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Transcript
```Double-Angle
Double-Angleand
and
11-5Half-Angle
11-5
Half-AngleIdentities
Identities
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
Double-Angle and
11-5 Half-Angle Identities
Warm Up
Find tan θ for 0 ≤ θ ≤ 90°, if
1.
2.
3.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Objective
Evaluate and simplify expressions by
using double-angle and half-angle
identities.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
You can use sum identities to derive the
double-angle identities.
sin 2θ = sin(θ + θ)
= sinθ cosθ + cosθ sinθ
= 2 sinθ cosθ
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
You can derive the double-angle identities for
cosine and tangent in the same way. There are
three forms of the identity for cos 2θ, which are
derived by using sin2θ + cos2θ = 1. It is common
to rewrite expressions as functions of θ only.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 1: Evaluating Expressions with Double-Angle
Identities
Find sin2θ and tan2θ if sinθ =
and 0°<θ<90°.
Step 1 Find cosθ to evaluate sin2θ = 2sinθcosθ.
Method 1 Use the reference angle.
In Ql, 0° < θ < 90°, and sinθ =
x2 + 22 = 52
Use the Pythagorean
Theorem.
Solve for x.
r=5
θ
x
Holt McDougal Algebra 2
y=2
Double-Angle and
11-5 Half-Angle Identities
Example 1 Continued
Method 2 Solve cos2θ = 1 – sin2θ.
cos2θ = 1 – sin2θ
cosθ =
Substitute
Simplify.
Holt McDougal Algebra 2
for cosθ.
Double-Angle and
11-5 Half-Angle Identities
Example 1 Continued
Step 2 Find sin2θ.
sin2θ = 2sinθcosθ
Apply the identity for sin2θ.
Substitute
for cosθ.
Simplify.
Holt McDougal Algebra 2
for sinθ and
Double-Angle and
11-5 Half-Angle Identities
Example 1 Continued
Step 3 Find tanθ to evaluate tan2θ =
.
Apply the tangent ratio identity.
Substitute
for cosθ.
Simplify.
Holt McDougal Algebra 2
for sinθ and
Double-Angle and
11-5 Half-Angle Identities
Example 1 Continued
Step 4 Find tan 2θ.
Apply the identity for tan2θ.
Substitute
Holt McDougal Algebra 2
for tan θ.
Double-Angle and
11-5 Half-Angle Identities
Example 1 Continued
Step 4 Continued
Simplify.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Caution!
The signs of x and y depend on the quadrant for
angle θ.
sin
cos
Ql
+
+
Qll
+
–
Qlll
–
–
QlV
–
+
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 1
Find tan2θ and cos2θ if cosθ =
270°<θ<360°.
and
Step 1 Find tanθ to evaluate tan2θ =
.
Method 1 Use the reference angle.
In QlV, 270° < θ < 360°, and cosθ =
12 + y2 = 32
Use the Pythagorean
Theorem.
x=1
Solve for y.
θ
r=3
Holt McDougal Algebra 2
y= –2√ 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 1 Continued
Step 2 Find tan2θ.
tan2θ =
Apply the identity for tan2θ.
Substitute –2
Simplify.
Holt McDougal Algebra 2
for tanθ.
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 1 Continued
Step 3 Find cos2θ.
cos2θ = 2cos2θ – 1
Apply the identity for cos2θ.
Substitute
Simplify.
Holt McDougal Algebra 2
for cosθ.
Double-Angle and
11-5 Half-Angle Identities
You can use double-angle identities to prove
trigonometric identities.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 2A: Proving identities with Double-Angle
Identities
Prove each identity.
sin 2θ = 2tanθ – 2tanθ sin2θ Choose the right-hand
side to modify.
= 2tanθ (1– sin2θ)
= 2tanθ cos2θ
= 2(tanθcosθ)cosθ
= 2sinθcosθ
= sin2θ
Holt McDougal Algebra 2
Factor 2tanθ.
Rewrite using 1 –sin2θ =
cos2θ.
Regroup.
Rewrite using tanθcosθ
= sinθ.
Apply the identity for
sin2θ.
