Practice and Problem-Solving Exercises
A Practice
Use an angle sum identity to derive each double-angle identity. A See Problems 1, 2, and 3.
-
sin
2
θ
=
2
sin
θ
cos
θ
sine 2 theta equals 2 sine theta cosine theta
-
tan
2
θ
=
2
tan
θ
1
−
tan
2
θ
tangent 2 theta equals . fraction 2 tangent theta , over 1 minus , tangent squared , theta end fraction
Use a double-angle identity to find the exact value of each expression.
-
sin
240
°
sine , 240 degrees
-
cos
120
°
cosine , 120 degrees
-
tan
120
°
tangent , 120 degrees
-
sin
90
°
sine , 90 degrees
-
cos
240
°
cosine , 240 degrees
-
tan
240
°
tangent , 240 degrees
-
cos
600
°
cosine , 600 degrees
-
sin
600
°
sine , 600 degrees
Use a half-angle identity to find the exact value of each expression. A See Problems 4.
-
cos
15
°
cosine , 15 degrees
-
tan
15
°
tangent , 15 degrees
-
sin
15
°
sine , 15 degrees
-
sin
22
.
5
°
sine 22 . 5 degrees
-
cos
22
.
5
°
cosine , 22 . 5 degrees
- tan
22
.
5
°
22 . 5 degrees
-
cos
90
°
cosine , 90 degrees
-
sin
7
.
5
°
sine 7 . 5 degrees
Given
cos
θ
=
−
4
5
cosine theta equals negative , 4 fifths and
90
°
<
θ
<
180
°
,
90 degrees less than theta less than 180 degrees comma find the exact value of each expression. See Problem 5.
-
sin
θ
2
sine , theta over 2
-
cos
θ
2
cosine , theta over 2
-
tan
θ
2
tangent , theta over 2
-
cot
θ
2
co-tangent , theta over 2
Given
cos
θ
=
−
15
17
cosine theta equals negative , 15 over 17
and
180
°
<
θ
<
270
°
,
180 degrees less than theta less than 270 degrees comma find the exact value of each expression.
-
sin
θ
2
sine , theta over 2
-
cos
θ
2
cosine , theta over 2
-
tan
θ
2
tangent , theta over 2
-
sec
θ
2
secant , theta over 2
B Apply
-
Think About a Plan Triangle ABC is a right triangle with right angle C. Show that
cos
2
B
2
=
a
+
c
2
c
.
cosine squared . b over 2 , equals . fraction eh plus c , over 2 c end fraction . .