• 5.1 Fundamental Identities Statements like “ tan sin / cos ” and “ csc 1/ sin ”
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Transcript • 5.1 Fundamental Identities Statements like “ tan sin / cos ” and “ csc 1/ sin ”
•
5.1 Fundamental Identities
Statements like “ tan sin / cos ” and
“ csc 1/ sin ” are trigonometric identities
because they are true for all values of the variable for
which both sides of the equation are defined.
• The set of all such values is called the domain of validity of
the identity.
• Basic Trigonometric Identities
– Some trigonometric identities follow directly from the
definitions of the six basic trigonometric functions.
– These basic identities consists of the reciprocal identities
and the quotient identities.
Basic Trigonometric Identities
• Reciprocal Identities
1
1
1
csc
sec
cot
sin
cos
tan
1
sin
csc
1
cos
sec
1
tan
cot
Basic Trigonometric Identities
• Quotient Identities
sin
tan
cos
cos
cot
sin
Basic Trigonometric Identities
• Pythagorean Identities
cos sin 1
2
2
1 tan sec
2
2
cot 1 csc
2
2
Using Identities
• Find sin and cos if tan 5 and
cos 0.
sec 1 tan 1 5 1 25 26
2
2
sec 26
cos 1
26
tan 5
sin
5
cos
2
sin 5cos
1
5
26
5 26
sin
26
Cofunction Identities
sin cos
2
cos sin
2
tan cot
2
cot tan
2
sec csc
2
csc sec
2
Odd-Even Identities
sin( x) sin x
cos( x) cos x
tan( x) tan x
csc( x) csc x
sec( x) sec x
cot( x) cot x
Using More Identities
• If cos 0.34 , find sin 2 .
sin
2 sin
2
cos
0.34
Simplifying by Factoring and Using Identities
sin x sin x cos x.
3
• Simplify the expression
2
sin x sin x cos x
3
2
sin x sin x cos x
2
sin x(1)
sin x
2
Simplifying by Expanding and Using Identities
sec x 1 sec x 1 / sin 2 x.
• Simplify the expression
sec x 1 sec x 1 / sin x
2
1
sec x 1 tan x sin x
2
2
2
2
sin x
sin x cos x sin x
2
2
1
2
sec x
2
cos x
2
Simplifying by Combining Fractions and Using Identities
• See Example 5 on p. 448 – 449.
Solving a Trigonometric Equation
• See Example 6 and Example 7 on p. 449 – 450.
More Practice!!!!!
• Homework – Textbook p. 451 – 452 #1 – 31
ODD.