7.3.1 – Product/Sum Identities
• So far, we have talked about modifying angles
in terms of addition and subtraction
• Not included within that was the case of
multiply certain angles by values
• Specifically, what if we double it?
Double-Angle Identities
• Double Angle identities will allow us to find
trig values for when we double the angle of
interest
Sine/Cosine
• Let a be an angle; radians or degrees
• sin(2a) = 2sin(a)cos(a)
• cos(2a) = cos2(a) – sin2(a)
= 2cos2(a) – 1
= 1 – 2sin2(a)
Tangent
2 tan( a)
• Tan(2a) =
2
1  tan (a)
• Just as before, we typically we try to use
angles from the unit circle we know about
(from our chart)
• Similar to problems from the last section, we
must be able to use the given identities with
or without an angle
• Example. Given that sin(x) = 1/√5, and tan(x) is
positive, determine the value of cos(2x),
sin(2x) and tan(2x)
• Example. Given that cos(x) = -2/√5 and that
sin(x) is positive, determine the values for
cos(2x), sin(2x) and tan(2x).
Proving Identities
• Also using product identities, we may verify or
prove other identities
• Still may need to use previous identities (have
those handy, or use the reference page from
the back of the book)
2 tan( x)
• Example. Show sin(2x) =
2
1  tan ( x)
Half-Angle Identities
x
1  cos( x)
sin  
2
2
x
1  cos( x)
cos  
2
2
x 1  cos( x)
sin( x)
tan 

2
sin( x)
1  cos( x)
• Using these identities, we can rewrite angles,
similar to before
• Example. Determine the exact value of
sin(π/8)
– What angle is π/8 half of?
• Example. Determine the exact value for
tan(7π/12)
• Assignment
• Pg. 576
• 1, 3, 5, 12, 19, 20, 22, 24