Trigonometry: The Reference Triangle by: javier
Trigonometry: The Reference Triangle
by: javier
THE FUNCTIONS SECOND ACT
THE FUNCTIONS SECOND ACT big picture
THE FUNCTIONS: BIG PICTURE
THE FUNCTIONS: BIG PICTURE
▶ Made to describe ratios, ALL THE RATIOS
C b
hyp b b
A
b
β b
adj
opp
b
B
THE FUNCTIONS: BIG PICTURE
▶ Made to describe ratios, ALL THE RATIOS ▶ LEAD to solving of triangles, amazing applications. sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
C b
hyp b b
A
b
β b
adj
opp
b
B
THE FUNCTIONS: BIG PICTURE
▶ Made to describe ratios, ALL THE RATIOS ▶ LEAD to solving of triangles, amazing applications. sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
▶ That was great already, we did amazing things with that. hyp
C b
b b
A
b
β b
adj
opp
b
B
THE FUNCTIONS
THE FUNCTIONS second act
THE FUNCTIONSsecond act
THE FUNCTIONSsecond act
▶ THE FUNCTIONS were later found useful in sound transmission, image data transmission, data corrections algorithms, data compression, and a million other applications.
THE FUNCTIONSsecond act
▶ THE FUNCTIONS were later found useful in sound transmission, image data transmission, data corrections algorithms, data compression, and a million other applications. ▶ The key feature was to extend their definition beyond the angles 0 ≤ β ≤ 90o
THE FUNCTIONSsecond act
▶ THE FUNCTIONS were later found useful in sound transmission, image data transmission, data corrections algorithms, data compression, and a million other applications. ▶ The key feature was to extend their definition beyond the angles 0 ≤ β ≤ 90o ▶ How do we extend the life of the functions beyond 0 ≤ β ≤ 90o ? We use ”THE REFERENCE TRIANGLE”!
THE REFERENCE TRIANGLE
THE REFERENCE TRIANGLE a first look
THE REFERENCE TRIANGLEa first look
THE REFERENCE TRIANGLEa first look ▶ Every angle has one.
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis.
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis. ▶ Rotate it counter-clock wise for positive angles, clockwise for negative angles.
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis. ▶ Rotate it counter-clock wise for positive angles, clockwise for negative angles. ▶ Drop a perpendicular TO THE x-axis
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis. ▶ Rotate it counter-clock wise for positive angles, clockwise for negative angles. ▶ Drop a perpendicular TO THE x-axis ▶ label the sides, including signs, stand at origin to identify, opposite, hypothenuse and adjacent sides
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis. ▶ Rotate it counter-clock wise for positive angles, clockwise for negative angles. ▶ Drop a perpendicular TO THE x-axis ▶ label the sides, including signs, stand at origin to identify, opposite, hypothenuse and adjacent sides ▶ now the functions can go way beyond the original restrictive domain 0 ≤ β ≤ 90o
THE REFERENCE TRIANGLE
THE REFERENCE TRIANGLE some examples
EXAMPLES: The Reference Triangle reference triangle for 150o
2
EXAMPLES: The Reference Triangle reference triangle for 150o
2
EXAMPLES: The Reference Triangle reference triangle for 240o
2
EXAMPLES: The Reference Triangle reference triangle for 240o
2
EXAMPLES: The Reference Triangle reference triangle for 330o
2
EXAMPLES: The Reference Triangle reference triangle for 330o
2
EXAMPLES: The Reference Triangle reference triangle for −210o
2
EXAMPLES: The Reference Triangle reference triangle for −210o
2
EXAMPLES: The Reference Triangle reference triangle for −135o
2
EXAMPLES: The Reference Triangle reference triangle for −135o
2
EXAMPLES: The Reference Triangle reference triangle for 90o
2
EXAMPLES: The Reference Triangle reference triangle for 90o
2
EXAMPLES: The Reference Triangle reference triangle for 180o
2
EXAMPLES: The Reference Triangle reference triangle for 180o
2
EXAMPLES: The Reference Triangle reference triangle for 390o
2
EXAMPLES: The Reference Triangle reference triangle for 390o
2
ALL THE VALUES on ALL FAMOUS ANGLES
sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
ALL THE VALUES on ALL FAMOUS ANGLES
sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
csc β =
hyp opp
sec β =
hyp adj
cot β =
adj opp
ALL THE VALUES on ALL FAMOUS ANGLES
sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
csc β =
hyp opp
sec β =
hyp adj
cot β =
adj opp
0o , 30o , 45o , 60o , 90o , 120o , 150o , 180o , 210o , 225o ,
ALL THE VALUES on ALL FAMOUS ANGLES
sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
csc β =
hyp opp
sec β =
hyp adj
cot β =
adj opp
0o , 30o , 45o , 60o , 90o , 120o , 150o , 180o , 210o , 225o , you should know how to evaluate any function at any famous angle with nothing new memorize!
