Download Day X 5.5

In the book of life, the
answers aren't in the
back.
Charles M. Schulz
(1922 – 2000)
an American cartoonist,
who wrote the comic
strip Peanuts
Chapter 5
Analytic Trigonometry
Day X.
Multiple-Angle
and
Product-to-Sum
Formulas (5.5)
Standard 11.1
Students demonstrate an
understanding of half-angle
and double-angle formulas
for sines and cosines
Double- Angle
Identities
The most commonly used
multiple-angle formulas
are the double-angle
formulas.
sin 2θ = sin (θ + θ)
State the sum identity for
sine.
sin( + ) =
sin  cos  + cos  sin 
Substitute θ in for  and
.
sin(
θ + θ) =
sin θ cos 
 sin θ
=
θ + cos θ
2 sin θ cos θ
sin 2θ = 2 sin θ cos θ
We have just derived the
double-angle formula for
sine. You do the same for
cosine and tangent.
cos 2θ =
2
cos
θ–
2
sin
2 tan θ
tan 2θ =
2
1 – tan θ
θ
cos 2θ has 3 forms.
2
cos
2
sin
cos 2θ =
θ–
θ
2
2
cos θ = 1 – sin θ
2
= 1 – 2 sin θ
2
2
sin θ = 1 – cos θ
2
= 2 cos θ - 1
Example 1
Finding the Exact
Values
Use the figure to find the
exact value of the trig
functions.
1
θ
3
sin 2θ = 2 sin θ cos θ
3
1
=2

10 10
= 3/5
Your Turn
1
3
θ
tan 2θ =
¾
1
3
θ
sec 2θ =
5/4
Example 2
More Exact Values
Find the exact values of
sin 2u, cos 2u, and tan
2u if cos u = -2/3 and
/2 < u < .
5
3
-2
sin 2θ = 2 sin θ cos θ
-45/9
5
3
-2
cos 2θ =
2
cos
θ–
-1/9
2
sin
θ
5
3
-2
2
tan
θ
tan 2θ =
2
1 – tan θ
45
or
sin
2θ
tan 2θ =
cos 2θ
What do you get when
you cross an elephant
with a Volkswagen?
A little car with a big
trunk.
Half- Angle Identities

1
–
cos

sin = 
2
2
1
+
cos


cos = 
2
2

tan = 
2
1 – cos 
1 + cos 
Example 3
Using Half-Angle
Identities
Find the exact values of
the sine, cosine, and
tangent of the given
angle using the half-angle
formulas.

30
sin
= sin 15 = sin
2
12
30
sin
= + 1 – cos 30
2
2
30
sin
=
2
=
2 - 3
2
2
3/2
1 – cos
30
2
2 - 3
=
4
=
2 - 3
= 2 - 3
4
2
Your Turn
2
2
cos 67.5 =
2
tan 165 = -2 + 3
Standard 11.2
Students can use those
formulas to prove and/or
simplify other
trigonometric identities.
Example 4
Verify the Identity
x
tan = csc x – cot x
2
Your Turn
sin 2x – cos 2x = sec x
sin x
cos x