure 2: y = sin(x)
Clearly, the graph fails the HLT. However, we can still define an inverse function if we have a piece
where:
• the graph of y = sin x is 1-1
• the graph of y = sin x covers its entire range
The standard interval used is
h π πi
− ,
2 2
Definition: The inverse sine function (or arcsine function) is defined by:
y = sin−1 (x) or y = arcsin(x)
if an only if
π
π
sin(y) = x AND −1 ≤ x ≤ 1, − ≤ y ≤
2
2
Comment: The graph of y = sin−1 (x) is the reflection of the graph of y = sin x over the line y = x
on the restricted domain.
2
y
1
-Π
- Π2
Π
2
Π
x
-1
-2
Figure 3: y = sin(x)
Π
y
Π
2
-2
-1
1
- Π2
-Π
Figure 4: y = arcsin(x)
x
Example 1: Evaluate sin−1
√ !
3
.
2
1
Example 2: Evaluate arcsin − √ .
2
Example 3: Is sin−1 (sin π) = π?
2) Inverse Cosine Function
Notations: f (x) = cos−1 (x) or f (x) = arccos(x)
2
y
1
-2 Π - 32Π
-Π
- Π2
Π
2
Π
3Π
2
x
2Π
-1
-2
Figure 5: y = cos(x)
Again, we choose an interval where y = cos x is one-to-one and covers the entire range:
[0, π]
Definition: The inverse cosine function (or arccosine function) is defined by:
y = cos−1 (x) or y = arccos(x)
if an only if
cos(y) = x AND −1 ≤ x ≤ 1, 0 ≤ y ≤ π
2
y
1
- Π2
Π
2
Π
3Π
2
x
-1
-2
Figure 6: y = cos(x)
3Π
2
y
Π
Π
2
-2
-1
1
- Π2
Figure 7: y = arccos(x)
2
x
√ !
3
Example 4: Evaluate arccos −
.
2
Example 5: Evaluate cos−1 (3).
3) Inverse Tangent Function
Notations: f (x) = tan−1 (x) or f (x) = arctan(x)
y
4
2
-2 Π - 32Π
-Π
- Π2
Π
2
Π
3Π
2
x
2Π
-2
-4
Figure 8: y = tan(x)
Definition: The inverse tangent function (or arctangent function) is defined by
y = tan−1 (x) or y = arctan(x)
if an only if
π
π
tan(y) = x AND − < y <
2
2
Π
y
Π
2
-4
-2
2
4
- Π2
-Π
Figure 9: y = arctan(x)
x
Example 6:
Evaluate tan−1 (−1)