3.6: HalfAngle Identities
Learning Objectives
 Apply the half angle identities to expressions, equations and other identities.
 Use the half angle identities to find the exact value of trigonometric functions for certain angles.
Just as there are double angle identities, there are also half angle identities. For example:
Deriving the Half Angle Formulas
In the previous lesson, one of the formulas that was derived for the cosine of a double angle is:
Solving this for
Example 1: Determine the exact value of
Solution: Using the half angle identity,
Plugging this into a calculator,
Example 2: Use the half angle identity to find exact value of
Solution: since
One of the other formulas that was derived for the cosine of a double angle is:
Example 3: Given that the
Solution: Because
Example 4: Use the half angle formula for the cosine function to prove that the following expression is an identity:
Solution: Use the formula
The half angle identity for the tangent function begins with the reciprocal identity for tangent.
The half angle formulas for sine and cosine are then substituted into the identity.
At this point, you can multiply by either
So, the two half angle identities for tangent are
Example 5: Use the halfangle identity for tangent to determine an exact value for
Solution:
Example 6: Prove the following identity:
Solution: Substitute the double angle formulas for
Solving Trigonometric Equations Using Half Angle Formulas
Example 7: Solve the trigonometric equation
Solution:
Then
Points to Consider
 Can you derive a third or fourth angle formula?
 How do
12sinx andsin12x differ? Is there a formula for12sinx ?
Review Questions
 Find the exact value of:

cos112.5∘ 
sin105∘ 
tan7π8 
tanπ8 
sin67.5∘ 
tan165∘

 If
sinθ=725 andθ is in Quad II, findsinθ2,cosθ2,tanθ2  Prove the identity:
tanb2=secbsecbcscb+cscb  Verify the identity:
cotc2=sinc1−cosc  Prove that
sinxtanx2+2cosx=2cos2x2  If
sinu=−813 , findcosu2  Solve
2cos2x2=1 for0≤x<2π  Solve
tana2=4 for0∘≤a<360∘  Solve the trigonometric equation
cosx2=1+cosx such that0≤x<2π .  Prove
sinx1+cosx=1−cosxsinx .
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Date Created:
Feb 23, 2012Last Modified:
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