Double-Angle and
11-5 Half-Angle Identities
Example 2B: Proving identities with Double-Angle
Identities
cos2θ = (2 – sec2θ)(1 – sin2θ)
cos2θ = (2 – sec2θ)(1 – sin2θ)
= (2 – sec2θ)(cos2θ)
Choose the right-hand side
to modify.
Rewrite using 1 – sin2θ =
cos2θ.
= 2cos2θ – 1
Expand and simplify.
= cos2θ
Apply the identity for
cos2θ.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Choose to modify either the left side or the right
side of an identity. Do not work on both sides at
once.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 2a
Prove each identity.
cos4θ – sin4θ = cos2θ
(cos2θ – sin2θ)(cos2θ + sin2θ) =
(1)(cos2θ) =
Factor the left side.
Rewrite using
1 = cos2θ + sin2θ and
cos2θ = cos2θ – sin2θ.
cos2θ = cos2θ Simplify.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 2b
Prove each identity.
Rewrite tan θ ratio identity
and Pythagorean identity.
Reciprocal sec θ identity
and simplify fraction.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 2b Continued
Prove each identity.
Simplify.
Double angle identity.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
You can use double-angle identities for cosine to derive
the half-angle identities by substituting for θ. For
example, cos2θ = 2 cos2θ – 1 can be rewritten as cosθ = 2
cos2 – 1. Then solve for cos
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Half-angle identities are useful in calculating exact
values for trigonometric expressions.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 3A: Evaluating Expressions with Half-Angle
Identities
Use half-angle identities to find the exact value
of cos 15°.
Positive in Ql.
Cos 30° =
Simplify.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 3A Continued
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 3B: Evaluating Expressions with Half-Angle
Identities
Use half-angle identities to find the exact value
of
.
Negative in Qll.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 3B Continued
Simplify.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 3B Continued
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 3a
Use half-angle identities to find the exact value
of tan 75°.
tan
(150°)
Positive in Ql.
Simplify.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 3a Continued
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 3b
Use half-angle identities to find the exact value
of
.
Negative in Qll.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 3b Continued
Simplify.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 4: Using the Pythagorean Theorem with HalfAngle Identities
Find cos
and tan
if tan θ =
and 0<θ<
Step 1 Find cos θ to evaluate the half-angle
identities. Use the reference angle.
In Ql, 0 < θ <
242 + 72 = x2
and tanθ =
Pythagorean Theorem.
Solve for the missing
side x.
Thus, cosθ =
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 4 Continued
Step 2 Evaluate cos
x
θ
Choose + for cos
where 0 < θ <
Evaluate.
Holt McDougal Algebra 2
24
7
Double-Angle and
11-5 Half-Angle Identities
Example 4 Continued
Simplify.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Example 4 Continued
Step 3 Evaluate tan
Choose + for tan
0<θ<
Evaluate.
Holt McDougal Algebra 2
where
Double-Angle and
11-5 Half-Angle Identities
Example 4 Continued
Simplify.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 4
Find sin
and cos
if tan θ =
and 0 < θ < 90.
Step 1 Find cos θ to evaluate the half-angle
identities. Use the reference angle.
In Ql, 0 < θ <
42 + 32 = r
and tanθ =
2
Pythagorean Theorem.
r=
Thus, cosθ =
Holt McDougal Algebra 2
.
Solve for the missing
side r.
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 4 Continued
Step 2 Evaluate cos
r
θ
Choose + for cos
where 0 < θ <
Evaluate.
Simplify.
Holt McDougal Algebra 2
3
4
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 4 Continued
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 4 Continued
Step 3 Evaluate sin
Choose + for sin
< 90°.
Evaluate.
Holt McDougal Algebra 2
where 0 < θ
Double-Angle and
11-5 Half-Angle Identities
Check It Out! Example 4 Continued
Simplify.
Holt McDougal Algebra 2
Double-Angle and
11-5 Half-Angle Identities
Lesson Quiz: Part I
1. Find cos
and cos 2θ if sin θ =
2. Prove the following identity:
Holt McDougal Algebra 2
and 0 < θ <
Double-Angle and
11-5 Half-Angle Identities
Lesson Quiz: Part II
3. Find the exact value of cos 22.5°.
Holt McDougal Algebra 2
```
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