Transcript 6.5 – Inverse Trig Functions

6.5 – Inverse Trig Functions
Review/Warm Up
• 1) Can you think of an angle ϴ, in radians, such that
sin(ϴ) = 1?
• 2) Can you think of an angle ϴ, in radians, such that
cos(ϴ) = -√3/2
• 3) From precalculus, do you remember how to solve
for the inverse function if y = 2x3 + 1?
• 4) How can you verify whether two functions are
inverses of one another? Use the inverse you found for
the function above.
• 5) Say you know all three sides from a right triangle.
Can you think of a way to determine the other missing
degree angles?
• Like other functions from precalculus, we may
also define the inverse functions for trig
functions
• In the case of trig function, why would the
inverse be useful?
• Say you know sin(ϴ) = 0.35
– Do we know an angle ϴ off the top of our heads
that would give us this value?
• The inverse is there for us to now determine
unknown angles
The Inverse Functions
• There are two ways to denote the inverse of
the functions
• If y = sin(x), x = arcsin(y)
• OR
• If y = sin(x), x = sin-1(y)
•
•
•
•
Similar applies to the others
If y = cos(x), x = arccos(y)
OR
If y = cos(x), x = cos-1(y)
• If y = tan(x), x = arctan(y)
• OR
• If y = tan(x), x = tan-1(x)
Finding the inverse
• To find the inverse, or ϴ of each function, we
generally will use our graphing calculator to
help us
• Example. Evaluate arccos(0.3)
• Example. Evaluate tan-1(0.4)
• Example. Evaluate sin-1(-1)
• In the case of inverse trig functions, f-1(f(x))
and f(f-1(x)) is not necessarily = x
• Always evaluate trig functions as if using order
of operations; inside of parenthesis first
• Example. Evaluate arcsin(sin(3π/4))
– Do we get “x” back out?
• Example. Evaluate cos(arctan(0.4))
•
•
•
•
Assignment
Pg. 527
5-33odd
40, 41