Download Inverse Trig.

Inverse Trig.
We know how to find side measures...
x
15
5
But, how do we find angle measures?
We use:
So, using our calculators...
our calculator displays
What does this mean?
and
are the notations for inverse sine
The notation sin­1(x) is consistent with the inverse function notation f­1(x)
The arcsin x notation comes from the association of a central angle with its intercepted arc length on a unit circle.
So, arcsin x means the angle (or arc) whose sin is x.
Both sin ­1(x) and arcsin(x) commonly used.
NOTE OF CAUTION!!!!
SIN­1 X DENOTES THE INVERSE SINE FUNCTION
NOT THE RECIPROCAL WHICH IS WHICH IS CSC X!!
So, by definition...
y = arcsin x if and only if sin y = x
Sin x must be between ­90 < x < 90
Tan x must be between ­90 < x < 90
Cos x must be between 0 < x < 180
If the domains are not restricted, we do not have inverses that are functions.
Let's take a look at the graphs to see why...
y = sin x
x = sin y or y = sin­1 x
y
x
x
y
Fails Horizontal line test.
Inverse is NOT a function!
Sin x is NOT a One­to­One Function.
Fails vertical line test!
We can make sin­1 (x) a function by restricting the domain of sin(x)
y
Over this interval:
1
­π
­1
π
Sin x has an inverse function on the interval from [­π/2, π/2]
x
1) sin x is increasing
2) sin x takes on it's
full range of values
3) sin x is one­to­one
y = arcsin(x) - vs. - f(x) = sin(x)
y = sin
-1
x
- OR - y = arcsin x
Find the values of :
does
not
exist
y = arccos(x) - vs. - f(x) = cos(x)
y = cos
-1
x - OR - y = arccos x
Find the values of :
does
not
exist
y = tan
-1
x - OR - y = arctan x
Find the values of :
Six Trig Functions with Domain and Range
RAN GE
Inverse
Function
Domain
Interval
Quadrants of
Unit Circle
y = arcsin x [-1, 1]
[-π , π]
2 2
I and IV
y = arccos x [-1, 1]
[0, π ]
I and II
y = arctan x (-∞ , ∞ )
(-π , π)
2
2
I and IV
Example:
Evaluate each function without using a calculator.
b) arcsin (0)
a) tan­1 (√3)
Example:
Use an inverse trigonometric function to write θ as a function of x.
5
θ
x+2
Example:
Use the properties of inverse trigonometric
functions to evaluate the expressions.
a) cos[arccos(­0.1)]
b) arcsin(sin 3 π)
Example:
Find the exact value of the expression.
a) sec(arcsin 4/5)
b) tan[arcsin(­3/4)]
Example:
Write an algebraic expression that is equivalent to the expression.
a) sin(arctan x)
b) Various Solving Methods
Solving trig with linear methods :
Solve: 3 tan θ - √3 = 0 over the interval [0, 360 o).
Solving trig equations by factoring :
Solve: cos θcotθ = -cos θ over the interval [0, 360 o).