Product to sum and sum to product formulae
Product to sum and sum to product formulae
1/1
Let’s use what we know about angle addition to simplify the following:
cos(a − b) + cos(a + b) =
Product to sum and sum to product formulae
2/1
Let’s use what we know about angle addition to simplify the following:
(
+
cos(a − b) + cos(a + b) =
)+(
=
Product to sum and sum to product formulae
−
)
2/1
Let’s use what we know about angle addition to simplify the following:
cos(a − b) + cos(a + b) =
( cos(a) cos(b) + sin(a) sin(b)) + (
=
Product to sum and sum to product formulae
−
)
2/1
Let’s use what we know about angle addition to simplify the following:
cos(a − b) + cos(a + b) =
( cos(a) cos(b) + sin(a) sin(b)) + ( cos(a) cos(b) − sin(a) sin(b))
=
Product to sum and sum to product formulae
2/1
Let’s use what we know about angle addition to simplify the following:
cos(a − b) + cos(a + b) =
( cos(a) cos(b) + sin(a) sin(b)) + ( cos(a) cos(b) − sin(a) sin(b))
= 2 cos(a) · cos(b)
Product to sum and sum to product formulae
2/1
Let’s use what we know about angle addition to simplify the following:
cos(a − b) + cos(a + b) =
( cos(a) cos(b) + sin(a) sin(b)) + ( cos(a) cos(b) − sin(a) sin(b))
= 2 cos(a) · cos(b)
For you: Simplify
cos(a − b) − cos(a + b) =
and
sin(a − b) + sin(a + b) =
Product to sum and sum to product formulae
2/1
Let’s use what we know about angle addition to simplify the following:
cos(a − b) + cos(a + b) =
( cos(a) cos(b) + sin(a) sin(b)) + ( cos(a) cos(b) − sin(a) sin(b))
= 2 cos(a) · cos(b)
For you: Simplify
cos(a − b) − cos(a + b) = 2 sin(a) sin(b)
and
sin(a − b) + sin(a + b) = 2 sin(a) cos(b)
Product to sum and sum to product formulae
2/1
Products of Trigonometric functions
Theorem (The Product to sum rule)
For all a and b
1
2 · (cos(a − b) + cos(a + b))
1
sin(a) sin(b) = 2 · (cos(a − b) − cos(a + b))
sin(a) cos(b) = 12 · (sin(a − b) + sin(a + b))
cos(a) cos(b) =
Product to sum and sum to product formulae
3/1
Products of Trigonometric functions
Theorem (The Product to sum rule)
For all a and b
1
2 · (cos(a − b) + cos(a + b))
1
sin(a) sin(b) = 2 · (cos(a − b) − cos(a + b))
sin(a) cos(b) = 12 · (sin(a − b) + sin(a + b))
cos(a) cos(b) =
Use these formulas to compute
cos(5π/12) · cos(π/12),
sin(5π/12) · sin(π/12),
sin(5π/12) · cos(π/12).
I’ll do one and you’ll do the other two.
Product to sum and sum to product formulae
3/1
Products of Trigonometric functions
Theorem (The Product to sum rule)
For all a and b
1
2 · (cos(a − b) + cos(a + b))
1
sin(a) sin(b) = 2 · (cos(a − b) − cos(a + b))
sin(a) cos(b) = 12 · (sin(a − b) + sin(a + b))
cos(a) cos(b) =
Use these formulas to compute
2 cos x+y
· cos x−y
,
2
2
x+y x−y 2 sin 2 sin 2 ,
2 sin x+y
cos x−y
2
2
I’ll do one and you’ll do the other two. We’ll get some interesting formulas
out of this exercise.
Product to sum and sum to product formulae
4/1
sum to product rule
Theorem (The sum to product rule)
For all a and b
a+b
cos a−b
2
2
a−b
cos(a) − cos(b) = −2 sin a+b
sin
2
2
a−b
sin(a) + sin(b) = 2 sin a+b
cos
2
2
a−b
sin(a) − sin(b) = 2 cos a+b
sin
2
2
cos(a) + cos(b) = 2 cos
Compute sin(5π/12) + sin(π/12)
Product to sum and sum to product formulae
5/1
mixing sin and cos in the sum to product formula
Theorem (The sum to product rule)
For all a and b
a+b
cos a−b
2
2
a−b
cos(a) − cos(b) = −2 sin a+b
sin
2
2
cos a−b
sin(a) + sin(b) = 2 sin a+b
2
2
sin(a) − sin(b) = 2 cos a+b
sin a−b
2
2
cos(a) + cos(b) = 2 cos
Is there a nice formula for sin(x) + cos(x)?
Product to sum and sum to product formulae
6/1
mixing sin and cos in the sum to product formula
Theorem (The sum to product rule)
For all a and b
a+b
cos a−b
2
2
a−b
cos(a) − cos(b) = −2 sin a+b
sin
2
2
cos a−b
sin(a) + sin(b) = 2 sin a+b
2
2
sin(a) − sin(b) = 2 cos a+b
sin a−b
2
2
cos(a) + cos(b) = 2 cos
Is there a nice formula for sin(x) + cos(x)?
A trick: Remember that sin(x) = cos(π/2 − x)
Product to sum and sum to product formulae
6/1
mixing sin and cos in the sum to product formula
Theorem (The sum to product rule)
For all a and b
a+b
cos a−b
2
2
a−b
cos(a) − cos(b) = −2 sin a+b
sin
2
2
cos a−b
sin(a) + sin(b) = 2 sin a+b
2
2
sin(a) − sin(b) = 2 cos a+b
sin a−b
2
2
cos(a) + cos(b) = 2 cos
Is there a nice formula for sin(x) + cos(x)?
A trick: Remember that sin(x) = cos(π/2 − x)
Write as a product
sin(x) + cos(x) = cos(π/2 − x) + cos(x)
Product to sum and sum to product formulae
6/1
mixing sin and cos in the sum to product formula
Theorem (The sum to product rule)
For all a and b
a+b
cos a−b
2
2
a−b
cos(a) − cos(b) = −2 sin a+b
sin
2
2
cos a−b
sin(a) + sin(b) = 2 sin a+b
2
2
sin(a) − sin(b) = 2 cos a+b
sin a−b
2
2
cos(a) + cos(b) = 2 cos
Is there a nice formula for sin(x) + cos(x)?
A trick: Remember that sin(x) = cos(π/2 − x)
Write as a product
sin(x) + cos(x) = cos(π/2 − x) + cos(x)
exit quiz
Use the formula you get on x = π/6 to recover a formula for cos(π/12).
Product to sum and sum to product formulae
6/1