hprec8 2
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8.2: Inverse Trig Functions
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Review Special Angles
•Evaluate inverse trig functions.
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ReviewThe special angles are:
6
3
60° or
4
45° or
30° or
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Consider the first two special angles in degrees...
30°
60°
Long Side
Short
Sid
e
Hypotenus
e
Review
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And in radians:
Long Side
Short
Sid
e
Hypotenus
e
Review
6
3
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Important IdeaIn a 30° ,60° ,
triangle:•the short side is one-
half the hypotenuse.
•the long side is times the short side.
3
( 6) ( 3)( 2)90°
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Important IdeaIn a 45° ,45° ,90° triangle
:•The legs of the triangle are equal.
•the hypotenuse is times the length of the leg.
2
( 4) ( 4)( 2)
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Try ThisFind the length of the missing sides: 30
°
60°1
2
1 3 or 3
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Try ThisFind the length of the missing sides
1
12
45°
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Important IdeaMany trig functions can be solved without graphing by using special angles and inverse trig functions. A special angle solution will be an exact solution whereas a graphing solution is only approximate.
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DefinitionTrig Function
Inverse Trig Function
siny x 1sin x y
cosy x 1cos x y
tany x 1tanx y
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Important Idea
arcsin y
arccos y
arctan y
In some books:1sin y
1cos y
1tan y
instead ofinstead ofinstead of
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ExampleFind the exact value without using a calculator;
1 1sin
2 1 2
cos2
1sin 1
1tan (1) 1 3cos
2
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Example
Find all values of x in the interval for which:
2cos
2x
0 360x
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Example
Find all values of x in the interval for which:
2cos
2x
0 2x
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Try ThisFind all values of x in the interval for which: 1
sin2
x
0 360x
210 & 330x
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Try ThisFind all values of x in the interval for whichtan 1x
0 360x
45 & 225x
Hint: in what quadrants is the tangent positive?
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Example
Write each equation in the form of an inverse relation:
4tan
5
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Example
Write each equation in the form of an inverse relation:
1cos
3
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Try This
Write each equation in the form of an inverse relation:sin 1x
1sin 1 or arcsin1x x
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ExampleFind the value of x in the for which:
cos .6328x
Leave your answer in degrees to the nearest tenth.
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Try ThisFind the value of x in the for which:sin .6328x Leave your answer in degrees to the nearest tenth.
x=39.3
Can you find another value for x?
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DefinitionThe calculator will provide only the Principal Values of inverse trig functions:
1sin x
1cos x
1tan ( )x 90 ,90 2, 2
0,180 0,or
or
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ExampleEvaluate the expression. Assume all angles are in quadrant 1.
1sin arcsin
2
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ExampleEvaluate the expression. Assume all angles are in quadrant 1.
3sin arccos
2
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Try ThisEvaluate the expression. Assume all angles are in quadrant 1. 1
os arcsinc2
3
2
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Racetrack curves are banked so that cars can make turns at high speeds. The proper banking angle,, is given by:2
tanv
gr
where v is the velocity of the car, g is the acceleration of gravity & r is the radius of the turn. Find when r=1000ft & v=180 mph.
Example
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Lesson Close
In order to have an inverse trig function, we must restrict the domain so that duplicate values are eliminated.