right triangle trigonometry solving right triangles prepared by title v staff: daniel judge,...
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Right Triangle TrigonometrySolving Right Triangles
Prepared by Title V Staff:Daniel Judge, Instructor
Ken Saita, Program Specialist
East Los Angeles College
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Consider a Right Triangle.
Note – a is the leg opposite c is the leg opposite our right angle b is the leg adjacent to
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So that we have the following right triangle.
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The six trigonometric ratios are defined as follows:
opp hypsin ( ) = ; csc ( ) =
hyp opp
adj hypcos ( ) = ; sec ( ) =
hyp adj
opp adj tan ( ) = ; cot ( ) =
adj opp
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What are the six trigonometric ratios for ?
Note – We need the length of one of the legs of our right triangle.
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Use the Pythagorean Theorem . . .
2 2 23 ? 5
29 ? 25
2? 16 ; ? 4 ; ? 4
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For this triangle we get:
hyp
adj
opp
4 5sin( ) csc( )
5 4
3 5cos( ) ; sec( )
5 3
4 3tan( ) ; cot( )
3 4
;
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Notice we have another angle at .
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We can obtain the six trigonometric ratios for ,
HYP
opp
adj
3 5sin( ) ; csc( )
5 3
4 5cos( ) ; sec( )
5 4
3 4tan( ) ; cot( )
4 3
hyp
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Together the model looks as follows.
HYP
adj / opp
opp / adj hyp
With + = 90°
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Recall the 45º - 45º - 90º Special Triangle.
What are the six trigonometric ratios for 45º?
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hypopp 45º
adj 45º
adj x 1 2sin(45 ) ; ; ;
hyp 2x 2 2
adj x 1 2cos(45 ) ; ; ;
hyp 2x 2 2
opp xtan(45 ) ; ; 1
adj x
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hypopp 45º
adj 45º
2csc(45 )
2sec(45 )
cot(45 ) 1
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Recall the 30º - 60º - 90º special triangle.
What are the six trigonometric ratios for 30º ?What are the six trigonometric ratios for 60º ?
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For 60º hyp
opp 60º / adj 30º
opp 30º / adj 60º
opp x 3 3sin(60 ) ; ;
hyp 2x 2
adj x 1cos(60 ) ; ;
hyp 2x 2
opp x 3tan(60 ) ; ; 3
adj x
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Thus, hyp
opp 60º / adj 30º
opp 30º / adj 60º
2 2 3csc(60 ) ;
33
sec(60 ) 2
1 3cot(60 ) ;
33
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For 30º hyp
opp 60º / adj 30º
opp 30º / adj 60º
opp x 1sin(30 ) ; ;
hyp 2x 2
adj x 3 3cos(30 ) ; ;
hyp 2x 2
opp x 1 3tan(30 ) ; ; ;
adj x 3 3 3
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Thus, hyp
opp 60º / adj 30º
opp 30º / adj 60º
csc(30 ) 2
3cot(30 ) ; 3
3
2 2 3sec(30 ) ;
33
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Summary
sin() cos() tan()
30º
45º 1
60º
1 2
2 2
3 2
3 2
2 2
1 2
3 3
3
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End of Right Triangle Trigonometry
Title V East Los Angeles College
1301 Avenida Cesar ChavezMonterey Park, CA 91754
Phone: (323) 265-8784
Email Us At:[email protected]
Our Website:http://www.matematicamente.org
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