Mathematical Investigations IV
Name:
Mathematical Investigations IV
Trigonometry - Beyond the Right Triangles
Half-Angle Formulas
If we can look at double angles, we can also look at half angles.
Recall: cos(2) = cos2 – sin2
and also
cos(2) = 2 cos2 – 1 = 1 – 2 sin2
cos(2 = 2 cos2 - 1
cos(2 = 1 - 2 sin2
1 + cos(2 = 2 cos2
2 sin2 = 1 – cos(2
1  cos(2
1  cos(2
= cos2
sin2 =
2
2
1  cos(2
1  cos(2
±
= cos
±
= sin
2
2
Let  =
cos

2
Important Note:
Now, tan

2


.
2
Let  =
1 cos 
2
sin

2


.
2
1 cos 
2
In each case, you must make the appropriate choice of "+" or "–",
depending on the quadrant that the resulting angle /2 is in.
sin
=
cos

2 =

2
1 cos
2
1 cos 

2

=

1 cos
1 cos 
OR
1 cos1 cos 
1  cos1 cos 
= ±
= ±
1 cos 2 
= ±
= ±
1 cos 2 
= ±

=
1 cos2 
= ±
sin 2 
1 cos 
sin 
=
Trig. 10x.1
1 cos1  cos
1  cos1  cos
1 cos2 
1  cos
2
sin 2 
1  cos
2
sin 
1  cos
Rev. S06
Mathematical Investigations IV
Name:
Thus,
1.
2.
tan
2

1 cos 
1 cos 
or
tan
 1 cos 

2
sin 
or
tan

2

sin 
1 cos 
Use the half-angle formulas to find exact values for each of the following.
a.
sin 22.5°
b.
cos 22.5°
c.
tan 22.5°
d.
tan 75°
e.
cos 75°
f.
sin 75°
Fill in the blanks below given that
a.
3.

< 2 <
b.

<  < 
2

<
<
2
Let  be in the third quadrant. Find the value of tan
Trig. 10x.2
c.
<
3
<
2

4
if sin =  .
2
5
Rev. S06
Mathematical Investigations IV
Name:
Without using your calculator, find exact values, where possible, for each of the following:
(Check your answers with the other students in your group.)
4.
Find sin



, cos , and tan
for each diagram.
2
2
2

(8,15)


(24,-7)
(-8,-6)
5.
Let cos  =
8

where sin < 0. Find the value of tan + tan  + tan 2.
17
2
Trig. 10x.3
Rev. S06