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You can use sum identities to derive the double-angle identities.
sin 2θ = sin(θ + θ)
= sinθ cosθ + cosθ sinθ
= 2 sinθ cosθ
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.
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
θ
r = 5y = 2
x
Use the PythagoreanTheorem.
Solve for x.
Example 1 Continued
Method 2 Solve cos2θ = 1 – sin2θ.
cos2θ = 1 – sin2θ
cosθ =Substitute for cosθ.
Simplify.
Example 1 Continued
Step 2 Find sin2θ.
sin2θ = 2sinθcosθ Apply the identity for sin2θ.
Simplify.
Substitute for sinθ and
for cosθ.
Example 1 Continued
Step 3 Find tanθ to evaluate tan2θ = .
Apply the tangent ratio identity.
Simplify.
Substitute for sinθ and
for cosθ.
The signs of x and y depend on the quadrant for angle θ.
sin cos
Ql + +
Qll + –
Qlll – –
QlV – +
Caution!
Find tan2θ and cos2θ if cosθ = and 270°<θ<360°.
Method 1 Use the reference angle.
Check It Out! Example 1
Step 1 Find tanθ to evaluate tan2θ = .
In QlV, 270° < θ < 360°, and cosθ =
12 + y2 = 32 Use the Pythagorean Theorem.
Solve for y. θ
r=3
x=1
y= –2√ 2
Check It Out! Example 1 Continued
Step 2 Find tan2θ.
Apply the identity for tan2θ.
Simplify.
tan2θ =
Substitute –2 for tanθ.
Check It Out! Example 1 Continued
Step 3 Find cos2θ.
cos2θ = 2cos2θ – 1 Apply the identity for cos2θ.
Simplify.
Substitute for cosθ.
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) Factor 2tanθ.
= 2tanθ cos2θRewrite using 1 –sin2θ = cos2θ.
= 2(tanθcosθ)cosθ Regroup.
= 2sinθcosθRewrite using tanθcosθ
= sinθ.
= sin2θ Apply the identity for sin2θ.
Example 2B: Proving identities with Double-Angle Identities
cos2θ = (2 – sec2θ)(1 – sin2θ)
cos2θ = (2 – sec2θ)(1 – sin2θ)
= (2 – sec2θ)(cos2θ)
= 2cos2θ – 1
= cos2θ
Choose the right-hand side to modify.
Rewrite using 1 – sin2θ = cos2θ.
Expand and simplify.
Apply the identity for cos2θ.
Choose to modify either the left side or the right side of an identity. Do not work on both sides at once.
Helpful Hint
Check It Out! Example 2a
cos4θ – sin4θ = cos2θ
(cos2θ – sin2θ)(cos2θ + sin2θ) =
(1)(cos2θ) =
cos2θ = cos2θ
Factor the left side.
Rewrite using 1 = cos2θ + sin2θ and cos2θ = cos2θ – sin2θ.
Simplify.
Prove each identity.
Check It Out! Example 2b Prove each identity.
Rewrite tan θ ratio identity and Pythagorean identity.
Reciprocal sec θ identity and simplify fraction.
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
Example 3A: Evaluating Expressions with Half-Angle Identities
Use half-angle identities to find the exact value of cos 15°.
Positive in Ql.
Simplify.
Cos 30° =
Example 3B: Evaluating Expressions with Half-Angle Identities
Use half-angle identities to find the exact value
of .
Negative in Qll.
Check It Out! Example 3a Use half-angle identities to find the exact value of tan 75°.
tan (150°)
Positive in Ql.
Simplify.
Example 4: Using the Pythagorean Theorem with Half-Angle 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 < and tanθ =
242 + 72 = x2
Thus, cosθ =
Pythagorean Theorem.
Solve for the missing side x.
Check It Out! Example 4
Step 1 Find cos θ to evaluate the half-angle identities. Use the reference angle.
42 + 32 = r 2 Pythagorean Theorem.
Solve for the missing side r.
Find sin and cos if tan θ = and 0 < θ < 90.
In Ql, 0 < θ < and tanθ =
r =
Thus, cosθ = .
Check It Out! Example 4 Continued
r4
3θ
Step 2 Evaluate cos
Evaluate.
Choose + for cos
where 0 < θ <
Simplify.
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