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14-4 Sum and Difference Identities
Check it out
Does the sin(75) =sin(45)+sin(30) ?
Holt Algebra 2
14-4 Sum and Difference Identities
Holt Algebra 2
14-4 Sum and Difference Identities
Example 1A: Evaluating Expressions with Sum and
Difference Identities
Find the exact value of cos 15°.
Write 15° as the difference
45° – 30° because
cos 15° = cos (45° – 30°)
trigonometric values of
45° and 30° are known.
Apply the identity for cos (A – B).
= cos 45° cos 30° + sin 45° sin 30°
Evaluate.
Simplify.
Holt Algebra 2
14-4 Sum and Difference Identities
Example 1B: Proving Evaluating Expressions with
Sum and Difference Identities
Find the exact value of
.
Write
as the sum of
Apply the identity for
tan (A + B).
Holt Algebra 2
14-4 Sum and Difference Identities
Example 1B Continued
Evaluate.
Simplify.
Holt Algebra 2
14-4 Sum and Difference Identities
Check It Out! Example 2
Prove the identity
.
Apply the identity for
cos A + B.
Evaluate.
= –sin x
Holt Algebra 2
Simplify.
14-4 Sum and Difference Identities
Check It Out! Example 1b
Find the exact value of each expression.
Write
as the sum of
because
trigonometric values of
and
are known.
Apply the identity for
sin (A – B).
Holt Algebra 2
14-4 Sum and Difference Identities
Check It Out! Example 1b Continued
Find the exact value of each expression.
Evaluate.
Simplify.
Holt Algebra 2
14-4 Sum and Difference Identities
Example 3: Using the Pythagorean Theorem with
Sum and Difference Identities
Find cos (A – B) if sin A =
if tan B =
with 0 < A <
and
with 0 < B <
Step 1 Find cos A, cos B, and sin B.
Use reference angles and the ratio
definitions sin A = and tan B = Draw a
triangle in the appropriate quadrant and
label x, y, and r for each angle.
Holt Algebra 2
14-4 Sum and Difference Identities
Example 3 Continued
In Quadrant l (Ql),
0° < A < 90° and
sin A = .
r=3
A
x
Holt Algebra 2
In Quadrant l (Ql),
0°< B < 90° and tan B =
r
y=1
B
x=4
y=3
.
14-4 Sum and Difference Identities
Example 3 Continued
r=3
A
r
y=1
x
y=3
B
x=4
x2 + 12 = 32
32 + 42 = r2
Thus, cos A =
Thus, cos B =
and sin A =
and sin B =
Holt Algebra 2
.
14-4 Sum and Difference Identities
Example 3 Continued
Step 2 Use the angle-difference identity to find
cos (A – B).
cos (A – B) = cosAcosB + sinA sinB
Apply the identity for
cos (A – B).
Substitute
for
cos A, for cos
B, and
cos(A – B) =
Holt Algebra 2
Simplify.
for sin B.
14-4 Sum and Difference Identities
Check It Out! Example 3
Find sin (A – B) if sinA = with 90° < A < 180°
and if cosB = with 0° < B < 90°.
In Quadrant ll (Ql),
90< A < 180 and
sin A = .
In Quadrant l (Ql),
0< B < 90° and cos B =
r=5
y=4
A
x
Holt Algebra 2
y
r=5
B
x=3
14-4 Sum and Difference Identities
Check It Out! Example 3 Continued
r=5
y=4
y
r=5
A
x
B
x=3
x2 + 42 = 52
52 – 32 = y2
Thus, sin A =
Thus, cos B =
and cos A =
Holt Algebra 2
and sin B =
14-4 Sum and Difference Identities
Check It Out! Example 3 Continued
Step 2 Use the angle-difference identity to find
sin (A – B).
sin (A – B) = sinAcosB – cosAsinB
sin(A – B) =
Holt Algebra 2
Apply the identity for
sin (A – B).
Substitute for sin
A and sin B, for
cos A, and for
cos B.
Simplify.
14-4 Sum and Difference Identities
Lesson Quiz: Part I
1. Find the exact value of cos 75°
2. Prove the identity sin
3. Find tan (A – B) for sin A =
cos B =
with 0 <B<
Holt Algebra 2
= cos θ
with 0 <A<
and