by: javier
THE FUNCTIONS SECOND ACT
THE FUNCTIONS SECOND ACT big picture
THE FUNCTIONS: BIG PICTURE
THE FUNCTIONS: BIG PICTURE
▶ Made to describe ratios, ALL THE RATIOS
C b
hyp b b
A
b
β b
adj
opp
b
B
THE FUNCTIONS: BIG PICTURE
▶ Made to describe ratios, ALL THE RATIOS ▶ LEAD to solving of triangles, amazing applications. sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
C b
hyp b b
A
b
β b
adj
opp
b
B
THE FUNCTIONS: BIG PICTURE
▶ Made to describe ratios, ALL THE RATIOS ▶ LEAD to solving of triangles, amazing applications. sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
▶ That was great already, we did amazing things with that. hyp
C b
b b
A
b
β b
adj
opp
b
B
THE FUNCTIONS
THE FUNCTIONS second act
THE FUNCTIONSsecond act
THE FUNCTIONSsecond act
▶ THE FUNCTIONS were later found useful in sound transmission, image data transmission, data corrections algorithms, data compression, and a million other applications.
THE FUNCTIONSsecond act
▶ THE FUNCTIONS were later found useful in sound transmission, image data transmission, data corrections algorithms, data compression, and a million other applications. ▶ The key feature was to extend their definition beyond the angles 0 ≤ β ≤ 90o
THE FUNCTIONSsecond act
▶ THE FUNCTIONS were later found useful in sound transmission, image data transmission, data corrections algorithms, data compression, and a million other applications. ▶ The key feature was to extend their definition beyond the angles 0 ≤ β ≤ 90o ▶ How do we extend the life of the functions beyond 0 ≤ β ≤ 90o ? We use ”THE REFERENCE TRIANGLE”!
THE REFERENCE TRIANGLE
THE REFERENCE TRIANGLE a first look
THE REFERENCE TRIANGLEa first look
THE REFERENCE TRIANGLEa first look ▶ Every angle has one.
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis.
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis. ▶ Rotate it counter-clock wise for positive angles, clockwise for negative angles.
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis. ▶ Rotate it counter-clock wise for positive angles, clockwise for negative angles. ▶ Drop a perpendicular TO THE x-axis
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis. ▶ Rotate it counter-clock wise for positive angles, clockwise for negative angles. ▶ Drop a perpendicular TO THE x-axis ▶ label the sides, including signs, stand at origin to identify, opposite, hypothenuse and adjacent sides
THE REFERENCE TRIANGLEa first look ▶ Every angle has one. ▶ Start on with a segment, any positive length, on right side of the x axis. ▶ Rotate it counter-clock wise for positive angles, clockwise for negative angles. ▶ Drop a perpendicular TO THE x-axis ▶ label the sides, including signs, stand at origin to identify, opposite, hypothenuse and adjacent sides ▶ now the functions can go way beyond the original restrictive domain 0 ≤ β ≤ 90o
THE REFERENCE TRIANGLE
THE REFERENCE TRIANGLE some examples
EXAMPLES: The Reference Triangle reference triangle for 150o
2
EXAMPLES: The Reference Triangle reference triangle for 150o
2
EXAMPLES: The Reference Triangle reference triangle for 240o
2
EXAMPLES: The Reference Triangle reference triangle for 240o
2
EXAMPLES: The Reference Triangle reference triangle for 330o
2
EXAMPLES: The Reference Triangle reference triangle for 330o
2
EXAMPLES: The Reference Triangle reference triangle for −210o
2
EXAMPLES: The Reference Triangle reference triangle for −210o
2
EXAMPLES: The Reference Triangle reference triangle for −135o
2
EXAMPLES: The Reference Triangle reference triangle for −135o
2
EXAMPLES: The Reference Triangle reference triangle for 90o
2
EXAMPLES: The Reference Triangle reference triangle for 90o
2
EXAMPLES: The Reference Triangle reference triangle for 180o
2
EXAMPLES: The Reference Triangle reference triangle for 180o
2
EXAMPLES: The Reference Triangle reference triangle for 390o
2
EXAMPLES: The Reference Triangle reference triangle for 390o
2
ALL THE VALUES on ALL FAMOUS ANGLES
sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
ALL THE VALUES on ALL FAMOUS ANGLES
sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
csc β =
hyp opp
sec β =
hyp adj
cot β =
adj opp
ALL THE VALUES on ALL FAMOUS ANGLES
sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
csc β =
hyp opp
sec β =
hyp adj
cot β =
adj opp
0o , 30o , 45o , 60o , 90o , 120o , 150o , 180o , 210o , 225o ,
ALL THE VALUES on ALL FAMOUS ANGLES
sin β =
opp hyp
cos β =
adj hyp
tan β =
opp adj
csc β =
hyp opp
sec β =
hyp adj
cot β =
adj opp
0o , 30o , 45o , 60o , 90o , 120o , 150o , 180o , 210o , 225o , you should know how to evaluate any function at any famous angle with nothing new memorize!
