Section 5.2: Angle Sum/Difference Formulas
The formulas in this section can be regarded as the ‘grandparents’ of a LOT of
other trig. identities you’ll use in the future. ALSO, these formulas will help you
unlock the use of the difference quotient for sine and cosine in calculus 1.
ALL THE FORMULAS MENTIONED IN THIS SECTION
DO NOT HAVE TO BE COMMITTED TO MEMORY!
Angle Sum and Angle Difference Formulas
... for sine:
sin(α + β ) = sinα cos β + cos α sin β
sin(α − β ) = sinα cos β − cos α sin β
... for cosine:
cos(α + β ) = cos α cos β − sinα sinβ
cos(α − β ) = cos α cos β + sinα sinβ
... for tangent:
tanα + tanβ
tan(α + β ) =
1 − tanα tan β
tanα − tan β
tan(α − β ) =
1 + tanα tan β
ex) Use the fact that 105° = 60° + 45° to determine the exact values of:
a) sin(105°)
b) cos(105°)
ex) What is the exact value of tan(15°) ?
ex) Evaluate cos 35π cos 415π + sin 35π sin 415π without using your calculator.
ex) Verify the identity: sin(270°− θ ) =−cos θ
ex) Verify the identity:
ex) Given sinα =
a) sin(α − β )
cos(α − β )
= 1 + cot α cot β
sinα sin β
12
4
and tan β =
( α and β are acute angles), evaluate:
13
3
b) cos(α − β )
c) tan(α − β )
ex) Given sec α = −3 (in quad. III) and csc β = 2 (quad II), evaluate;
a) sin(α + β )
b) sec(α + β )
(You can leave this answer UN‐rationalized)
15
ex) If the point (− 178 , 17
) were rotated by 45 ° clockwise, what would the
coordinates of the rotated point be?
Rewriting Products as Sums (and vice versa) Formulas
Another more immediate use of these ‘grandparent’ formulas is providing a way of
rewriting a product of sines and cosines in the form of a sum or difference.
Product to Sum Formulas
sinα cos β = 12 sin(α + β ) + sin(α − β )
cos α sin β = 12 sin(α + β ) − sin(α − β )
cos α cos β = 12 cos(α + β ) + cos(α − β )
sinα sinβ = 12 cos(α − β ) − cos(α + β )
ex) Rewrite the products as sums (or differences)
a) sin(3x)cos(2 x)
b) sin(18t )sin(22t )
ex) Verify the identity: 2cos( 32θ )sin( 2θ ) = sin(2θ ) − sin(θ)
Sum to Product Formulas
sin A + sinB = 2sin( A+2 B )cos( A−2 B )
sin A − sinB = 2cos( A+2 B )sin( A−2 B )
cos A + cos B = 2cos( A+2 B )cos( A−2 B )
cos A − cos B =−2sin( A+2 B )sin( A−2 B )
ex) Rewrite
sin(7 x ) − sin(5x )
as a single trig. function.
cos(7 x ) + cos(5x )