Download Unit 9: Inverse Trigonometric Functions π π sin ( ) or arcsin( ) y x y x

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AMHS Precalculus - Unit 9
Unit 9: Inverse Trigonometric Functions
The Inverse Sine Function
In order for the sine function to have an inverse that is a function, we must first restrict its domain to
  
  2 , 2  so that it will be one-to-one and therefore have an inverse that is a function.


y  sin( x)
Domain: [
 
, ]
2 2
Range:
y  sin 1 ( x) or
y  arcsin( x)
Domain:
Range:
The range of the arcsine function can be visualized by:
The arcsine function ( arcsin ( x )), or inverse sine function ( sin 1 ( x) ), is defined by
y  arcsin( x) if and only if x  sin( y) where 1  x  1 and 
In other words, the arcsine of the number x is the angle y where 

2

2
 y
 y
Ex. 1: Find the exact value of the given expression.
a) arcsin (
c)
1
)
2
sin 1 (-1)
1
b) sin (
3
)
2
d) arcsin ( 
2
)
2

2

2
.
whose sine is x .
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AMHS Precalculus - Unit 9
e) arcsin(sin(
3
))
4
f)
1
cos(arcsin( ))
2
The Inverse Cosine Function
In order for the cosine function to have an inverse that is a function, we must first restrict its domain to
[0,  ] .
y  cos( x)
Domain: [0,  ]
y  cos1 ( x) or
Range:
Domain:
y  arccos( x)
Range:
The range of the arccosine function can be visualized by:
The arccosine function ( arccos ( x )), or inverse cosine function ( cos1 ( x) ), is defined by
y  arccos( x) iff x  cos( y) where 1  x  1 and 0  y   .
In other words, the arccosine of the number x is the angle y where 0  y   whose cosine is x .
Ex. 2: Find the exact value of the given expression.
a)
1
arccos (  )
2
b) cos (
c)
cos1 (-1)
d) arccos ( 
e) arccos(cos(
5
))
4
1
f)
3
)
2
2
)
2
1
cos(arcsin( ))
3
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AMHS Precalculus - Unit 9
The Inverse Tangent Function
In order for the tangent function to have an inverse that is a function, we must first restrict its domain to
(
 
, ).
2 2
y  tan( x)
Domain: (
y  tan 1 ( x) or
 
, )
2 2
Range:
y  arctan( x)
Domain:
Range:
The range of the arctangent function can be visualized by:
The arctangent function ( arctan ( x )), or inverse tangent function ( tan 1 ( x) ), is defined by
y  arctan( x) iff x  tan( y) where 1  x  1 and 

2
 y

2
In other words, the arctangent of the number x is the angle y where 
.

2
 y
Ex. 3: Find the exact value of the given expression.
a)
tan 1 (1)
c)
arctan(tan )
7
b) arctan( 3)

1
4
d) sin(arctan( ))

2
whose tangent is x .
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AMHS Precalculus - Unit 9
Ex. 4: Write the given expression as an algebraic expression in x .
sin(tan 1 x)
“Algebraic” solutions to Trigonometric Equations
Solutions for basic Trigonometric equations.
1. cos( x)  c , ( 1  c  1 )
x  cos1 (c)  2 n and x   cos1 (c)  2 n
a) Solve : cos x  0.6
b) Solve : 8cos x  1  0
2.
sin( x)  c , ( 1  c  1 )
a) Solve : sin x  0.75
b) Solve: 3sin 2 x  sin x  2  0
x  sin 1 (c)  2 n and x  (  sin 1 (c))  2 n
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AMHS Precalculus - Unit 9
3.
tan( x)  c
x  tan 1 (c)   n
a) Solve: tan x  3.6
b) Solve: sec2 x  5tan x  2
Angle of inclination
If L is a nonvertical line with angle of inclination  ( 0    180 ), then tan  = the slope of L .
Ex. 1: Find the angle of inclination of a line of slope
5
.
3
Ex. 2: Find the angle of inclination of a line of slope -2.
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AMHS Precalculus - Unit 9
Law of Sines and Law of Cosines – techniques for solving general triangles.
When we are given two angles and an included side (ASA), two angles and a non-included side (AAS), or
two sides and a non-included angle (SSA), we can find the remaining sides and angles using the Law of
Sines.
Law of Sines
sin  sin  sin 


a
b
c
Ex 1: A telephone pole makes an angle of 82 with the ground. The angle of elevation of the sun is 76 .
Find the length of the telephone pole if its shadow is 3.5m. (assume that the tilt of the pole is away
from the sun and in the same plane as the pole and the sun).
SSA – The ambiguous case. When given two sides and a non-included angle, there are three different
scenarios:
a) No triangle
b) One, unique triangle
c) Two different triangles (since you will be solving for an angle with SSA, see if another triangle is
possible by subtracting the acute angle found with arcsine from 180 )
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AMHS Precalculus - Unit 9
Ex. 2: Solve the triangle: a  2, c  1,   50
Ex. 3: Given a triangle with a = 22 inches, b =12 inches and  = 35 , find the remaining sides and
angles.
Ex. 4: Solve the triangle: a  6, b  8,   35 .
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AMHS Precalculus - Unit 9
Law of Cosines
We use the law of cosines when we are given three sides (SSS) or two sides and an included angle (SAS).
a2  b2  c2  2bc cos 
b2  a 2  c2  2ac cos 
c2  a 2  b2  2ab cos 
Ex. 5: Find all the missing angles of a triangle with sides a  8, b  19, c  14 .
Ex. 6: A ship travels 60 miles due east and then adjusts its course 15 northward. After traveling 80
miles in that direction, how far is the ship from its departure?