Transcript Fundamental Identities

5
Trigonometric
Identities
Copyright © 2009 Pearson Addison-Wesley
5.1-1
5 Trigonometric Identities
5.1 Fundamental Identities
5.2 Verifying Trigonometric Identities
5.3 Sum and Difference Identities for Cosine
5.4 Sum and Difference Identities for Sine
and Tangent
5.5 Double-Angle Identities
5.6 Half-Angle Identities
Copyright © 2009 Pearson Addison-Wesley
5.1-2
5.1 Fundamental Identities
Fundamental Identities ▪ Using the Fundamental Identities
Copyright © 2009 Pearson Addison-Wesley
1.1-3
5.1-3
Fundamental Identities
Reciprocal Identities
Quotient Identities
Copyright © 2009 Pearson Addison-Wesley
1.1-4
5.1-4
Fundamental Identities
Pythagorean Identities
Negative-Angle Identities
Copyright © 2009 Pearson Addison-Wesley
1.1-5
5.1-5
Note
In trigonometric identities, θ can be
an angle in degrees, an angle in
radians, a real number, or a variable.
Copyright © 2009 Pearson Addison-Wesley
1.1-6
5.1-6
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT
Example 1
If
and θ is in quadrant II, find each function
value.
(a) sec θ
Pythagorean
identity
In quadrant II, sec θ is negative, so
Copyright © 2009 Pearson Addison-Wesley
1.1-7
5.1-7
Example 1
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT (continued)
(b) sin θ
Quotient identity
Reciprocal identity
from part (a)
Copyright © 2009 Pearson Addison-Wesley
1.1-8
5.1-8
Example 1
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT (continued)
(b) cot(– θ)
Reciprocal identity
Negative-angle
identity
1 3
cot( )  
5 5
3
Copyright © 2009 Pearson Addison-Wesley
1.1-9
5.1-9
Caution
To avoid a common error, when
taking the square root, be sure to
choose the sign based on the
quadrant of θ and the function being
evaluated.
Copyright © 2009 Pearson Addison-Wesley
1.1-10
5.1-10
Example 2
EXPRESSING ONE FUNCITON IN
TERMS OF ANOTHER
Express cos x in terms of tan x.
Since sec x is related to both cos x and tan x by
identities, start with
Take reciprocals.
Reciprocal identity
Take the square
root of each side.
The sign depends on
the quadrant of x.
Copyright © 2009 Pearson Addison-Wesley
1.1-11
5.1-11
Example 3
REWRITING AN EXPRESSION IN
TERMS OF SINE AND COSINE
Write tan θ + cot θ in terms of sin θ and cos θ, and
then simplify the expression.
Quotient identities
Write each fraction
with the LCD.
Pythagorean identity
Copyright © 2009 Pearson Addison-Wesley
1.1-12
5.1-12
Caution
When working with trigonometric
expressions and identities, be sure
to write the argument of the function.
For example, we would not write
An argument such as θ
is necessary.
Copyright © 2009 Pearson Addison-Wesley
1.1-13
5.1-13