6.7 Inverse Trig Functions
We know what the graph of y = sin x looks like.
The inverse graph entails swapping the x and y coordinates, the y-axis is labeled in terms
of π, and the x-axis is labeled using integers.
To view this graph as a function, we need to
only concentrate on a portion of this graph
that passes the vertical line test and the graph
encompasses points on the interval [-1, 1].
By examining the y-axis interval [0, π], this
portion does not pass the vertical line test.
By examining the y-axis interval [-π/2, π/2],
this portion does pass the vertical line test,
and therefore represents the graph of the
inverse function of sin x.
The equation y = sin-1x = arcsin x represent the inverse function of sin x.
The equation y = cos-1x = arcos x represent the inverse function of cos x.
The equation y = tan-1x = arctan x represent the inverse function of tan x.
y = sin-1x
Example Graph y = sin-1x – π/2
y = cos-1x
y = tan-1x
Try the following:
Graph.
y = 2 cos-1x
y = tan-1 (x – 1)
y = cos-1x + π
Answers:
The inverse functions are used when you want to know an angle measure given its
trigonometric value. For instance, sin-1( ½ ) represents what angle measure can you take
the sine of whose value is ½ . We look at this as sin x = ½ where x is on the interval [π/2, π/2]. Hence, x = π/6.
Example Evaluate cos-1(- 3 / 2 )
Think about cos x = - 3 / 2 . x = 150˚ = 5π/6
cos-1(- 3 / 2 ) = 5π/6
Try the following:
Evaluate.
tan-1(1)
sin-1(- 3 / 2 )
cos-1(-1)
sin-1(1)
Answers:
π/4 ; -π/3 ; π ; π/2
Example Evaluate sin-1(sin 2π/3)
Order of operation states to do parentheses first. Think about sin 2π/3 in the parentheses
and evaluate. sin 2π/3 = 3 / 2
Now we have sin-1(sin 2π/3) = sin-1( 3 / 2 ) = π/3
Try the following:
Evaluate.
cos-1(cos π/4)
tan-1(tan 5π/3)
sin-1(cos 7π/6)
Answers:
π/4 ; -π/3 ; -π/3