inverse trigonometric functions are used in roads, algebra homework help

Use the information on page 826 to explain how inverse trigonometric functions are used in road design. Include a few sentences describing how to determine the banking angle for a road and a description of what would have to be done to a road if the speed limit were increased and the banking angle was not changed.

13-7
Inverse Trigonometric
Functions
Main Ideas
. Solve equations by
using inverse
trigonometric
functions.
• Find values of
expressions involving
trigonometric
functions
New Vocabulary
principal values
Arcsine function
Arccosine function
Arctangent function
GET READY for the Lesson
When a car travels a curve on a horizontal
road, the friction between the tires and the
road keeps the car on the road. Above a
certain speed, however, the force of friction
will
not be great enough to hold the car in
the curve. For this reason, civil engineers
design banked curves.
The proper banking angle 8 for a car
making a turn of radius r feet at a velocity
v in feet per second is given by the equation
5. In order to determine the
appropriate value of e for a specific curve,
you need to know the radius of the curve,
the maximum allowable velocity of cars
making the curve, and how to determine
the angle given the value of its tangent.
tan 8 =
Solve Equations Using Inverses Sometimes the value of a
trigonometric function for an angle is known and it is necessary to find
the measure of the angle. The concept of inverse functions can be applied
to find the inverse of trigonometric functions.
In Lesson 88, you learned that the inverse of a function is the relation in
which all the values of x and y are reversed. The graphs of y = sin x and
its inverse x = sin y, are shown below.
10
10
My
10
–
Notice that the inverse is not a function, since it fails
the vertical line test. None of the inverses of the
trigonometric functions are functions.
We must restrict the domain of trigonometric
functions so that their inverses are functions. The
values in these restricted domains are called
principal values. Capital letters are used to
distinguish trigonometric functions with restricted
domains from the usual trigonometric functions.
806 Chapter 13 Trigonometric Functions