Expressions of the form A sin x + B cos x
We can use the sum and difference identities to rewrite expressions of the form A sin x +
B cos x as something simpler. The trick is to simultaneously transform the A into something
that looks like sin φ and transform the B into something that looks like cos φ. Then we
can use a sum identity. We do this by imagining a point in the xy-plane with coordinates
(A, B). If φ is the angle between the line connecting (A, B) with the origin and the x-axis,
then
A
B
cos φ = √
sin φ = √
2
2
2
A +B
A + B2
Thus we can rewrite, using the sum identity:
√
A
B
2
2
A +B √
sin x + √
cos x
A sin x + B cos x =
A2 + B 2
A2 + B 2
√
A2 + B 2 (cos φ sin x + sin φ cos x)
=
√
=
A2 + B 2 sin(x + φ)
Example: Express 3 sin x + 4 cos x in the form k sin(x + φ)
√
√
solution: Here, k = A2 + B 2 = 32 + 42 = 5 also, sin φ = 45 , cos φ =
radians.
Thus,
3 sin x + 4 cos x = 5 sin(x + 0.927)
3
5
so φ = 0.927
7.3
Double-Angle, Half-Angle, and Product-sum Formulas
We state the double-angle identites:
sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x
= 1 − 2 sin2 x
= 2 cos2 x − 1
tan 2x =
2 tan x
1 − tan2 x
Example: If cos x = − 23 and x is in quadrant II, find cos 2x and sin 2x.
solution:
2
1
2
8
cos 2x = 2 cos x − 1 = 2 −
−1= −1=−
3
9
9
2
sin 2x = 2 sin x cos x
√
Since x is in quadrant II, we can sub in sin x = 1 − cos2 x:
r
r
√
2
s
4
2
4
4 5
4 5
2
=−
1− −
1− =−
=−
sin 2x = 2 −
3
3
3
9
3 9
9
Formulas for Lowering Powers
sin2 x =
1 − cos 2x
2
tan2 x =
cos2 x =
1 + cos 2x
2
1 − cos 2x
1 + cos 2x
Half-angle formulas
sin
u
2
r
=±
r
u
1 − cos u
1 + cos u
cos
=±
2
2
2
u 1 − cos u
tan
=
2
sin u
The ± is chosen in the first two depending on what quadrant
u
2
is in.
Example: Find the exact value of sin 22.5◦ .
solution: Since 22.5◦ is half of 45◦ , we use the half-angle formula for sin with u = 45◦ .
Since 22.5◦ is in the first quadrant, we choose the + sign.
◦
45
◦
sin 22.5 = sin
2
r
1 − cos 45◦
=
2
s
√
1 − 22
=
2
s
√
2− 2
=
4
Example: Fine tan
u
2
if sin u =
2
5
and u is in quadrant II.
solution: From the formula,
tan
u
2
=
1 − cos u
sin u
We know sin u = 25 , and from the usual pythagorean identity we know
p
p
2
cos u = ± 1 − sin u = − 1 − sin2 u
We choose the negative because cos is negative in the second quadrant. Hence
u
1 − cos u
tan
=
2
sinpu
1 + 1 − sin2 u
=
qsin u
1 − ( 52 )2
1+
=
1+
=
21
25
2/5
5
=
2
=
2/5
q
5+
5+
√ !
21
5
√
21
2
Product-to-Sum Formulas
1
[sin(u + v) + sin(u − v)]
2
1
cos u sin v =
[sin(u + v) − sin(u − v)]
2
1
cos u cos v =
[cos(u + v) + cos(u − v)]
2
1
sin u sin v =
[sin(u − v) − sin(u + v)]
2
sin u cos v =
Example: Express sin 3x sin 5x as a sum of trig functions.
solution: We use the fourth product-to-sum formula:
1
1
1
sin 3x sin 5x = [sin(3x − 5x) − sin(3x + 5x)] = (cos(−2x) − cos 8x) = (cos(2x) − cos 8x)
2
2
2
Sum-to-Product Formulas
sin x + sin y =
sin x − sin y =
cos x + cos y =
cos x − cos y =
x+y
x−y
2 sin
cos
2
2
x+y
x−y
2 cos
sin
2
2
x+y
x−y
2 cos
cos
2
2
x−y
x+y
sin
−2 sin
2
2
Example: Write sin 7x + sin 3x as a product.
solution: We use the first formula:
7x + 3x
7x − 3x
sin 7x + sin 3x = 2 sin
cos
= 2 sin 5x cos 2x
2
2