### 7.1

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Trigonometric
Identities and
Equations
7.1-1
Trigonometric Identities and
7 Equations
7.1 Fundamental Identities
7.2 Verifying Trigonometric Identities
7.3 Sum and Difference Identities
7.4 Double-Angle and Half-Angle Identities
7.5 Inverse Circular Functions
7.6 Trigonometric Equations
7.7 Equations Involving Inverse
Trigonometric Functions
7.1-2
7.1 Fundamental Identities
Fundamental Identities ▪ Using the Fundamental Identities
1.1-3
7.1-3
Fundamental Identities
Reciprocal Identities
Quotient Identities
1.1-4
7.1-4
Fundamental Identities
Pythagorean Identities
Negative-Angle Identities
1.1-5
7.1-5
Note
In trigonometric identities, θ can be
an angle in degrees, an angle in
radians, a real number, or a variable.
1.1-6
7.1-6
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
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
1.1-7
7.1-7
Example 1
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
(b) sin θ
Quotient identity
Reciprocal identity
from part (a)
1.1-8
7.1-8
Example 1
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
(c) cot(– θ)
Reciprocal identity
Negative-angle
identity
1.1-9
7.1-9
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.
1.1-10
7.1-10
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
Take reciprocals.
Reciprocal identity
Take the square
root of each side.
±
+±±±±±

±
=
+
1.1-11
The sign depends on
7.1-11
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
1.1-12
7.1-12
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.