Download MATH 2412 Sections 4.7 and 4.8 Section 4.7 Inverse Trigonometric

MATH 2412 Sections 4.7 and 4.8
Section 4.7 Inverse Trigonometric Functions
Defining the Inverse Sine Function
Restricting the domain
−1
Definition:
=
y sin
=
x or y arcsin x...
Special Angles and the Inverse Sine Function:
Example. Find the exact value of the following:
 1
a) arcsin  − 
 2
b) sin −1 (1)
c) sin −1
3
2
Defining the Inverse Cosine Function
Restricting the domain
−1
Definition:
=
y cos
=
x or y arccos x...
Special Angles and the Inverse Sine Function:
Example. Find the exact value of the following:
a) arccos ( 0 )
b) cos −1 ( −1)
Defining the Inverse Tangent Function
Restricting the domain
−1
Definition:
=
y tan
=
x or y arctan x...
c) cos −1
1
2
Special Angles and the Inverse Tangent Function:
Example. Find the exact value of the following:
(
a) arctan − 3
)
b) tan −1 (1)
c) tan −1 −
The Graphs of Inverse Sine, Inverse Cosine, and Inverse Tangent
3
3
Solving Equations Containing Inverse Trigonometric Expressions
Example:
Solve for x :
 3 π
arccos( − x ) + arcsin  = .
 2  2
Example:
Solve for x :
 1  5π
.
2 tan −1 ( x ) + cos −1   =
2 3
Applications Using Inverse Trigonometric Functions
Example: A 680 foot rope anchors a hot air balloon. Use an inverse trigonometric function to
express the angle θ as a function of the height h. Find θ if h = 500 feet.
Example: Rainbows are created when sunlight of different wavelengths (colors) is refracted and
reflected in raindrops. The angle of elevation θ of a rainbow is always the same. It can be
θ 4β − 2α , where sin a =k sin β and a =
59.4° and k =
1.33 is the index of
shown that =
refraction of water. Use the given information to find the angle of elevation θ of a rainbow.
Example. As the moon revolves around the earth, the side that faces the earth is usually just
partially illuminated by the sun. The phases of the moon describe how much of the surface
appears to be in sunlight. An astronomical measure of phase is given by the fraction F of the
lunar disc that is lit. When the angle of the sun, earth, and moon is θ ( 0 ≤ θ ≤ 360° ) , then
=
F
1
(1 − cos θ )
2
Determine the angles θ that correspond to the following phases:
(a) F = 0 (new moon)
(b) F = 0.25 (a crescent moon)
(c) F = 0.5 (first or last quarter)
(d) F = 1 (full moon)
Inverse Tangent Application
Students: please work in groups of 2 or 3 on exercises 51 and 53 on page 385
Section 4.8 Various Applications Problems
Angle of Elevation/Depression
Example: From the sun deck of the Whalewatcher's Resort at Kehei, Maui, an observer watches
a whale moving directly toward the resort. If the observer is 200 feet above the water and if the
angle of depression from the observer to the whale changes from 15° to 32° during the period of
observation, approximate the distance that the whale travels.
Navigation
Example: A prop plane leaves the airport traveling at 215 mph at a heading of 65. 4° at the same
time a jet plane leaves the airport traveling at 480 mph at a heading of 335. 4°. Find the
distance between them after two hours and thhe bearing of the prop plane from the jet plane.
Simple Harmonic Motion
The position of a weight attached to a spring is modeled by s (t ) = −3cos(20π t ) ,
in inches and where t is the time measured in seconds.
(a) Find the amplitude and the period.
(b) Find the frequency.
(c) Graph 2 periods of s (t ) .
s (t )
t
Miscellaneous
Problem: For a certain electrical circuit, the voltage E is modeled by E = 5.1cos(80π t ) ,
where t is the time measured in seconds.
(a) Find the amplitude and the period.
(b) Find the frequency (the number of cycles completed in one second).
(c) Graph 2 periods of E.
E
t