Download 6.4 TRIGONOMETRIC ANGLES

2.2 Trigonometric Angles and Basic Identities
HOMEWORK: Sec 2.2: 3-10, 13-16, 18, 19, 23, 24, 28-33,
37-54, 57-60
Read Sec 2.3
sin = y / r = opp / hyp
cos  = x / r = adj / hyp
tan  = y / x = opp / adj (x  0)
csc  = r / y = hyp / opp (y  0)
sec  = r / x = hyp / adj (x  0)
cot  = x / y = adj / opp (y  0)
SOHCAHTOA
I. Finding the values of the six trig function of a triangle

3

2
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2.2 Trigonometric Angles and Basic Identities
II. Given the value of one trig function, find the others. Assume
all angles are QI.
Example 1: cos( )  23
Example 2: cot   3
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2.2 Trigonometric Angles and Basic Identities
FUNDAMENTAL IDENTITIES:
**** Very Important to MEMORIZE these ASAP ****
RECIPROCAL IDENTITIES:
1
1
1
tan  
csc  
cos
cot 
sin 
1
1
1
cos 
cot  
sin  
sec 
tan 
csc 
sec  
TANGENT/COTANGENT QUOTIENT:
tan  sin cot  cos
cos
sin
PYTHAGOREAN IDENTITIES:
sin 2  cos2 1
1 tan2  sec2
1 cot  csc2
Example: Use trig identities to find the values of the other trig
functions if sin  = ½ if θ in QI
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2.2 Trigonometric Angles and Basic Identities
Example (to try at home): Use trig identities to find the values
of the other trig functions if tan  = ½ if θ in QI
Work the problem again using the triangle method:
Answer:
sin  
5
2 5
1
5
, cos  
, tan   , sec  
, csc   5 , cot   2
5
5
2
2
Avoid common mistakes: Note: tan  = ½ DOES NOT MEAN
sin θ = 1 and cos θ = 2! The denominators canceled! You need
to use Pythagorean ID’s!
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2.2 Trigonometric Angles and Basic Identities
Example: Use trig identities to verify the following identities
tan (cos  cot )  sin 1
sec2 1  sin 2
sec2
Watch for common mistakes: Never drop variables! I.e.
cos is not the same as cos θ!
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2.2 Trigonometric Angles and Basic Identities
COMPLEMENTARY: Two angles are complementary if their
sum is  or 90º
2
Example: Find the complement of 
6
SUPPLEMENTARY: Two angles are supplementary if their
sum is  or 180º
Example: Find the supplement of 
6
** Important Note: You MUST leave answer in the same
angle mode as the original angle unless directed to
convert.
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2.2 Trigonometric Angles and Basic Identities
Complementary Angle Theorem
Co-functions of complementary angles are equal

sin θ = cos(90º- θ) or sin θ = cos(   )
2

cos θ = sin(90º- θ) or cos θ = sin(   )
2

tan θ = cot(90º- θ) or tan θ = cot (   )
2

cot θ = tan(90º- θ) or cot θ = tan (   )
2

csc θ = sec(90º- θ) or csc θ = sec (   )
2

sec θ = csc(90º- θ) or sec θ = csc (   )
2
Examples: cos(40º)=sin(50º) because
tan(30º)=cot(60º) because
sec(25º)=csc(65º) because
sin(  )  cos( ) because
3
6
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2.2 Trigonometric Angles and Basic Identities
Example: Use identities to simplify the following without
a calculator
cos 55

tan 35 
 sin 2 (13 )  cos2 (13 )
cos 35
(sec2 (24 )  1) sec2 (66 ) cos2 (24 )  1
Note: You MUST have all ID’s MEMORIZED and
practice enough problems to be able to recognize them
quickly!
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